COLLOCATION AND GALERKIN METHODS FOR THE APPROXIMATE SOLUTION OF LINEAR AND NONLINEAR DIFFERENTIAL EQUATIONS
Dimitrios Politis
A thesis submitted for the degree of Master of Science in the School of Mathematics University of New South Wales
1996 Abstract
In this thesis we obtain a fourth order rate of convergence for the approximate so lution and establish the regularity of solutions to a class of second order boundary value problems of the form
(1.la) -u" + ao(x)u' + a1(x)u = f(x)
(1.lb) u(0) = u(l) = 0 by the collocation method with uniform mesh on a partition IIN of J. In (1.1) Lis a linear differential operator, f is assumed in L2(J) and u denotes the unique solution contained in the sobolev space H 2 (J). More precisely we prove with the use of Cea's Lemma that the Galerkin method for (1.1) converges in the L2-norm, that is, IIUN - ullL2 -+ 0 if the following hold: (i) L is a coercive operator (ii) the approximation operators L-;/PNL are uniformly bounded (iii) the projection operators PN satisfy PNu-+ u as N-+ oo.
Condition (iii) is equivalent to requiring PN -+ I (N -+ oo) where J is the identity operator on H 2 (J). We obtain convergence of the Galer kin approximate equations PN LUN = PN f to the exact solution u of Lu= f provided;
IIPNL- L
IIPNJ - /11£2 -+ 0
We prove that if cubic B-splines are employed for the approximation space, then convergence in the maximum norm is bounded above by the modulus of continuity of u and proceed to show w( u; h) -+ 0, as h -+ 0.
An important L2 global error estimate is derived for the convergence of the collocation method at the two special Gauss-Legendre collocation points in each subinterval if UN(x) are cubic B-splines with C1(J) continuity. This result is
11 summarised as follows:
llu - UNIIV(I) = O(h4 ) for sufficiently small h.
Uniqueness of a solution and hence existence for both the Collocation and Galerkin methods applied to (1.1) is established. In particular, we prove if the residual involving UN vanishes precisely at the N + I collocation points which are the roots of an orthogonal polynomial, then the resulting system of algebraic equations are uniquely solvable and the following global collocation error estimates are achieved:
11u; - u"IIL~(I) < CEN(u") 11uM>cx) - uCi>(x)IILCO(/) < CEN(u") for j = 0, 1 where EN( u") is the error in best uniform approximation to u" chosen from elements in the closed subspace XN. A numerical illustration is given which demonstrates that UN and its deriva tive exhibit uniform convergence to u of fourth and third order, respectively, predicted by the convergence estimates contained in Chapters 2,3,4. Moreover, superconvergence for the collocation method, at the nodes of fourth order is ob tained in agreement with (3.32).
In Chapter 4, we establish a uniform error bound at the nodes Xi of IIN for the Galerkin method using a Green's function approach and show that
provided u E H 11 (I) n HJ(/). We derive an optimal estimate of the error e = u - UN for any x E J, namely
A superconvergence error estimate of the H 1-Galerkin method in the approx imation space SPo(IIN, r, 2) at the nodes Xi for r > 3 is proved. In this result, UN is known to satisfy a variational formulation and if u E H 11 (I) n HJ(J), 2 $ s $ r + 1 then the rate of convergence of UN(x) to
lll u(x) is
We study the general second order nonlinear boundary value problem u" = f*( x, u, u') and prove that if v is the exact solution of the transformed equation v = Tv and VN solves the approximate equations P;,TvN = VN then we have
The quantity IIP;,v-vllL~ can then be made arbitrarily small due to the denseness assumption
for all v E L!.
IV Contents
Abstract ii
1 INTRODUCTION 1 1.1 Outline of thesis . 1 1.2 Background . . . 3 1.3 Notations, basic assumptions and preliminary results 9 1.4 Description of the collocation and galerkin methods 17
2 SPLINE APPROXIMATION THEORY 20 2.1 Spline approximation and projection methods .... 20
3 THE COLLOCATION METHOD 33 3.1 Error estimates for the approximate solution of boundary value problems by the collocation method using cubic splines and global polynomials ...... 33
4 THE GALERKIN METHOD 53 4.1 Optimal convergence of the H 1-Galerkin approximation with a piecewise spline space. Existence and uniqueness of the Galerkin solution ...... 53
4.2 A convergence theorem for the approximate solution by the H 1- Galerkin method of second order nonlinear differential equations 71
References 82
V Chapter 1
INTRODUCTION
1.1 Outline of thesis
Our main concern in this thesis is the approximate solution by the Collocation and Galerkin methods of the linear two point boundary value problem
{I.la) -u" + ao(x)u' + a1(x)u = f(x), 0 {I.lb) u(0) = u{l) = 0 We assume a0 , a1 are in C00{J) and that f = 0 implies u = 0. Equation {1.1) is known to occur frequently in applications of physics and engineering, especially in the study of vibratory and oscillatory motion of mechanical systems and in electrical circuit theory. We note that from elliptic regularity theory, in general, if f E Hr(J) then u is the unique solution of {1.1) contained in Hr+2(J) for any positive integer r ~ 0. This function space and others are defined later in the introduction. We shall use C as a generic constant throughout this thesis, unless it is specif ically stated otherwise. Note that the constant C may not necessarily take the same value at each occurrence. The introduction is mainly devoted to a descrip tion of the Collocation and Galerkin methods including showing high-order error 1 estimates for the cubic Hermite approximation of the exact solution u(z) of (1.1) and discussing research background material. Chapter 2 is concerned primarily on an overview of the projection method for solving boundary value problems as well as developing spline error results. There we claim conditions for the convergence of the Finite Element Colloca tion and Galerkin methods in terms of finite-dimensional operators. In Chapter 3, we study the collocation method for the approximate solution of (1.1) using cubic B-spline functions i and develop optimal convergence of fourth order. This result lays particular stress to the two Gauss-Legendre collocation points in each subinterval Ji of [O, 1). N We then move to show that this C1-collocation approximation UN = I: Cii i=l 1s unique for the coefficients Ci by solving a resulting linear algebraic sys- tem of the form Ac = f ·when UN(x) E SPo(IIN,3,2) is directly substi tuted into equation (1.1). The C 1-collocation method is defined as seeking an UN(x) E SPo(IIN, 3, 2) such that LUN(e;,k) = J(e;;k) evaluated at the collocation points e;,k for j = 1, 2, 3, ... , N; k = 1, 2. We prove a superconvergence phenomena occurring at the knots of the colloca tion approximation in the maximum norm by using a quasi-interpolant technique. We establish global collocation error estimates and in particular show that the sequence of approximate solutions UN(x) together with its first derivative converge in the maximum norm to the exact solution u(x) and its corresponding derivative, as N --. oo. A similar argument shows that UJ:,( x) converges to u"( x) with respect to the L! norm. A numerical illustration is provided for a boundary value problem of type (1.1) using cubic B-splines for the approximation space which shows O(h4 ) and O(h3 ) rates of convergence for the maximum errors in u and u', respectively. In Chapter 4, we examine the Finite Element Galerkin method for the ap proximate solution of (1.1) and a nonlinear ordinary differential equation. For the linear case, we apply orthogonal projection to show that the H 1- Galerkin approximation to u(x) is the function UN E SPo(IIN,3,2) satisfying the 2 fundamental variational equation (LUN, v") - (/, v") = 0 for all v in the trial space. S11o(IIN, 3, 2) is the set of all real-valued functions s( x) of degree < 3 on each subinterval [xi,Xi+i] of I such that s(O) = s(l) = 0 and with s(x) E C2(/). We find a uniform error bound at the nodes Xi of our partition which is order ( s+ 2) accurate, for 1 < s :5 4. We then derive an optimal estimate of the Galerkin error which states that for any x E / and u E H•(J), we have We next show that if u E H•(J) n HJ(/) is the true solution of (1.1) and UN satisfies the variational formulation ( 4.38), then there exists a positive constant C such that From this we obtain a superconvergence error estimate of the H 1-Galerkin method for higher degree splines, namely 0(hr+l) convergence for r > 3. A feature of this chapter is if T satisfies certain conditions then the approximate solutions VN of the Galerkin equations VN = PNTvN converge to the exact solution v of the nonlinear operator equation v = Tv with a rate given by llv - vNIILj :5 CIIPNv - vllLj· Once this estimate has been established, it is then a relatively easy task to show that IIPNv - vllLj-+ 0, as N-+ oo. We examine the discrete H 1-Galerkin method for the same equation and show that the approximations VN achieve h2 convergence in the L2-norm. 1.2· Background Two well-known projection methods which give global continuous approximate solutions to the boundary value problem (1.1) are the methods of Collocation and Galerkin. The accuracy of these methods using arbitrary basis functions to determine the finite-dimensional subspace in which the approximate solutions are constructed shall be examined later in the sequel. In some practical problems, interpolating polynomials are not suitable for use as an approximation. Since in order to obtain a good approximation to an 3 arbitrary function f(x) by an n'th degree interpolating polynomial, it may be necessary to use a large value of n. Unfortunately, polynomials of high degree often have a very oscillatory be haviour which is not desirable in approximating functions of reasonable smooth ness. Another disadvantage of high degree global polynomial interpolation is that a fluctuation over a small portion of the interval can propagate to large fluc tuations over the entire domain, thus restricting their use when approximating many smooth functions that arise as solutions to actual Science and Engineering problems governed by differential equations. 1 In [34], Runge has shown that given the continuous function f(x) = -- 1 +x2 with ( n + 1) uniformly distributed nodes in [-5, 5] and its interpolating poly- nomial of degree n,pn(x), constructed at these points; then the maximum error 11/(x) - Pn(x)IILoo(/) becomes arbitrarily large as n -+ oo (to be more precise) sup 1/(x) - Pn(x)I-+ oo for 3.64 < lxl < 5, k ~ 0). n>k - In 1916; Bernstein [4] proved that for /(x) = lxl together with equidistant interpolation nodes in [-1, 1), then 1/(x) - Pn(x)I diverges everywhere with the exception of the single points x = -1, 0, 1. Both Faber [16] and Erdos [15] settled the question by demonstrating that for any fixed interpolation node array, it is possible to find a continuous function whose interpolating polynomial of degree n is such that llf(x) - Pn(x)IIL00(1) f-+ 0 as n-+ oo. The latter author obtains this result via a Lebesgue constant argument and show ing its unboundedness. Marcinkiewicz [30] proved for every continuous function f(x) there exists at least one set of nodes such that the interpolation polynomial obtained from it converges uniformly to f(x). Necessary and sufficient conditions are known for the convergent behaviour of equidistant interpolation, see Lanczos [26] for a more complete discussion. These illustrations show that while interpo lation is often useful, it must be used with some care. Most of the interest in piecewise polynomial functions has centred on spline 4 functions Sp(IlN, k, 11 ). The beginning of the theory of spline functions is credited to Schoenberg (37], de Boor (10], Ahlberg, Nilson and Walsh [1]. Widely used computer software for applying spline functions to approximate solutions of linear boundary value problems is based on the programs developed in de Boor (10]. Adaptations of these are available in the IMSL and NAG numerical analysis libraries. The idea of piecewise polynomial approximation spaces has been used suc cessfully by Varga, Ciarlet et. al (6] for the solution of second order boundary value problems. They employ a weak formulation or Galerkin method instead of the collocation method. In the nonlinear case, these residual integrals must be replaced and evaluated by quadrature sums with the number of quadrature points equal to the number of basis elements. If quadrature is required at each time step, which is commonly the case, then lengthy computations will arise. This difficulty is eliminated by a collocation method since no quadratures are involved. The use of projection methods with spline functions for numerical approxi mation of linear and nonlinear boundary value problems with linear boundary conditions has been studied by many authors, see (27], (28], (35], and others. Numerical methods for solving two-point boundary value problems are con sidered thoroughly in [24]. Among these methods are the shooting method and finite-difference methods of various orders. de Boor [8] has shown that nodal cubic spline collocation (i.e.; collocation at the nodes of the partition) for (1.1) is only second order accurate, whereas other projection methods utilizing the same spline space yielded fourth order approximations. In a pioneering work, de Boor and Swartz [9] demonstrated that an optimal fourth order rate of convergence for cubic spline approximation is attained by collocating (1.1) at the two Gauss-Legendre points in each subinterval of IlN, This approach is called orthogonal spline collocation and yields high order approximations tothe exact solution. Orthogonal collocation has been developed 5 and used with success in problems of viscous· fluid flow, heat and mass transfer and diffusion with chemical reaction which are partial differential equation ap plications. Particularly, Zamani and Sun [48] have used bicubic Hermite splines in the solution of the nonlinear Poisson's equation modelling .a compressible, ir rotational flow problem which arises in aerodynamics. The resulting nonlinear algebraic equations were solved iteratively to 0.01 % accuracy using a successive approximation method advocated by Chan et al (5]. The orthogonal spline collo cation method for solving (1.1) involves determining a cubic spline function s(x) satisfying (I.la) at a discrete set of points as well as the Dirichlet conditions (I.lb). For example, one requires that s(x) E C[O, 1] to be of degree less than or equal to k on each subinterval of J and determine s( x) from the conditions that Ls(xi;) = f(xi;) ·· where Xij E (xi, Xi+i] for 1 ~ i < N, j = 1, 2 [ s(0) = s(l) = 0. The Galerkin and collocation methods are both optimal, in the sense that un der smoothness assumptions on the coefficients of equation (1.1) both can exhibit fourth order convergence llu - UNIIL2(1) = O(h4 ) in the L2-norm for continuous piecewise cubic splines. Collocation at the special Gauss-Legendre points is an example of this where one achieves optimal rates of convergence in the stronger L00 norm, i.e.; llu-UNIIL i = 0, 1, 2, ... , N, j = 0, 1; the error in the approximation to the solution and its derivative at the knots of IIN is O(h4 ). Results such as the latter are known as superconvergent since the rate of convergence at certain isolated points is faster than is possible globally. Russell and Christiansen (36] examined adaptive mesh strategies for use in Orthogonal spline collocation procedures. More recently, Houstis, Christara and Rice [19] 6 formulated an extrapolated quadratic spline collocation method for the solution of (1.1) and derived an optimal order O(h3-;) global estimate for the j-th derivative of the error, j = 0, 1, 2. The most significant virtue of the collocation procedure is its ease in applica tion. For instance, the matrix elements of the discretized equations are evaluated directly unlike the Galerkin method which require considerable effort to evaluate numerical integrations for each component element of the matrix A. Also the bandwidth of A in the linear system of algebraic equations Ac= f for the co efficients Ci(i = 1, 2, 3, ... , N) is smaller in the collocation method than that for the Galerkin method when the same degree splines are used. For the collocation method, the number of nonzero terms in a row of A is equal to the number of nonzero basis functions at the corresponding knot. In general, polynomial splines of odd degree 2N + 1 render the bandwidth of N in the case of collocation as compared to 2N + 1 for the Galerkin procedure. Thus we observe that the higher-order convergence of the Galerkin method is obtained at the expense of higher-order computational complexity. The collocation method which is known for its accuracy and efficiency requires no quadrature routines and is relatively easy to code. Given the set of fixed nodes xi( i = 0, 1, 2, ... , N) and a uniform mesh spacing h, then a result of Sharma and Meir [39] shows that if u( x) E C 2 ( I) is the exact so lution to (1.1) and UN E Sp(IIN, 3, v) satisfies UN(x;) = u(x;) j = 0, 1, 2, 3, ... , N as well as UJv(O) = u'(0), UJv(l) = u'(l) we have for each x E J lu(r)(x) - ut>(x)I ~ 5h 2-rw(f"; h), r = o, 1, 2 where w(f"; h) is the modulus of continuity of f"(x) on [O, l]. For an optimal H2 rate of convergence of the collocation method see [12] who prove that if u E Ha(J) n HJ(/) then where r is the degree of spline polynomial used for the approximate solution of boundary value problems. Hybrid methods which combine advantages of Galerkin 7 methods, with advantages of collocation methods have been analysed by Diaz [11). The Collocation-Galerkin method introduced by Diaz have computational com plexity intermediate between the Galerkin method and the collocation method. They have the advantage over the Galerkin method in that the integrals to be evaluated are simpler because they involve only piecewise linear functions. An other advantage is that there are fewer of them. In [20), Irodotou-Ellina a.nd Houstis have considered the numerical solution of a. fourth order linear boundary value problem by assuming a. high order pertur bation a.t the knots of the unique quintic-spline interpola.nt of the exact solution. In their analysis, optima.I 0( h6 ) global error estimates a.re obtained for a. uniform partition subject to a. B-spline ha.sis expansion of this interpolant which is contained in C10(I). It is known [see 45] that standard nodal collocation for the same problem with quintic splines produces only second order accuracy. Ascher, Christiansen a.nd Russell [2] have used the collocation ideas presented in [46] a.nd [35] to develop a. computer software code called COLSYS designed to approximate the solution of linear a.nd nonlinea.r boundary value problems. This code implements orthogonal spline collocation with B-spline bases. In COLSYS, approximate solutions are computed on a sequence of automatically selected meshes from initial uniform meshes until a. user-specified set of toler ances is satisfied. Mesh selection and error estimation algorithms a.nd strategies incorporated in COLSYS are an important feature in its success for obtaining accurate approximate solutions to ordinary differential equations with boundary layers (narrow regions in which the solution changes very rapidly). 8 1.3 Notations, basic assumptions and prelimi nary results Let X be a real vector space over the field of real numbers JR. A norm on X is a non-negative real-valued function II · II : X --+ JR+ with the properties {Nl) 114>II ~ O; 114>II = 0 if and only if 4> = 0 (N2) II 0 4>II = 1°1114>II {N3) · 114> + v,11 ~ 114>II + llv,11 for all q,, tJ, E X and all o: E JR. On such a space we introduce a metric p defined by p( 4>, tJ,) = 114> - v,II which is called the distance between 4> and v,. If X is complete in this metric, we call X a Banach space. Completeness, means that if { 'Pn} is a sequence of elements of X such that n-+oo there exists an element 4> in X such that n-+oolim 114> - 'Pnll = 0. In a normed space the distance dist{x, Xn) from an element ( E X to a subspace Xn of X is defined to be For any positive integer N, let IIN: 0 = xo < x1 < · · · < XN-1 < XN = 1 be a partition of our spatial domain J = [O, 1] and set hi = Xi - Xi-l where h = max hi for i = 1, 2, ... , N. The sequence of nodes { x0 , xi, x 2 , ••• , XN} are known as the knots or meshpoints of IIN. The ith element is contained between Xi-1 and Xi. We make the natural assumption that h --+ 0 as N --+ oo and that the partition IIN is uniform. 9 An inner product (-, ·) on a linear space X defines a norm by ll'PII = (cp, cp)112 for all cp E X. If X is complete with respect to this norm it is called a Hilbert space and for an arbitrary interval [a, b] we define its associated L2 inner-product by (1.2) (cp,'fP) = 1b cp(x)tp(x)dx We formulate the second order differential equation (1.1) in operator notation Lu= f. In the next paragraphs we give some important properties that L is assumed to have and this information will be used later in the convergence proofs. Our linear operator L : .X ~ Y is assumed bounded, that is there exists a positive number C such that for all cp EX. L is bounded if and only if IILII satisfies the condition IILII = sup IIL'PIIY < oo 114>ll=l L is continuous at an x 0 E X if for every f > 0 there is a 8 > 0 such that IILx - Lx0 II < f for all x E X satisfying llx - xoll < 8. The inverse operator L-1 exists and is said to be continuous if and only if there exists a C > 0 such that (1.3) for all In the Hilbert space X suppose the bilinear form ( Lcp, then by the Cauchy-:-Schwarz inequality we arrive at a useful result of functional analysis Cllcpll 2 ~ Re(Lcp, cp) ~ IILcpllll'PII 10 known as the Lax-Milgram theorem, see inequality (1.3). From this result follows the boundedness of the inverse IIL-1 11 < ~ and the surjectivity of L. Our linear transformation L from a Hilbert space X onto a Hilbert space Y is said to be completely continuous if for every sequence {Xn} with uniformly bounded norms in X (that is llxnll < c for c > 0 and all n) there exists a subsequence { Xn;} and an element y in Y, such that { Lxn;} converges to y. A completely continuous ( or compact) operator is bounded, however, the con verse is not necessarily true. Denote by Pn(I) the set of functions on I which are polynomials of degree at most n. We employ the notation En(/) for the distance in the uniform norm from / E X to the subspace Pn consisting of polynomials of degree less than or equal ton. Thus An alternative useful notation for En(/) is the following: A v E Pn is called a best approximation to f with respect to Pn if (1.4) En(/)= II/ - vii= inf II/ - ull ueP,. i.e; u E Pn has smallest distance from f. We shall establish some theorems of Jackson type which state that the error in best uniform approximation En(/) converges to zero all the faster when / is assumed a smooth function. The set of functions which are m(m ~ 0) times continuously differentiable on I we denote by cm(/). By L00 (I) we denote the vector space consisting of all continuous functions (x)IILoo(J) max l(x)I < oo = xel For 1 ~ p < oo, L11(I) denotes the class of real-valued functions (x )1 11 dx )1111 exists and is finite. 11 The norm on IJ'(J) is given by ll For the special case p = 2, L2(J) is the Hilbert space of square integrable functions with respect to the usual inner-product (1.2). Whether f is continuous or not, we define its modulus of continuity by the equation (1.5) w(f; 6) = max 1/(x) - /(t)I l:i:-tl~cS If / E C(J), then it is uniformly continuous, hence w(f; 6) < f (t: arbitrarily small). Let r be an integer ~ 1, then the Sobolev space Hr(J) of order r on J is the space of all functions whose distributional derivatives of order less than and including r belong in L2 (J). Hr(J) is the closure of the space C00(1) in the norm r ] 1/2 [ ll The closed subspace HJ ( I) of H 1 ( I) arising as a test space in the Galerkin nu merical solution to (1.1) are the functions Sp(IIN,p,q) = {s E C9 (J): s(x)l1; E P,,(h), 1 :5 i :5 N} 12 These functions are polynomial splines of degree p over IlN. H p = 3 and q = 0, 1 or 2 then s( x) is known as a cubic spline over I. The subspace of functions in Sp(IlN,P, q) that satisfy the homogeneous (or zero) boundary conditions (1.1) will be denoted by SPo(IlN,P, q). Thus SPo(IlN,P, q) = {s E Sp(IlN,P, q): s(O) = s(l) = 0} Definition 1.1 Given a function defined on [O, 1] and a partition IlN of I, a cubic spline interpolant, s( x ), for satisfies the following conditions: (i) s is a cubic polynomial, denoted s; on the subinterval [x;, Xi+i] for all i = 0, 1, 2, ... , N - l (ii) s(x;) = (x;) for all i = 0, 1, ... , N (iii) Si+i (xi+i) = s;(Xi+i) for all i = 0, 1, 2, ... , N - 2 (iv) s~+l (x;+i) = sHxi+i) for all i = 0, 1, 2, ... , N - 2 (v) s~~ 1 (Xi+i) = s~'(xi+I) for all i = 0, 1, 2, ... , N - 2 (vi) one of the following set of boundary conditions is satisfied s"(xo) s" ( x N) = 0 (Free boundary or natural spline) or s'(x0 ) = '(xo) ands'(xN) = '(xN) (Clamped boundary) The cubic spline s;(x) in any subinterval [x;,Xi+i] for a uniform mesh (hi= h for all i) is given explicitly by the formula (1.6) + ( !i - y~h) (xi+l - x) for 0 ~ i ~ N- l where st(xi) = Y:', s~'(xi+i) = y;~ 1 , 0 ~ i ~ N - l and 13 If we fix the values of yg and y'f.v then the unknown coefficients yf, y~, ••• , y'f.v _1 are found by solving the tridiagonal ( N - 1) x ( N - 1) linear system Ay = b denoted by 4h h 0 0 0 0 0 yf h 4h h 0 0 0 0 bi - hyg y~ 0 h 4h h 0 0 0 b2 yg 0 0 h 4h h 0 0 - 0 0 0 h 4h h 0 0 0 0 0 0 h 4h h 0 ,, bN-2 YN-2 ,, bN-1 - hy'f.v YN-1 0 0 0 0 0 0 h 4h or what is the same hy:'-1 + 4hy;' + hy;~l = bi; b· = 6 ( D-i - D.i-1) ' h h where D.q>i = q>(Xi+i) - q>(xi) for 1:5i:5N-l. It is clear that the coefficient matrix A is diagonally dominant and therefore non-singular. It follows that for this particular coefficient matrix A, the linear system Ay = b possesses a unique solution for bnf:11• On choosing the natural spline condition yg = y'fv = 0 of Definition 1.1, (vi) and invoking (1.6) one can evaluate s(x) at any x = a on I= [O, 1]. In some practical problems, interpolating poly nomials are not suitable for use as an approximation. A major advantage in the use of cubic splines for the approximate solution of boundary value problems is the highly oscillatory behaviour inherent in an undesirable nth-degree interpolat ing polynomial approximation for large n is minimized as much as possible, with piecewise cubics while achieving the strong convergence property ll-sllL2(I) -+ 0. Cubic splines enjoy the approximation property that the true solution u to (1.1) can be approximated by these splines culminating in global L2 errors that are proportional to h4, that is, llu - sllL2(I) = O(h4 ) (refer (44]). 14 Derivatives of order j of this exact solution (smooth function) have approxima tion errors which are proportional to h4-;. Convergence is not often guaranteed if high degree interpolating polynomials are employed, as classical counterexamples indicate. Definition 1.2 The Hermite cubic spline space is given by Sp(ITN, 3, 1) = {( x) E C1(J); (x) is a cubic polynomial on each interval [xi, Xi+i], i = 0, 1, 2, ... , N-1 defined by ITN}. This class of approximating functions consist of piecewise cubic polynomials which have continuous first derivatives at the knots Xi, i = 1, 2, 3, ... , N -1. Since there are a total of N different cubic polynomials and each piecewise polynomial has 4 parameters with 2 matching continuity conditions at each interior knot, so the dimension is 4N -2(N -1) = 2N + 2. Thus dim Sp(ITN,3, 1) = 2N + 2. A convenient basis for generating the piecewise cubic Hermite polynomials Sp(ITN, 3, 1) is the set oflinearly independent functions {½(x), \Jf i(x) I i = 0, 1, 2, ... , N} given by 1 - 3x2 - 2x3 -1 :'.5 x :'.5 0 V(x)= 1-3x2 +2x3 0:'.5x:'.5l 0 elsewhere x(l + x) 2 -1 :'.5 x :'.5 0 \Jl(x) = x(l - x) 2 0 :'.5 x :'.5 1 0 elsewhere (1.7) X X· >- I It is assumed that Vo(x), \J1 0 (x) vanish to the left of x0 and VN(x), \Jf N(x) vanish 15 to the right of x N. From (1. 7), we observe that the set {V.{x), \Jli(x) I i = 0, 1, 2, ... , N} possesses the following properties: 1. Each V.(x) and \Jli(x) is continuous together with its first derivative on (0, 1]. 2. Each V.(x) and \Jli(x) is a cubic polynomial in each subinterval and they vanish outside the subinterval [xi-1, Xi+1] 3. V.(x;) = bi;, ¼'(x;) = 0 for O $ i < N, 0 < j $ N \Jli(x;) = 0, \JIHx;) = bi; for O $ i < N, 0 < j ~ N where bi; is the usual Kronecker delta. Any arbitrary element ( x) E Sp{IIN, 3, 1) can he written uniquely as N N (1.8) 2 2 1 (4) (1.9) u(x) - Lagrange interpolating polynomials L k ( x) by ¾(x) = (1 - 2(x - xi)L~(xi)]L~(x) and 16 where (1.10) for each k = 0, 1, 2, ... , N. It follows for a uniform partition as we are considering here and by elementary cal culus that the maximum value of l(x-xi)2 (x-Xi+i)2 1is attained at x = Xi +ti+l, hence that Extending this for O < x $ I, we obtain from equation (1.9) 1 h4 < max luC4)(t)I · -- ee(o,1) "' 4! 16 lu(4)({)1h4 max---- ee(o,1) 384 h4 (1.11) - 384 llu(4)( X) IILoo(J) Hence the cubic Hermite interpolant of the true solution u(x) of (1.1) is fourth order accurate. For a uniform partition, de Boor (10] demonstrates lu(4)({)1h3 llu'(x) - c/>'(x)IILoo(J) $ max for O < x $ 1 (E[0,1] 24 globally. This result is an order less than for the function values. While at the knots this order of convergence is regained, thus for i = 1, 2, ... , N - l. 1.4 Description of the collocation and galerkin methods The collocation method for approximately solving the differential equation (1.1) consists in seeking an approximate solution from a finite dimensional subspace by requiring that the single equation Lu = f is satisfied only at a specially chosen set of numbers {ti, t 2, ... , tN-2} C I called collocation points. In general, let 17 X and Y be Banach spaces and let XN C X and YN C Y = C(I) be such that dimXN = dim YN = N, with L : X -+ Y. Select a set of basis functions { tf,1, t/>2, ... , tf>N} for the subspace XN, i.e, XN = span{ tf,1 , ••• , tf>N }. Under this framework, the crux of the collocation method requires that an approximation N UN(x) = E Cit/>i(x) be substituted into the boundary value problem (1.1) to hold i=l at each of the collocation points ti for 1 < i < N - 2. Thus we need UN E XN to satisfy the collocation equations (1.12) (LUN )(ti) = f(ti), i = 1, 2, ... , N - 2 which leads to solving the linear algebraic system N (1.13) L Ci(Lt/>i)(t;) = f(t;), j = 1, 2, ... , N - 2 i=l N.. N LCit/>i{O) = o, LCit/>i{l) = 0 i=l i=l for the unknown coefficients { Ci }~1 of the approximation UN. If we solve (1.1) with a spline space Sp(ITN,3,2) obtaining {ti}~t1 where N + 3 = dimSp(ITN,3,2), collocation methods determine spline approximations UN E Sp(ITN, 3, 2) to the exact solution u by solving The parameters { Ci} in the collocation spline approximation are fully determined by requiring UN(x) to satisfy exactly N+3 conditions, that is, the differential equation at the N + 1 collocation points in [O, 1] as well as the two boundary conditions. One important question is how to select the special collocation points {ti} and basis functions 18 choices for the basis functions because of their smoothness properties. For high order convergence of the collocation method, it is natural to complement these spline selections UN by substituting them directly into (1.1) and evaluating the resulting linear system at optimal Gauss-Legendre collocation nodes, Chebyshev nodes or roots of certain orthogonal polynomials which are projected onto each subinterval /; of the domain I. The Galerkin method approximates the solution to (1.1) by solving for UN E SPo(IIN, 3, 2) = XN the equations (1.15) fo\-u;., + aoU1 + a1UN)v;(x)dx = fo 1 fv;dx i = 1, 2, 3, ... , N where {v;} is a basis for the space of test functions YN = span {v 1 , v2 , ••• , VN}. Hence for the two-point boundary value problem (1.1), the Galerkin approxi mation is defined to be an element UN E Sp0 (IIN, 3, 2) satisfying {1.16) (U1,v') + (aoU1,v) + (a1UN,v)- (f,v) = 0 for all v E Sp0 (IIN, 3, 2), where SPo(IlN, 3, 2) is a finite dimensional subspace of HJ(I). N Without loss of generality, substituting UN(x) - E Ci;(x) in the bilinear i=l form (1.16) with v = v; yields N N N (1.17) I:(~, vj)Ci + I:(ao~, v;)Ci + I:(a1;, v;)Ci - (f, v;) = 0 i=l i=l i=l for j = 1,2,3, ... ,N where(·,·) as before is the usual L2-inner product. Thus (1.16) reduces to a uniquely solvable system of linear algebraic equations for the unknown scalars c1 , c2, c3 , ••• , CN. The uniqueness and hence existence of the H 1-Galerkin solution UN is proved in Chapter 4. 19 Chapter 2 SPLINE APPROXIMATION THEORY 2.1 Spline approximation and projection methods The spline interpolation procedure yields a smoother function than the piecewise cubic Hermite interpolate and also depends on less parameters. If we have N uniform equally spaced subintervals on I, then Sp(IIN, 3, 2) has dimension N + 3 constituting three matching continuity constraints while the dimension of the Hermite space Sp(ITN, 3, 1) is 2N + 2. The most used approximating spaces are polynomial splines. The use of cu bic spline spaces is responsible for the popularity of projection methods, since by the Weierstrass theorem they have excellent approximation properties in the maximum norm. One observes that since dim Sp(IIN, 3, 2) is less than dim Sp(ITN, 3, 1), which means smooth spline methods result in smaller bandwidth matrices and hence less computational effort in solving the matrix system (1.17). For example in the Galerkin method, piecewise polynomial spaces with 20 well suited for numerical computation. Theorem 2.1 If u E HJ(J), then (2.1) For a proof see [47). Definition 2.1 Let XN be a subspace of a Banach space X. An operator PN is called a projection from X onto XN ie PN : X --+ XN, if PN is a bounded linear self adjoint and idempotent map such that P;; = PN Lemma 2.1 (2.2) Suppose X is the Banach space L2[0, 1) equipped with the L2-norm ll'PIIL2(I) = (I~ PNx E XN, one aims for the condition dist(x, XN) --+ 0 as N--+ ·oo, to be satisfied. This is known as the denseness property and if the subspaces XN possess it for all (2.3) (PNx, v) = (x, v) all with(·,·) the inner product in L2 (J). Introduce the notation (2.4} rN(x) = LUN(x) - f(x) This function rN(x) is called the residual in the Galerkin approximation of (1.1). We write (2.5) 21 for clarity. The coefficients {Ci }(i = 1, 2, ... , N) are chosen by forcing rN( x) to be ap proximately zero. If X = L2(1) and UN E XN as before with associated basis { 1, 2, ••• , 'PN }, then, Galerkin 's method requires the residual rN to satisfy the orthogonality condition (2.6) (rN,'P,) = 0, i = 1,2,3, ... ,N The orthogonal projection PNx has another interpretation; it is the unique solu tion of the minimization problem {2.7) If one constructs an orthonormal basis { t/J1, t/J2, ... , 'PN} for XN from the previous { 1, 2, ... , 'PN} such that {'Pi, t/J;) = Di; i,j = 1, 2, 3, ... , N where 8,; is the Kronecker delta. In this setting, an arbitrary ( E X can be written as a linear combination ( = C1tp1 + C2tp2 + "' + Ci'tpi + "' • Thus ((, 'Pi) - (Ct 'Pl + C2tp2 + ' ' ' + C,'tpi + ' ' ' , 'Pi) using the fact that 'Pi is an orthonormal basis. Hence the orthogonal projection N PN( = I: Ci'Pi i=l can now be written in the form N {2.8) PN( = L((, 'Pi)'Pi i=l The representation (2.8) shows that (2.9) PNrN = 0 if and only if (rN, 'Pi) = 0 for i = 1, 2, 3, ... , N 22 since the t/J.'s a.re a linear combination of these. It follows that the result PNrN = 0 is equivalent to (2.6) We have established that the projection method for (1.1) amounts to solving the Galerkin equations PNrN = 0 or what is the same, (2.10) where N (2.11) UN(x) = E Ci PNLUN = PNJ has a unique solution UN E XN for all n ~ N and if these solu tions converge to the exact solution u of Lu = f. The Galerkin projection method together with a B-spline basis or cubic Hermite splines for UN can be classified as local, in the sense that each basis function is a cubic piecewise polynomial on a small interval of the spatial domain outside of which it vanishes completely, i.e., the { q,1 , q,2, q,3 , ••• , sufficiently large and a positive constant C such that for all n ~ N the finite 23 dimensional operators are invertible and the operators L·i,1 PNL : X -+ X are uniformly bounded IILN1 PNLII < C Then (2.12) Proof. Since the sequence of bounded linear operators LN: X-+ Y are point wise bounded, that is, IILNull :5 C for each u E X, N E N then the sequence {LN }(NE N) are uniformly bounded by Theorem 2.2, ie IILNII :5 C. Conversely due to the identity L"'i/ PNL-,P = 'ljJ for all -,p E XN, we have the relation (2.13) which is the same as (2.14) UN - u - Li PN Lu - u - Li PN L-,p + 1/J (L·i/ PNL - l)(u -1/J). Applying the L2-norm to both sides yields IIUN - ullL2(J) - ll(Li PNL - i)(u -1/J)IIL2(I) < IILi PNL - Illllu -1PIIL2(1) (2.15) < (IILN1 PNLII + IIIll)llu -1/JIIL2(I) Recalling the assumption that our operators Li PNL are uniformly bounded and the norm of the identity operator is unity gives (2.16) Taking the infimum with respect to the subspace XN, we note that the left hand side of the inequality (2.16) stays the same since its value is less than the greatest lower bound. Now we are able to state an important result of approximation analysis: (2.17) 24 The denseness property shows that (2.18) as N-+oo. Combining this with the error estimate (2.17), we see that the sequence of Galerkin approximations UN of (2.10) converge to the true solution u(x) of the boundary value problem (1.1) provided N-+ oo. Theorem 2.4 [Case a0 = 0 in (1.1)] In a Hilbert space X, let L: X-+ X be a strictly coercive and bounded linear operator. Then the Galerkin method converges. Proof. Coerciveness shows that (2.19) for all where(·,·) is the L2 inner product (1.2). Recall that orthogonal projections are self adjoint, i.e, (2.20) Thus, (2.21) Since X is an inner product space then by utilizing the Cauchy-Schwarz inequal ity, we have, where 1 ) 1/2 llullL2(1) = ( lo u2 (x)dx Inequality (2.22) becomes (2.23) for all By (2.23), PNLUN = 0 implies UN = O, so that LN = PNL : XN -+ XN is bijective. 25 It remains to be shown that the Galerkin solution converges. This is done by using (2.23). Hence (2.24) CIILi PNLUNIIL2(J) :5 ll(PNL)LN1 PNLUNIIL2(J) - IIPNLUNIIL2(J) smce LNL"il - I. Recalling that every Hilbert space structure has IIPNIIL2(1) = 1, we see that: (2.25) CIILi PNLUNIIL2(J) < IIPNIIL2(J)IILIIL2(J)IIUNIIL2(J) - IILIIL2(J)IIUNIIL2(J) To prove convergence, it is only left to show that the approximation operators L·i/ PNL of Theorem 2.3 are uniformly bounded. Thus combining (2.24) and (2.25) yields Therefore, -lp LII < IILIIL2(J) (2.27) IILN N L2 (J) - C for N sufficiently large as required. All the assumptions of Theorem 2.3 have now been fulfilled. This completes the proof. The collocation method for approximately solving the single equation (2.28) Lu=f consists in differentiating the approximate solution N (2.29) UN(x) = I: Ci (2.31) i(O) = 'P,(1) = 0 i = 1, 2, 3, ... , N. Suppose the basis functions 'Pi in (2.29) are the Hermite piecewise cubic polyno mials of dimension 2N + 2 generated by (1.7). This means we need 2N + 2 relations to specify the approximate solution UN of u in equation (1.1). It is known that two of these conditions can be obtained directly from the boundary conditions. In this context, the crux of the collocation technique re quires that the remaining conditions be obtained by having equation (1.1) satis fied at 2N points. Since there are N subintervals / 1 of I, it is natural to locate two points in each subinterval. We shall employ as our collocation points the two points in each / 1 that are the affine images of the roots of the Legendre polynomial of degree 2. This is done in the following manner 1 (-l)kh; (2.32) !;,k = 2(x;-1 + x;) + 2v3 for j = 1, 2, 3, ... , N; k = 1, 2 where h; = x; - Xj-1· The !;,kin equation (2.32) are known as the Gauss-Legendre collocation nodes. (-l)k The factor v3 (k = 1, 2) of h; in ·(2.32) appears as the roots of the Legendre polynomial of order 2. Other choices for the !;,k are possible, such as Lobatto or Radau points, how ever, they are less commonly in use because they provide a lower order rate of convergence for the approximate solution. Roots of orthogonal polynomials such as the Gauss-Legendre collocation nodes are preferred for the !;,k in (2.32) since they possess optimal high order conver gence properties (see for example, de Boor and Swartz [91), i.e minimizing the error attained in the maximum norm. Another popular choice following the same line of reasoning is derived from the Chebyshev nodes c. = (x;_1 + x;) + (-l)(x; - x;-1) cos-'-(2_k_-_1...;..)_1r (2.33) ... ,,k 2 2 4 27 for j = 1, 2, 3, ... , N k = 1, 2. These special collocation points are the roots of the quadratic Chebyshev polynomial (2.34) transformed onto I= (0, 1). One can now determine the coefficients Ci(i = 1, 2, 3, ... , N) in (2.29) from the condition that the residual (evaluated at e;,1:) (2.35) should vanish at the fixed 2N points of collocation ti = e;,1: given either by (2.32) or (2.33) when the approximate solution (2.29) is substituted into equation (1.1). The condition r(e;,1:) = 0 for j = 1, 2, 3, ... , N; k = 1, 2 can now be rewritten as (2.36) These collocation equations to be solved have a unique solution Ci (i = 1, 2, 3, ... , N). Thus (2.29) is fully determined and one finds, see for instance (9), that the collo cation solution satisfies llu - UNIILco(J) --+ 0 at the optimal e;,A: as N--+ 00. In the collocation method, X is the Banach space C(O, 1] and PN = PZ is the interpolation operator for the e;,A: (with IIPZIILco(J) > 1, see McLean (291) whereas in the Galerkin method X = L2 (0, 1) and PN = P~ is the Hilbert space orthogonal projection onto a spline space XN given by (2.8) and satisfying IIP~IIL2(1) = 1. The linear system of algebraic equations arising from (2.35) or (2.36) can be written as follows (2.37) Ac=f where A is an (2N + 2) x (2N + 2) full matrix and f is a column vector of length 2N + 2 (the dimension of the i). This motivates us to use cubic B-splines to represent the collocation approximate solution UN. A B-spline is a particularly compact piecewise polynomial function. Bf A:) is a non-negative polynomial of degree (k - 1) that is non zero only on the interval (xi, Xi+A:), More precisely, the support of the i-th B-spline of order k is [xi, Xi+A:] and zero elsewhere. 28 This makes B-splines particularly attractive ha.sis functions since the influence of any particular B-spline coefficient extends only over four subintervals. It is convenient now to choose additional equidistant knots in order to extend IIN = [x0, XN] for the sole purpose of defining the B-spline ha.sis. Thus we have (2.38) For this knot sequence Xi of length M = N + 1 there will be exactly m = M -4 = N +3 4 4 4 4 B-splines of order 4, say Bf ), B~ ), Bi >, B! ), ••• , B~t3 that span the B-spline space over I= [O, 1) with C 2-smoothness. For i = -3, -2, -1, ... , N - 1 define (4) _ iH (x; - x)! (2.39) Bi (x) - (xiH - xi) ~ 1/J~( ·) 3=1 1 x, where and (t - x)+ is the truncated power function given by fort 2:= X (2.40) ( f - X )~ = { (f ~ X )' fort< X The cubic B-splines Bf 4>(x) defined on [x0, xN] satisfy the following important properties; (a) Bf4>(x) = 0 for x ~ (x;,X;+4) (b) 0 $ Bf 4>(x) $ 1 for x E [xo,XN] N-1 (4) (c) E B; (x) = 1 for x E [xo,xN] i=-3 (2.41) (d) For any cubic spline function s(x) of Definition 1.1 with knots { x0 , x1, x 2 , ••• , x N} there exists unique coefficients c; ( c1, c2, ••• , CN) such that N-1 (4) s(x) = E c;B; (x) xo < x $ XN i=-3 A proof of any of the above properties may be found in de Boor [10], Schoenberg [37]. 29 One can generate higher order B-splines for a specified knot sequence { x0 , xi, ... , XN} by using the following recursive relation BJm>(x) = ( x - Xi ) BJm-l)(x) + ( Xi+m+l - X ) Bttl)(x) m > l Xi+m - Xi Xi+m+l - Xi+l (2.42) if Xi < X < Xi+l Bt{z) = [: otherwise Here BJm>(x) is the i-th B-spline of order m (i.e degree m - 1) as a function of x over the interval I= [O, 1]. Hence definition (2.42) for a fixed knot sequence { x0 , xi, ... , XN} produces, in general, a different B-spline polynomial of order m in each subinterval of I. The special case m = 4 of interest to us creates the piecewise cubic B-splines {B-3, B_2, B_1, ••• , BN-2, BN-d which span XN and possess c<2> smoothness. Our next aim is to prove that the exact solution u(x) of equation (1.1) consid ered in C[O, 1] can be approximated to a high order precision by cubic B-splines. Theorem 2.5 Suppose u is a continuous function denned on [x0 , XN], then the spline function N-l (2.43) s(x) = L u(xi+2)Bf4>(x) i=-3 satisfi.es (2.44) llu - sllLOC>(/) = max lu(x) - s(x)I $ 3w(u; h) xo$.:i:$xN where w( u; h) is the modulus of continuity of u Proof Recalling (2.41b) and (2.41c) we have N-1 N-1 (2.45) 1s(x) - u(x)I = L u(xi+2)Bf4>(x) - u(x) L Bj4>(x) i=-3 .i=-3 N-1 L (u(xi+2) - u(x))BJ4>(x) i=-3 N-1 < L lu(xi+2) - u(x)IBJ4>(x) i=-3 Suppose x E [x;, x;+i ]. 0 n Iy B;-a,(4) B(;-4)2 , B<;-i,4> B<; 4> 1·1ve on t h e .mterv: al [x;, x;+i ] . Hence j (2.46) ls(x) - u(x)I < L lu(xi+2) - u(x)IBJ4>(x) i=j-3 30 For i E Li - 3, j] and assumed uniform partition IIN we have (2.47) Xi+2 - x < X;+2 - x; - (x;+2 - x;+i) + (x;+i - x;) = 2h - 3h Employing the definition of the modulus of continuity stated in (1.5) shows that (2.48) lu(xH2) - u(x)I < w(u;3h) < 3w(u; h). The last step in the inequality (2.48) follows from [33], where k = 3 is the degree of splines being used. Recall earlier the distance from u to the B-spline subspace Sp(IIN, 3, 2) con tained in a Banach space C ( I) was defined by (2.49) dist(u,Sp(IIN,3,2)) = inf llu - sllLco(J) aESp(IIN,3,2) where llullLoo((o l]) = max lu(x)I is the maximum norm. ' 0$:z:$1 In this framework, the error estimate (2.46) combined with (2.48) yields (2.50) dist( u, Sp(ITN, 3, 2)) < 3w( u; h) Since u is continuous, then from (1.5) (2.51) limw(u; h) = 0 h-+0 which implies that xi+2 -+ x. Hence, by increasing the number of knots, the upper bound in (2.50) can be made to approach zero ash-+ 0. This completes the proof. The collocation method together with a spline space Sp(IIN, 3, 2) of dimension 4 4 N + 3 and associated basis { B~~, B~J, B~ {, ••• , Bt~1} reduces (2.36) and (2.37) 31 to (N + 3) x (N + 3) linear equations to be solved, i.e: N-1 E Ci(LBi)(t;) = f(t;) j = 1, 2, 3, ... , N + 1 i=-3 -1 {2.52) E CiBi{O) = O i=-3 N-1 E CiBi{l) = O. i=N-3 Here the internal collocation points {ti} {i = 1, 2, 3, ... , N + 1) are chosen to be the roots of an orthogonal polynomial of degree N + 1 on the interval [O, 1) with a non-negative weight function p( s) satisfying [ 1 ds (2.53) lo p(s) < oo. The new matrix A formed by solving (2.53) has components i = 1, 2, 3, ... , N + 3 (2.54) j = 1, 2, 3, ... , N + 3 and is sparse of bandwidth 4. Hence solving the algebraic equations (2.52) for the coefficients Ci(i = -3, -2, -1, ... , N - 1) requires much less computational time and effort than for Hermite cubic splines mainly due to the abundance of zero entries in the matrix A. A theorem of Schoenberg and Whitney (see, (381) shows necessary and suffi cient conditions so that this interpolation matrix A is nonsingular; namely that the diagonal of Ai; in (2.54) should contain no zero elements. This theorem ensures that the linear system (2.52) is uniquely solvable for the coefficients Ci(i = -3, -2, -1, ... , N - 1) in the collocation approximate solution N-1 (2.55) UN(x) = I: Cin!4>(x) i=-3 globally for each x E (0, 1]. 32 Chapter 3 THE COLLOCATION METHOD 3.1 Error estimates for the approximate solu tion of boundary value problems by the · collocation method using cubic splines and global polynomials Collocation at the Gauss-Legendre points, which we refer to as orthogonal spline collocation is more accurate in the maximum norm when employed for the ap proximate solution of linear and nonlinear differential equations. In [8], de Boor demonstrated that nodal spline collocation is suboptimal, that is, spline colloca tion at the knots of ITN is suboptimal in each of the maximum, L2 and H 1 norms. Suboptimal in the sense that only second order accuracy (i.e O(h2)) is achieved for piecewise cubics and this rate is best possible for nodal spline collocation. A result of approximation theory states that when using piecewise cubic poly nomials for XN one should expect a rate of convergence of UN(x) to u(x) for (1.1) which is fourth order accurate, provided u E C4 [0, l]. In particular the proof that finite element subspaces of degree k - l(k = 4 for our case) achieve approximation of order hk to an arbitrary smooth function u and of order hk-a 33 to its derivatives of order s can he found in Strang (43). Our error analysis for the collocation method is done in the Banach space C(I) with general collocation points ti, t 2, ••• , tN E J such that (3.1) det[i(t;)] # 0. In the collocation technique, the projection operator PZ applied to u EX is re garded to he an element belonging in XN that interpolate u at the ti(i = 1, 2, ... , N), ie PZx E XN. To find UN E XN, one solves the linear system (1.13). Suppose v(x) also satisfies the boundary conditions of (1.1), we define the bilinear form A(u, v) : X x Y -+ IR by (3.2) A(u, v) = fo 1 u(x)Lv(x)dx = fo1 (Lu(x))v(x)dx with norm (3.3) If the exact solution u(x) is smooth and (1.1) is elliptic (i.e a1(x) > 0)we know that (3.4) as well as (3.5) We shall also need the notation k (3.6) llullw~(I) = ~ llu(t)ll£00 (1) l=O Definition 3.1 A function f(x) defined on an interval [a, b] and satisfying the condition (3.7) for all ti, t2 E [a, b) is said to satisfy a Lipschitz condition of exponent a. Thus we write f(x) E Lip a. In what follows we derive an L2 global error estimate for the convergence of the collocation method. 34 Theorem 3.1 Suppose (1.1) has a unique solution u(z) with sufficiently smooth coefficients. Suppose also we employ a cubicB-spline basis forUN(x) with C1(1) continuity and collocate at the two unique Gauss-Legendre points in each subinterval. Then UN( x) for sufficiently small h satisfies (3.8) . (u - UN) Proof. Let 'P be the solution of L'P = v = II U II . U - N L2(I) The Green's function for the linear operator L is smooth such that ll'P(j)IILoe>(I) < C for j = 0, 1, 2. Then where J = LUN satisfies !(e;,k) = f(!;,k) at the two Gauss-Legendre points in 2 each subinterval {x;,x;+il· If p;(x) = TI (x -!;,k), then by property 2 [7, p253] k=l r,+1 (3.10) l:c- p;(x)r(x) dx = 0 J for all polynomials r(x) of degree less than two. The fact that relation (3.10) holds for Legendre polynomials is a major reason for the success of the collocation procedure at the Gauss-Legendre points. On [x;,x;+1], f-f = p;(x)q;(x) = L(u-UN) which vanishes at the collocation points and IIP;(x)IILoe>(J) = O(h2 ) since IP;(x)I = Ix - !;,1llx -e;,21 The standard existence proof for the collocation solution UN shows that ll(u - UN)(;)IILoe>(J) = O(h2) for j = 0, 1, 2 from which follows the boundedness of the derivatives of UN and hence f. Thus we have (3.11) 35 If one expands t/,(z)q;(z) in a Taylor series about z, with k = 2 terms and invoke (3.10), we have N-1 ,:;+i [1/Jq ·](k) - ~ 1,:; P;(z)[ k! (z;)(z - x;tdx 1 (3.12) - E 1~;+i P;(x) [(1/Jq;);(x;) (x - x;) + (tf,q:)" (e)(x - x;)2] dx j=O ,:, 1. 2. Recalling that in (3.12) N-11,:;+1 (1/Jq;)'(x;) L P;(x)(x - x;)dx = 0 j=O ,:; by the orthogonality condition (3.10) where r(x) = (x - x;). Thus N-1 (tµq·)"(e) 1,:·+1 (3.13) llu - UNllvz(I) = ~ ;, ,:/ P;(x)(x - x;) 2dx. Since lx-x;I < h implies that (x-x;)2 < h2 and notingp;(x) = (x-e;,1)(x-e;,2) < h2. So that our integrand p;(x)(x - x;)2 < h4 where h = x; - x;_1. Hence (3.13) becomes (3.14) We have shown that (u - UN) is fourth order accurate in the L2-norm. This completes the proof. Our next aim is to show uniqueness of the C1 collocation approximation UNto (1.1). The C 1-collocation method consists in finding UN E SPo(IIN, 3, 1) such that (3.15) j=l,2,3, ... ,N; k=l,2. Assume for v E H2(J) n HJ(I), there exists a positive constant C such that (3.16) (3.16) is known as the regularity assumption on the second order differential equation (1.1) necessary to establish uniqueness. 36 To achieve this we make use of a result of Douglas and Dupont (12] which relates the exact £ 2 inner-product with the discrete version. A proof of this lemma may be found there. Suppose U* is another solution of the collocation equations (3.15). Lemma 3.1 There exists a constant C such that (3.17) l(L(UN - U*), L(UN - U*)} - IIL(UN - U*)lli2(J)I < ChllUN - U*ll~(I)· In (3.17) the discrete L2 inner product takes the form N 2 (3.18) (p, q) =EE hw1cp(e;,1c)q(e;,1c) j=lk=l with weight function w1c > 0, k = 1,2 and p E P1(/i), q E H 2 (/i), Let (L(UN - U*), L(UN - U*)) denote the usual inner product defined by IIL(UN-U*)lli2(1) = (L(UN-U*), L(UN-U*)) fo 1[L(UN(x)-U*(x)))2dx. (3.19) The crux in obtaining the error estimate (3.17) above lies primarily in applying the Peano Kernel Theorem and bounding the linear functional L given by 2 (3.20) L(q) = L hw1cp({;,1c)q({;,1c) -1 p(x)q(x)dx k=l Ii Combining (3.16) with Lemma 3.1, we have ll(UN - U*Jllk2(1) < CIIL[UN - U*Jlli2(1) (3.21) < C[(L(UN - U*), L(UN - U*)) + ChllUN - U*llk2(1)]. For h arbitrarily small (3.22) ll[UN - U*Jllk2(r) ::; C(L(UN - U*), L(UN - U*)). Employing the discrete inner product (3.18) to the right hand side of inequality (3.22) shows that (3.23) since L(UN - U*)({;,1c) = 0 i = 1, 2, 3, ... , N; k = I, 2. Thus UN = U* which proves the uniqueness and hence existence of the C 1-collocation approximation UN satisfying (3.15). 37 We now introduce an intermediate function called the quasi-interpolant u of the exact solution u. More precisely u is a cubic spline interpolant of u. The underlying idea behind this quasi-interpolant being that the differential operator L applied to u - u produces a small residual at the collocation points. Definition 3.2 Let T3,nN be the interpolation operator mapping C 1(1) onto Sp(IIN, 3, 1) determined by the conditions (3.24) (Ta,nNu)(x;) = u(x;), (Ta,nNu)'(x;) = u'(x;) j = 0, 1, 2, ... , N for u(x) having at least C1 continuity. Choose u = T3,nNu, where u E Sp(IIN, 3, 1). It is shown in (12] that if u E W!(J), then (a) L(u - u)(e;,k) :5 Cllullwg.,(I)h4 (3.25) (b) l(u - u)(xi)I :$ Ch4 llullwic1) (c) l(u - u)'(xi)I < Ch4 llullwi(I) Since L(u-UN )(e;,k) = 0, the quasi-interpolant u satisfies the modified collocation relations ( 1.14) (3.26) (a) L(UN - u)(e;,k) L( u - u)(e;,k) j = 1, 2, ... , N k = 1, 2. (b) (UN - u)(O) - (UN - u)(l) = 0. Combining (3.25a), (3.26a) and (3.22) entails (3.27) 11uN - ullH2c1> < C(L(UN - u), L(UN - u)) 112 < Cllullwg.,(I)h4 • Next we use a special case of Sobolev's inequality (see, (171) to obtain a lower bound for IIUN - ullH2([)• Thus, IIUN - ullL 00 (I) + ll(UN - u)'IIL00 (1) < ½11uN - ullHJ(I) + ½11cuN - u)'IIHJ(I) (3.28) < IIUN - ullH2(I) But the left hand side of this inequality is exactly the quantity ll(UN - u)llwJ..(I)· Hence by (3.27), one has (3.29) ll(UN - u)llwJ..(I) :$ Cllullwg.,(I)h4 38 connecting the quasi-interpolant u to the exact solution u. This is a global con vergence estimate. Taking into account (3.25b) yields (3.30) l[(u- u) + (u - UN)](xi)I < l(u - u)(xi)I + l(u - UN)(xi)I < Ch4 llullw:.,(I) + Ch"llullw:.,(I) since in (3.29). Hence (3.31) l(u - UN)(xi)I < Ch4 [2llullw:.,(I)] - Coh 4 llullw:.,(I) Finally, a similar argument on (3.25c) gives a bound for the derivative of the error, (3.32) Estimates (3.31) and (3.32) are examples of a superconvergence phenomena oc curring at the knots of the collocation approximation. Thus we see a higher rate of convergence is achieved at special isolated points of IIN, than what is possible globally with respect to the maximum norm. We Iiow turn our attention to establish two error estimates for the collocation method by invoking nonspline bases functions for the approximate solution. In this projection method, single continuous global polynomials on I of de gree N + 1 are used which satisfy the homogeneous boundary conditions (I.lb). We model the proofs in line with the functional analysis theory contained in Kantorovich and Akilov [22], Vainikko [46]. Firstly assume that the homogeneous equation Lu = 0 with Dirichlet condi tions (I.lb) has only the trivial solution u(x) = 0. Our aim is to seek approximate solutions of the boundary value problem (1.1) in the form N (3.33) UN(x) = L 01c1c(x). k=O 39 Let {wN(z)}~=O be a sequence of orthogonal polynomials of degree Non the interval [0, 1] with respect to the weight function p(x) and satisfying (2.53). The collocation method, in this case, requires that the residual error must vanish precisely at the N + 1 collocation points so, s1, s2, ... , SN given by the roots of the global polynomial WN+1(x), i.e.; Equation (3.34) leads to a system of algebraic equations in the unknowns ak N (3.35) I:[-Z(si) + ao(si) The vector a = is determined by solving the matrix system (3.36) 0:N and then substituting it into expression (3.33) for UN( x ). Theorem 3.2 Suppose that the coefficients a0 , a 1 and f of (1.la) are con tinuous on the interval [0, 1] and suppose that the boundary-value problem (1.1) possesses a unique solution u(x). Then for sufficiently large n ~ N, the system (3.35) will have a unique solution, and the sequence of approximate solutions UN(x) together with its first derivative tends, as N-+ oo, in the maximum norm to the solution u(x) and its corresponding derivative, while the sequence UK,(x) converges to u"(x) in the L! norm with weight p(x). The rates of convergence are summarised by the inequalities 112 (3.37) IIU~ - u"IILi(I) - (fo1 p(x)[U~(x) - u"(x)] 2dx) < CEN(u") (3.38) 11u;P(x) - u 40 Proof. Let G(x, t) denote the Green's function for u" = v(x). Analogously we can view the Green's function as that solution of the equation: (3.39) LG= 8(x-t) that is, (3.40) G:i::i: - a0 (x)G:i: - a 1(x)G = -8(x - t) with associated boundary conditions (3.41) G(O, t) = G(l, t) = 0. Here 8(x-t) is the Dirac delta function which can be thought of as a concentrated source at x = t with properties X =/:- t (a) O(x - t) = [: x=t (3.42) (b) 1 = J~00 8(x - t)dt (c) f(x) = J~= f(t)8(x - t)dt Suppose µ 1(x) and µ2 (x) are any two linearly independent solutions of the ho mogeneous form of equation (3.40). Consider w1(x) to be a linear combination of µ 1 (x) and µ 2 (x) which satisfies w1 (0) = 0 and let w2(x) be a linear combination of µ 1(x) and µ 2 (x) which ensures that w2(1) = 0. The Green's function for our two-point boundary value problem, Lu = f becomes Wt (:,;)U12(t) X < t (3.43) G(x, t) = [ W(w1,U12)l.s=o U12(:,;)w1 (t) X > t W(w1,U12)1.s=O where W(wi, w2 )l:i:=O is the Wronskian determinant of w1(x) and w2(x) evaluated at X = 0. Thus, - w1(0)w;(o) - w~(O)w2 (0) 41 Since w1 and W2 are linearly independent functions, its Wronskian is always non zero on I. The exact solution of (1.1) is represented by (3.45) u(x) = fo1 G(x, t)v(t)dt where G(x, t) is given as in (3.43). Setting VN(x) = UJ:,.(x), we have (3.46) UJ.j\x) = fol l}i~~~' t) VN(t)dt for j = 0, 1 and conditions (3.34) can be written as: i /1 a1-;G(s· t) -vN(si)+ ~ a;(si) lo axi-;' VN(t)dt-f(si) = 0 for i = 0, 1, 2, ... , N. ,=o (3.47) If P~ denotes the linear projection operator constructed at the nodes s0 , s1, s 2 , •.• , SN which assigns to each function its Lagrange interpolation poly nomial of degree N and taking into account that VN(x) is a polynomial of degree not exceeding N. We have PNVN = VN and (3.47) becomes (3.48) In (3.48) K is the integral operator with kernel 1 a1-iG(s,t) / 1 (3.49) K(s,t)=I:a;(s) a i-; , KvN= lo K(s,t)vN(t)dt. j=O X O Here, K is a compact linear operator mapping L![0, 1] onto C[0, 1], 1.e K : L![0, 1] -+ C[0, 1]. According to the Erdos-Turan theorem, the Lagrange interpolation polyno mial of degree N constructed at the si(i = 0, 1, 2, ... , N) converges in the L! norm to the function being approximated. The above paragraph means that PN : C[0, 1] -+ L![0, 1] tends strongly to the identity operator I : C[0, 1] -+ L![0, 1], imbedding the continuous functions into the Hilbert space L!(J). The unique solution v(x) = u"(x) of the operator equation (3.50) -v+Kv=f 42 corresponds to the exact solution u(x) of (1.1). Equations (3.48) and (3.50) are both considered in the Hilbert space L!(J) of square-integrable functions. All conditions of the Uniform-boundedness principle, Theorem 2.2 ( also known as the Banach-Steinhaus Theorem) are met and from this follows the conclusion that the norms of the projection operators PN are uniformly bounded (3.51) N = 1, 2, 3, ... If the sequence PN are post-multiplied by the completely continuous operator K, then PN K : L! -+ L! tends strongly to the operator I K = K : (3.52) as N-+ oo. Since from (3.50), (K - I)v = 0 implies V = 0 and using the fact that (K - n- 1 is continuous at an arbitrary fit follows (K -J)-1 is also bounded from L! onto L!. From condition (3.52), one deduces that for n ~ N (N sufficiently large), the approximation operators (PNK - I) of (3.48) are also invertible and uniformly bounded: (3.53) By (3.53), equation (3.48) has a unique solution vN(t) and therefore the unique solvability of the system (3.35). Beginning with (3.54) PN(K - I)v + (-v + PNv) - PN f + (-v + PNV ). Subtracting (3.48) from (3.54) yields (3.55) Introduce PN(x) to be an arbitrary polynomial of degree at most N, then by (3.55) and the aid of (3.51), (3.53) it is easy to see that v - VN = (I+ PNK)-1(-v + PNv) 43 (3.56) llv - VNIIPp(I) - 11(1 + PNKr1(-v + PNv)IIPp < 11(1 + PNK)-111Lill(-v + PNv)IILl < C2ll(-v + PNv)IILl - C2ll(-v + PN) - PN(-v + PN)IIPp < C2 (11(-v + PN )IILl + IIPN(-v + PN )IILJ) < C2 (11(-v + PN )IILl + Gill - v + PNIILi) on noting PNPN = PN· Looking closely at the term II - v + PNIIL2p above, we find that 1 112 (3.57) II - v + PNIILi - (fo p(x) [-v(x) + PN(x)] 2 dx) 1 112 < (lr:J1 I - v(x) + PN(x)l2 fo p(x)dx) 1 112 - o1r:li I - v(x) + PN(x)I (fo p(x)dx) . Substituting this expression into (3.56) leads to: It is left to verify the main convergence result (3.37). This is achieved by recalling the notations u" = v(x), UN(x) = VN(x) and observing the definition for the best uniform approximation given by (1.4). Combining this with (3.58), we have (3.59) where C is a positive constant described earlier. Our next task is to prove the claim (3.38). This is done by recalling the Green's function relations (3.45) and (3.46): Hence, 1 8iG(x t) (3.60) u}j>(x) - uCi>(x) - 1 a .' (vN(t) - v(t))dt for j = 0, 1 o x' _ fo1 Bi~~:, t) (U;:,(t) - u"(t))dt /1 Bi~(x, t) /;{i)(u;:,(t) - u"(t))dt lo ax,f;(i) 44 Applying both the maximum norm and Cauchy-Schwarz inequality to equation (3.60), we notice that: _ max / 1 lJiG(x, t) · J;{i)(U'/., - u")dt OS:zS:1 lo 8xj f;{i) (3.61) < max [ 11 lJiG(~, t) 2 ~] 1/2 [ 11 p(t)(U" - u")2dt] 1/2. OS:zS:1 lo 8xJ p(t) lo N Let C be the maximum of the quantity 2 1 2 [ [11fiG(x,t) dt] ' for each j = 0, 1. oT:ti lo ax; p( t) Hence by (3.61), we have: (3.62) Noticing (3.59) plus taking account of (3.61), we obtain for each j = 0, 1 and completes our proof as required. Suppose that the coefficients a0 ( x), a 1 ( x) including the inhomogeneous term f(x) of equation (1.1) are r times continuously differentiable on I, then by D. Jackson's second Theorem (see Natanson [32]) we have, Cw (..!..) Cw (u (3.64) sup lu(r)(x) - u 45 Substituting (3.64) into inequality (3.63) we obtain, E (u") < C (3.65) N - Nr+a {O £ 2 and maximum norms when each of (3.48), (3.50) are considered in C(J). In this paper, to achieve the above estimate Karpilovskaya takes into account the Bernstein result [4] for Chebyshev interpolation points, namely that, (3.66) where the quantity AN denotes the L~besgue constant. Suppose if the Legendre polynomial l dN+l PN+1(x) = 2N+1(N + 1)! dxN+l (x2 - l)N+l' Po(x) = 1 was employed instead of TN +1 ( x) and if one chooses the sf+1 ( i = 1, 2, 3, ... , N +l) to be the Gauss-Legendre collocation points !;,k with p2(x) = 1, then convergence of the approximate solutions in this situation are O(N-'"+2)1 accurate. The essence of the above estimate is the lemma that the interpolation projec tion operator PN obeys (3.67) max IAN(x)I 0$.t:$1 N+l PN+i(x) - maxI: xeI k=l Pf.r+t (si)(x - si) - IIPNIIL 00 < CVN 46 which is proved by Grunwald and Tura.n (18) provided the equations (3.48), (3.50) are considered in C(J). A Numerical Illustration As an example of (1.1) (see also Sloan, Tran, Fairweather [40)), consider the boundary-value problem (3.68) u" - 4u = 4cosh(l), x E (0, 1) with u(0) = u(l) = 0. It is easy to show by the Variation of Parameters method that u(x) = cosh(2x - 1) - cosh(l) is the unique solution to (3.68). We take the partition IIN to be uniformly spaced, so that h = ~ and estimate the maxi mum errors in u and u' over the discrete set consisting of the multiples of 0.01 beginning at x = 0. All computations were carried out on an Sequent Symmetry S81 computer in double precision(~ 16 decimal digits) utilizing an ATS Fortran compiler with UNIX operating system. For various intervals N into which [0, 1] is divided (N = 10, 20, 40, 80,160), we obtain the exact values of the B-spline coefficients Ci in the collocation solu tion UN(x) of (2.55) for any x E [0, 1]. Our numerical approximations to (3.68) were obtained using a modified version of the de Boor subroutines [10) dealing with cubic B-spline computations and a matrix system solver namely (MAIN, SOLIN, BSPLVD, BSPLVB, INTERV, COL). The actual B-spline parameters of (2.55) are found by solving the linear system (2.52) for the cases when there are N = 10, 20, 40 subintervals of J. These results are documented in Table 3 below. We omit writing the cases N = 80,160 for condensation purposes, however, they have been done. The columns labelled eoc show the empirical orders of con vergence computed from the errors with mesh spacing h and 2h by the formula lleNIIL 00 (I)) 1 o: = log10 ( II II /log10(2)· el!. LCO(I) 2 The results in Table 1 show clearly O(h4 ) and O(h3 ) rates of convergence for the maximum errors in u and u', respectively. 47 The current notation lleNIILoo(J) especially for the numerical results of Table 1 are given by the formula for C = 0, 1, 2, ... , 100 which is taken over discrete values of x within the domain I. The numerical experiments of Table 2 clearly show that for a uniform partition, the approximate first derivatives eAr at the knots exhibit O(h4 ) superconvergence (whereas O(h3 ) is normally expected), but this has not been proved theoretically. 48 L00 errors in UN and Uk for the differential equation (3.68) (uniform mesh) N llu- UNIIL00 eoc llu' - UkllLoo eoc 10 0.595 X 10-5 0.197 X 10-3 20 0.357 X 10-6 4.06 0.248 X 10-4 2.99 40 0.227 X 10-7 3.98 0.309 X 10-5 3.00 80 0.137 X 10-s 4.05 0.386 X 10-6 3.00 160 0.900 X 10-lO 3.93 0.468 X 10-7 3.04 Table 1 Absolute errors in Ufv at knots for the differential equation (3.68) (uniform partition) lu' - Ufvl N X = 0.0 eoc X = 0.3 eoc X = 0.8 eoc 10 0.2903 X 10-5 0.1011 X 10-4 0.1320 X 10-4 20 0.1804 X 10-6 4.01 0.6023 X 10-6 4.07 0.9375 X 10-6 3.82 40 0.1126 X 10-7 4.00 0.3772 X 10-7 4.00 0.5917 X 10-7 3.99 80 0. 7029 X 10-9 4.00 0.2357 X 10-s 4.00 0.3697 X 10-s 4.00 Table 2 49 Actual B-spline coefficients c; for the collocation solution UN(x) of (3.68) (uniform mesh) BCOFF(-3) 0.2555488 N=lO BCOFF(-2) -0.0102544 BCOFF(-1) -0.2145311 BCOFF(0) -0.3654911 BCOFF(l) -0.4691899 BCOFF(2) -0.5297900 BCOFF(3) -0.5497234 BCOFF(4) -0.5297900 BCOFF(5) -0.4691899 BCOFF(6) -0.3654911 BCOFF(7) -0.2145311 BCOFF(8) -0.0102544 BCOFF(9) 0.2555488 Table 3a 50 Actual B-spline coefficients Ci for the collocation solution UN(x) of (3.68) (uniform mesh) BCOFF(-3) 0.1226595 ·N=20 BCOFF(-2) -0.0025697 BCOFF(-1) -0.1123807 BCOFF(0) -0.2078728 BCOFF(l) -0.2900018 BCOFF(2) -0.3595895 BCOFF(3) -0.4173324 BCOFF(4) -0.4638085 BCOFF(5) -0.4994828 BCOFF(6) -0.5247125 BCOFF(7) -0.5397500 BCOFF(8) -0.5447458 BCOFF(9) -0.5397500 BCOFF(lO) -0.5247125 BCOFF(ll) -0.4994828 BCOFF(12) -0.4638085 BCOFF(13) -0.4173324 BCOFF(14) -0.3595895 BCOFF(15) -0.2900018 BCOFF(16) -0.2078728 BCOFF(17) -0.1123807 BCOFF(18) -0.0025697 BCOFF(19) 0.1226595 Table 3b 51 Actual B-spline coefficients c; for the collocation solution UN(x) of (3.68) (uniform mesh) BCOFF(-3) 0.0600457 BCOFF(12) -0.4981776 N=40 BCOFF(-2) -0.0006428 BCOFF(13) -0.5120972 BCOFF(-1) -0.0574744 BCOFF(14) -0.5234388 BCOFF(0) -0.1105912 BCOFF(15) -0.5322308 BCOFF(l) -0.1601261 BCOFF(16) -0.5384951 BCOFF(2) -0.2062028 BCOFF(17) -0.5422475 BCOFF(3) -0.2489367 BCOFF(18) -0.5434972 BCOFF(4) -0.2884345 BCOFF(19) -0.5422475 BCOFF(5) -0.3247951 BCOFF(20) -0.5384951 BCOFF(6) -0.3581093 BCOFF(21) -0.5322308 BCOFF(7) -0.3884604 BCOFF(22) -0.5234388 BCOFF(8) -0.4159244 BCOFF(23) -0.5120972 BCOFF(9) -0.4405699 BCOFF(24) -0.4981776 BCOFF(lO) -0.4624586 BCOFF(25) -0.4816452 BCOFF(ll) -0.4816452 BCOFF(26) -0.4624586 BCOFF(27) -0.4405699 BCOFF(28) -0.4159244 BCOFF(29) -0.3884604 BCOFF(30) -0.3581093 BCOFF(31) -0.3247951 BCOFF(32) -0.2884345 BCOFF(33) -0.2489367 BCOFF(34) -0.2062028 BCOFF(35) -0.1601261 BCOFF(36) -0.1105912 BCOFF(37) -0.0574744 BCOFF(38) -0.0006428 BCOFF(39) 0.0600457 Table 3c 52 Chapter 4 THE GALERKIN METHOD 4.1 Optimal convergence of the H 1-Galerkin ap proximation with a piecewise spline space. Existence and uniqueness of the Galerkin solution The aim of Galerkin's method is to solve the boundary-value problem Lu = f where L : X --+ Y (H2(1) n HJ(J) --+ L2 (1)) is a continuous bounded linear operator from a Hilbert space X onto a Hilbert space Y. We consider a global continuous approximate solution to this boundary value problem. Let XN C X and YN C Y be closed subspaces with dimXN =/- dim YN, in general, and let PN : Y--+ YN(P'tT = PN) be the orthogonal projection such that IIP~IIL2 (I) = 1. The Galerkin method approximates the solution to the two point boundary value problem (I.la), (I.lb) by seeking a UN E Spo(IIN,3,2) = Sp(IIN,3,2)n {v: v(O) = v(l) = O} which solves the set of equations (4.1) fo 1 (-u; + aoU1 + a1UN) 53 For general ( E X, define the projection PNC to be the solution of the mini mization problem: (4.2) Since XN C L2 is finite dimensional, it can be shown that this problem has a solution and by XN being an inner product space with respect to the L2 inner product (·,·),the solution can be shown to be unique. N Let UN(x) = E Ci;(x) be an approximate solution to (1.la) and (I.lb). i=l The solutions UN E XN are elements of a finite-dimensional subspace (4.3) of dimension N whose basis elements { i}f:, 1 satisfy the boundary conditions (l.lb). Then UN E XN is a solution to the projection method, generated by XN and PN for the exact equation Lu = f if and only if (4.4) (LUN, x") = (!, x") for all x" E YN = SPo(ITN, l, 0). By the best approximation theorem, equation ( 4.4) is equivalent to (4.5) PN(LUN - f) = 0 where i = x" in (4.1) We henceforth call (4.4) the H 1-Galerkin equation. We are interested in the special case where Solving ( 4.4) is equivalent to the linear algebraic system N (4.6) ~ck(Lk,;) = (!,;), j = l,2,3, ... ,N k=l for the coefficients c1, c2, c3 , ••• , CN where ( ·, ·) is the usual L2 inner product de fined by (4.10). 54 N Thus the Galerkin solution UN( x) = E Cii( x) is determined by forcing the i=l residual (LUN - f) to be orthogonal to each basis function. The continuous Galerkin method reduces to determining Ci from the equations (4.7) fo1LUN(x)i(x)dx = fo1f(x)i(x)dx for i = 1,2,3, ... ,N resulting in the above linear system (4.6) which we denote by A c = g with matrix coefficients (4.8) The column vector g has components (4.9) 9; = fol J;dx j = 1, 2, 3, ... , N. The matrix A is symmetric and positive definite whenever the operator Lin (1.1) is elliptic (i.e; a 1(x);?: 0). Letting (u,v) denote the L2 inner product (4.10) (u,v) = fo1u(x)v(x)dx define the H 1-Galerkin approximation to the exact solution u of the linear two point boundary value problem (1.1) as the function UN E SPo(IIN, 3, 2) satisfying the condition ( 4.11) (LUN, v") = (f, v") for all v E Sp0 (IIN, 3, 2). SPo(IIN, 3, 2) is called the space of trial functions, whereas Sp0 (IIN, 1, 0) denotes the space of test functions. The approximation space Sp0 (IIN, 3, 2) is defined to be the set of cubic splines on IIN satisfying the boundary conditions ( 1.1 b). Likewise SPo(IIN, 1, 0) C C[0, 1] will denote the set of continuous piecewise linear functions defined on the uniform partition IIN of I and vanishing at the endpoints x = 0 and 1. We assume L : H2 (J) n HJ(J) -+ L2(I) is a continu ously invertible operator so one ensures that Lu = f has a unique solution u in H2 (J) n HJ(J). We introduce some definitions of convergence and stability of the finite element Galerkin method. 55 Definition 4.1 (CONVERGENCE) Galerkin's method is said to be con vergent if there exists-an No E N such that VN > No and for every f, we have a unique approximate solution UN E S.Po(IIN, 3, 2) and (4.12) as N-+oo where u is the exact solution of Lu= f. Definition 4.2 (STABILITY) The Galerkin method is said to be stable if VN ~ N0 we have a unique solution UN of the Galerkin approximation equations (4.11) for each f(x) contained in L2 (J) and if the sequence of projection operators P~ : H 2 (J)nHJ(J) -+ SPo(IIN, 3, 2) defined by (2.8) are uniformly bounded, that JS (4.13) µ := sup IIP.ZII < oo. N~No Definition 4.3 The formal adjoint L * of L in (1.1) is defined by L*v = - ~: - ! (a0 v) + a1v Vu, v E H 2 n HJ. L is called self adjoint if (u,Lv) = (L*u,v) implies both (i) L = L* and (ii) L,L* satisfy the equiva lent boundary condition manifolds for all u, v E H 2 n HJ. L is called positive definite if it is self adjoint and satisfies (Lu, u) > 0 for all u E H 2 n HJ with u #0. Remark: Thus L in (1.1) is not formally self adjoint unless a0 = 0 where the scalar or inner product is (u, v) = JcJ u( x )v( x )dx. For sufficiently smooth u E HJ ( I) we have (4.14) u(x) = -(Lu, G(x, ·)) where G(x,!) denotes the Green's function for (1.1). Let r( u, v) be the particular bilinear form r : H 1 ( J) x H 1 ( I) -+ lR given by (4.18) so that for all u,u1,u2 E H 1(J) and v,v1,v2 E H 1(J) and a scalar c we have (a) r(u1 + u2, v) = r(u1, v) + r(u2, v) (b) r(u,v1 + v2) = r(u,vi) + r(u,v2) (4.15) (c) r(cu,v)=cr(u,v) (d) r(u,cv)=cr(u,v). 56 If u is the solution of (1.1) then clearly (4.16) (Lu,v) = (f,v) for all v E HJ(J), and on integrating by parts and applying the Dirichlet boundary conditions we have (4.17) r( u, V) = (/, V) for all v E HJ(J), where (4.18) r( , t/J) - (', t/J') + (ao', t/J) + (a1 , t/J) - fo\ In what follows, we establish a uniform error bound at the nodes Xi of our partition. Suppose x E Sp0 (ITN, 3, 2) then (4.19) (u - UN )(xi) - r((u - UN), G(xi, ·)) - r(e, G(xi, ·)) - r(e, G(xi, ·) - x) because r(e,x) = 0 for all X E SPo(IlN,3.,2). Thus le(xi)I = jr(e, G(xi, ·) - x)I - l(e', Ge(xi, ·) - x') + (aoe', G(xi, ·) - x) + (a1e, G(xi, ·) - x)I < lle'IIL2 IIGe(xi, ·) - x'IIL2 + c1lle'IIL2IIG(xi, ·) - XIIL2 + c2llellL2IIG(xi, ·) - XIIL2 by the Cauchy-Schwarz inequality and since a0 ( x ), a 1 ( x) are continuous and there fore bounded on I. - lle'IIL2[11Ge(xi, ·) - x'IIL2 + c1IIG(xi, ·) - XIIL2] + c2llellL2IIG(x,, ·) - XIIL2 < lle'IIL2CIIG(xi, ·) - XIIH1 + CllellL2IIG(xi, ·) - XIIL2 < lle'IIL2CIIG(xi, ·) - xllH1 + CllellL2IIG(x,, ·) - XIIH1 - CIIG(x,, ·) - xllH1[11ellL2 + lle'IIL2] (4.20)::; CIIG(xi,·)- XIIH1llellH1 57 where C is a generic constant which does not necessarily take on the same value at each occurrence. By the smoothness properties of G(xi, ·), since a0 and a 1 E C00(l), it follows that as well as and hence there exists a Lagrange interpolant say x E SPo(IIN, 3, 2) such that ( 4.21) By Theorem 4.1 which we prove later, we have and certainly for cubic splines (4.22) for where h -+ 0 as the number of subintervals N goes to infinity. Hence combining the inequalities ( 4.20), ( 4.21 ), ( 4.22) we obtain l(u - UN)(xi)I - le(xi)I < CIIG(xi, ·) - XIIH1 llellH1 < Ch3 llellH1 < Ch8+211ullH•, 1 $ s $ 4; i = 0,1,2, ... ,N where u E H 8 (I) n HJ(/) is the exact solution to the boundary value problem (1.1). Our aim next is to derive an optimal estimate of the error e = u - UN (see also [17]). Theorem 4.1 Let u be the solution to (1.1) and UN the Galerkin solution satisfying r(UN,v) = (f,v) for all v E SPo(IIN,3,2). Suppose u E H 8 (l) where 1 $ s $ 4. 58 Then for h sufficiently small, Proof Let E HJ(/) be the solution of L* = e, x E J (O) = (1) = 0 where L* = -" - (a0 )' + at. L * denotes the formal adjoint of the linear operator L. We now invoke a result, namely ( 4.37), which is derived in the forthcoming uniqueness proof for the adjoint equation L *'if; = Y. In this context, it is given by (4.23) Since f( u, V) = (/1 V) and r(UN,v) = (f,v) for all v E Sp(IIN, 3, 2). Hence subtracting yields r(e, v) = 0. Using the fact that f(e,e) = f(e,u- X), X E Sp(IIN,3,2) it follows that (e', e') - f(e, e) - (a0 e', e) - (a1e, e) - r(e,u- x)- (aoe',e)- (a1e,e) or lle'lli2 = r(e,u- x)- (aoe',e)- (a1e,e). 59 Due to the boundedness of the bilinear form and the fact that a0 , a 1 are both bounded on I, then the Cauchy-Schwarz inequal ity implies/that Recalling (4.23) we obtain lle'll12 < CllellH1llellv + Cllelli1h2 + CllellH1llu - XIIH1 lle'll12 < c11e11i1h + c11e11i1h 2 + CllellH1llu - XIIHl Adding the quantity llelli2 to both sides of the inequality gives and hence (4.24) for h sufficiently small so that (4.25) For u E H 3 (/) n HJ(/), 2 ~ s ~ 4, the spline approximation theorem says that there exists a x E Sp(IIN, 3, 2) such that which implies because and hence that (4.26) 60 The combination of (4.23) with (4.26) yields as well as (4.28) Adding (4.27) and (4.28) proves the results of Theorem 4.1 for h sufficiently small. We have just demonstrated that the global convergence rate obtained in Theorem 4.1 above is of optimal order under the hypotheses that u(x) is required to satisfy. At this stage we examine the uniqueness of the Galerkin approximation to (1.1). In order to show that there exists a unique solution UN E Sp(ITN, 3, 2) of the H 1-Galerkin method (4.29) f(UN,v) = (f,v), for all v E Sp(IIN,3,2) and any f E L2(/) we use a technique described in [13] and adapted by Fair weather [17]. Suppose U1 and U2 are two different solutions of (4.29) and set Y = U1 - U2. Combining f(Ui,v) - (f,v) with f(U2,v) (f,v) we obtain (4.30) r(Y, v) = 0 for all v E Sp(ITN, 3, 2) Let 1/J E HJ (/) be the exact solution of L*'I/J = Y, X E J, t/J(O) = 1/J(l) = 0, where L * denotes the formal adjoint operator of L defined by L*u = -u" - (aou)' + a1u. 61 From the smoothness assumptions on the coefficients a0 and a 1 and since IIN is uniform, then by regularity we have (4.31) An integration by parts on (L*u, v) for v E HJ(/) gives (4.32) IIYlli2 - (L*tt,, Y) - r(Y,tt,) - r(Y, tt, - x) where the last line follows on noting equation ( 4.30). Hence (4.33)IIYll12 - (Y',(tt,- x)') + (aoY',(tt,- x)) + (a1Y,tt,- x) < IIY'IIL2ll(tfa - x)'IIL2 + CIIY'IIL2lltfa- xllL2+ CIIYIIL2lltfa- xllL2 - IIY'IIL2[II( tfa - x)'IIP + Clltfa - xllP] + CIIYIIL2lltfa - xllP < CIIY'IIL2lltfa - xlln 1 + CIIYIIL2lltfa - XIIL2 < CIIY'IIL2lltfa - xlln 1 + CIIYIIL2lltfa - xlln 1 - Clltfa - xlln 1 [IIY'IIL2+ IIYIIL2] < Clltfa - xlln 1 IIYlln1 where on the third previous last line we have used the fact that lltfa-xllL2 ~ lltfa-xlln• for any tp, X· By the spline approximation Theorem, there exists a x E Sp(IIN, 3, 2) such that (4.34) and (4.35) Therefore on adding these inequalities (4.34), (4.35) we obtain lltfa - xlli1c1> < Ch2 [lltfa'lli2cn + lltfa"lli2cn] < Ch2 lltfalli2c1> 62 or (4.36) Thus, IIYlll2 < ChllYIIH1 ll'PIIH2 < ChllYIIHl IIYIIL2 by result ( 4.31 ). It follows that (4.37) Since YE Sp(IIN, 3, 2), then f(Y, Y) = 0. Hence on expanding the variational formulation of the adjoint problem we have f(Y, Y) = (Y', Y') + (a 0Y', Y) + (a1Y, Y) = 0 that is IIY'lli2 = -(aoY', Y) - (a1Y, Y) Applying the-Cauchy-Schwarz inequality to bound the right hand side gives Combining this inequality with ( 4.37) leads to IIYllk1 - IIYlli2 + IIY'll12 :::; C [IIY'IIL2 IIYIIL2+ IIYllh] - C [IIY'IIL2 + IIYIIL2] IIYIIL2 < CIIYIIHl . hllYIIHl that is, Hence it follows that, for h sufficiently small 63 or Y=0 as required. This completes the proof. We have thus established that Y = U1 - U2 = 0, i.e.; U1 = U2. And with this, the weak form (4.29) to the H1-Galerkin method has a unique approximate solution belonging in the closed subspace Sp(ITN, 3, 2) of cubic splines. We shall turn our attention to derive an error bound in the Sobolev H 2-norm for the H1-Galerkin approximation to (1.1) satisfying the relation (4.38) (LUN, v") = (f, v") for all v E Sp(ITN, 3, 2). Here the functions v" are continuous piecewise linear on the interval J. Demonstration of uniqueness and hence existence of the Galerkin approxima tion UN E Sp(ITN, 3, 2) satisfying ( 4.38) follows a similar argument as the work just completed. To estimate the accuracy of the solution UN of (4.38) we assume 1/J E H2(J)nHJ(J) is the solution of the adjoint problem (4.39) L*'I/J e 1/J(0) = 1/J(l) - 0 As before IIUNlli2 = (LUN, - x") where x" E Sp(IIN, 1, 0) and hence (4.40) Taking infimums of both sides and noting that IIUNlli2 is constant on I, shows that (4.41) Recalling the spline approximation theorem, if E H 2(J), then (4.42) inf II - vllp < C h2ll"IIL2 vESp(IlN,1,0) < Ch2 llIIH2 64 Linking (4.41) with (4.42) and using (4.31) with t/J and Y replaced by and UN, respectively, we obtain IIUNlli2 < IILUNIIL2 • Ch2 ll < IILUNIIL2Ch 2 IIUNIIL2 or what is the same (4.43) IIUNIIL2 < Ch2 IILUNIIL2 < Ch2 IIUNIIH2 On performing an integration by parts and taking into account the homoge neous boundary conditions, we find that -(U~,UN) < IIU~IIL2 IIUNIIL2 where the last line follows by the Cauchy-Schwarz inequality. Result ( 4.43) gives smce for any UN E Sp(IIN, 3, 2), or equivalently (4.44) Without loss of generality, if we replace UN bye in (4.43) and (4.44), respectively, we have (4.45) and (4.46) Consider the variational formulations of (1.1), namely: (LUN, v") = (f, v") for all v E SPo(IlN, 3, 2) and 65 (Lu, v") = (f, v") for all v E S11o(ITN, 3, 2). Subtracting these relations yields (Le, v") = 0, \Iv" E Sp(ITN, 1, 0). If X E Sp(ITN, 3, 2) then (Le, e") = (Le, (u - x)") where Le is represented by (4.47) Le= -(u - UN)"+ ao(x)(u - UN)'+ a1(x)(u - UN) that is, e" =-Le+ ao(x)e' + a1(x)e On forming the exact inner product with respect to e", we see that (4.48) 1le"lli2 = -(Le, (u - x)") + (aoe', e") + (a1e, e") Introducing the Cauchy-Schwarz inequality to the right hand side of ( 4.48), we have, The continuity of a0 , a1 on I implies that they are bounded on I, and this is incorporated in the right side of (4.49). Continuing in this manner, we obtain (4.50) lle"lli2 < Cllelln2ll(u - x)")IIL2 + C(hllelln2 + h2 llelln2)lle"IIL2 < Cllelln2ll(u - x)"IIL2 + C(h + h2)llelln2llelln2- The first term on the right side of the first line of ( 4.50) is due to the boundedness of L from H 2 (I)nHJ(I) into L2 (1) (i.e. IILellL2(1) ~ Cllelln2(1))- If the differential operator Lis considered acting from L2(J) into L2(J), then it can be shown that L is unbounded (see Mikhailov [31]). We have wisely avoided this scenario by judiciously choosing L: H 2(J) n HJ(/) --... L2 (I). 66 In the second term of (4.50) we have made use of lle"IIL2 < llell112- Adding the quantities lle'llh, llellh to both sides of the inequality ( 4.50) produces our desired result (4.51) llell~ - llell},2 + lle'lli2 + lle"lli2 < Cllellwll(u - x)"IIL2 + C(h + h2)llell~ + Ch4 llell~ It follows for h sufficiently small that and hence We summarize the previous claim in the following theorem. Theorem 4.2 Suppose x = UN E Sp(IIN, 3, 2) is uniquely chosen to satisfy the Spline Approximation theorem. Let u denote the true solution of (1.1) and UN satisfies the variational formu lation (4.38). Suppose also that u E H 8 (J} n HJ(/), 2 :5 s :5 4, then there exists a positive constant C such that (4.52) llu - UNIIH2{1) < Cll(u - x)"IIL2 - Cll(u - UN)"IIL2 < Ch2llu<4>IIL2 < Ch2llullH4 (I)· We establish a superconvergence error estimate of the H 1-Galerkin method in the approximation space SPo(IlN, r, 2), for r > 3. That is we consider quartic, quintic and higher degree splines to approximate u in each subinterval. We have the error at the nodes {Xi} satisfying (4.53) -(Le, G(xi, ·)) - -(Le,G(xi,·)-x") Since (Le, x'') = 0, X E SPo(IlN, r, 2) 67 as before. Hence (4.54) le(xi)I < IILellvillG(xi, ·) - x"IIL2 < 1lellH2IIG(xi, ·) - x"IIL2 where G(xi, ·) denotes the Green's function for any eof the two-point boundary value problem (1.1) and x" is a piecewise continuous polynomial of degree r - 2, which interpolates G(xi, ·). The smoothness properties of G(xi, ·) show that and hence (4.55) IIG(xi, ·) - x"IIL2 $ Chr-l since x" E SPo(IIN,r - 2,0). Suppose u E H 11 (I) n HJ(/), 2 $ s $ r + 1, we see from (4.52), (4.54) and (4.55) that (4.56) le(xi)I < CllellH2hr-l < Chr+lllullH4 for r > 3 This shows that the rate of convergence of UN(x) to u(x) at the nodes Xi is of order hr+l where h = k· In terms of operators, convergence of the projection method for (1.1) means that for all n ~ N the finite dimensional operators LN = PNL : XN --+ YN are invertible and that pointwise convergence {4.57) L"i,/ PNLu--+ u, N--+ oo, holds for all u E H2 {/) n HJ(/). In general, we can expect convergence only if the subspaces XN possess the dense ness property {4.58) inf llt/1 - ulln2(I)--+ 0 as N--+ oo for all u EX= H2 {/) n HJ(/) 1/,EXN o 68 Since LN = PNL is a. linear opera.tor between two finite dimensional spaces, carrying out the projection method reduces to solving a. finite dimensional linear system. Theorem 4.3 There is always a projection PN from a Hilbert space onto any one of its closed subspaces. Proof. [omit] (see [141). Theorem 4.4 Let L: H2(I)nHJ(I)--+ L2(1) be an injective linear bounded operator. Suppose XN = SPo(IIN, 3, 2) C H 2 (J) n HJ(J) is a finite dimensional subspace. Then for each f E L2 (J) there exists a unique element UN E XN such that (4.59) It is called the least squares solution of Lu = f with respect to XN and corre sponds to the H 1-Galerkin formulation for the subspaces XN and YN := LXN. Proof. Notice that UN is just the least squares solution of Lu = f with respect to XN if and only if LUN is a best approximation to f with respect to YN. It is well known that the best approximation from XN exists and is unique. Further, the injectivity of the operator L implies uniqueness of the least squares solution. However, the best approximation LUN is characterized by the orthogo nality condition (4.60) (LUN-f,x'') = 0 for all x" E YN = SPo(IIN,1,0) C C[O,l]. But this exactly yields the H 1-Galerkin formulation (4.4) for the subspaces XN and YN, Theorem 4.5 (CONVERGENCE) Let L : X --+ Y [H2(I) n HJ(J) --+ L2 (J)] be continuously invertible. Then the following are equivalent. (4.61) (i) The Galerkin method converges for every right hand side f E Y. (4.62) (ii) lim dist(f, YN) = 0 for all f E Y N-+oo (4.63) ( iii) \/N ~ No(No sufficiently large): Y = YN ffi YJ where YN C Y' then YJ = {y* E HJ(J) I (x,y*) \/x E YN} is the orthogonal complement of HJ(I) and Y' C HJ is the space of all continuous linear functionals 69 on HJ (or the topological dual of HJ). (4.64) (iv) Let RN: Y--+ YN be the projection along YJ, then p* = sup IIRNII < oo N~No (4.65) (v) If u EX then lim dist( u, XN) = 0 where N-+oo dist(u, SN)= inf llu - xllm(I)· :cEXN o Proof. (i) ==} (iii) From (i) we have that for every given f E Y and \/N > No there exists a unique UN E XN such that LUN - f = -ZN E YJ, or similarly (4.66) and ZN E YJ. We are required to show that YN n YJ = {0}. To do this suppose WN E YNnYJ, say WN = LxN for a XN E HJ. Now for this particular choice f = WN there are two solutions, namely, UN= XN and UN= 0, so XN = 0 and hence WN = 0. This means that we have shown ( 4.67) Since UN--+ u as N--+ oo it follows that (4.68) IILUN - !IIL2 < IILllngllUN - ullng < CIIUN - ullng --+ 0 as N --+ oo because from the hypothesis of the theorem, L is also uniformly continuous on HJ ( I) and therefore bounded on I. Hence O ~ dist(f, YN) ~ IILUN - fllL2(1) --+ 0 as N --+ oo which yields dist(f, YN) --+ 0 as N --+ oo. We have thus shown (i) ==} (ii), i.e, (4.61) ==} ( 4.62). From the first implication it follows that RN f --+ f \/f E Y = L 2(1) where RN is the projection from L2 onto Sp0 (ITN, 1, 0) along YJ = Sp}(IIN, 1, 0). 70 This implies that RN converges pointwise to the identity operator on Y, that is, f -+ RN f as N -+ oo for all / E Y. Due to the fact RN is a continuous linear mapping L2 (J) onto SPo(lIN, 1, 0) (chosen from the set F of all continuous linear operators on L2 such that for each (4.69) and on applying the Banach-Steinhaus theorem we get (4.70) We have proved (4.64). It remains to show that (ii)==> (v), i.e, (4.62) ==> (4.65). Let u E H 2 n HJ, XN E XN then (4.71) dist(u,XN) inf llu - xllH2 ~ llu - XNIIH2 for some XN zESN o o - IIL-1 L(u - XN)IIHJ < IIL-1 IIHgllLu - LxNIIHJ where LxN E YN. Since this holds for every x N E XN, hence for every LxN it follows (4.72) 0 ~ dist(u,XN) < IIL- 1 IIHgdist(Lu, YN) IIL- 1 IIHgdist(f, YN) --+ 0 by (ii) for\;// E Y as N--+ oo 4.2 A convergence theorem for the approximate solution by the H 1-Galerkin method of sec ond order nonlinear differential equations In the Banach space L! consider the nonlinear ordinary differential equation (4.73) u"(x) = f*(x,u,u') 71 with u E L! subject to the mixed boundary conditions (4.74) -\ou'(0) + µou(0) - a -\1u'(l) + µ1u(l) - P Suppose the function J* is continuous on the set D = {(x,y,y') I 0 ~ x ~ 1,-oo < y < 00,-00 < y' < oo} an d t h at t h e parti"al d er1vatives . . -aar an d -aar are also contmuous. on n . y y' Suppose also that the Dirichlet boundary conditions hold. This is the case when ,\0 = ,\1 =a= P = 0, µ0 = µ 1 = 1 in (4.74). If the following conditions are met (i) a J* (;~y' y') > 0 for all ( x, y, y') E D (ii) a constant C exists such that of*(;;~' y') < C for all (x, y, y') E D with u(0) = u(l) = 0, then it is known that the nonlinear ordinary differential equation (4.73) has a unique solution (see Keller [241). We shall first transform (4.73) and (4.74) to an operator equation of the form (4.75) Kv=v with nonlinear operator K, continuous over I - [0, l]. Define this operator K: L!--+ C by [1 [18G(x t) (4.76) Kv = f*(x, lo G(x, t)v(t)dt, lo ax' v(t)dt) on the ball llv - vollLi ~ f, of arbitrarily small radius f. Let G(x, t) above be the Green's function for d~2 on a manifold of L! defined by the homogeneous form of (4.74). The required solution to (4.73), (4.74) is constructed in two parts. The func tion u 1(x) defined by /1 [1 /1 8G(x t) (4.77) u1 (x) = lo f*(t, lo G(x,t)v(t)dx, lo ax' v(t)dx)G(x,t)dt is a solution of (4. 78) d2u1 = f* dx 2 72 such that (4.79) Aou~ (0) + µou1 (0) - 0 A1u~(l) + µ1u1(l) - 0 To solve our original problem (4.73), (4.74) we seek a further function u2, which is a solution of the equation (4.80) tPu2 = 0 for x E (0,1) dx 2 satisfying (4.81) Aou~(0) + µou2(0) - a A1u~(l) + µ1u2(l) - /3 The full solution to (4.73), (4.74) (see Stakgold [42]) is given by u = u1 + u2 where u1(x) = W(w':':J:~ls=o JC: w1(t)f*(t, f~ G(x, t)v(t)dx, J~ aGJ;,t>v(t)dx)dt + W(w::J:~ls=o f~ w2(t)f*(t, f~ G(x, t)v(t)dx, J~ aGJ;,t>v(t)dx)dt u2(x) = W(~;}>l~=o [w2(l)A1 - w~(l)µ1] (4.82) + W(;~>t=)w1(0)Ao - wao)µo] w1(x) = c1 + c2x; AoC2 + µoc1 = a w2(x) = c3 + c4x; µ1c3 + (A1 + µ1)c4 = /3 and W(w1,w2)lx=O is defined by (3.44). The operators (4.83) mapping L! into L! are completely continuous because Gk(k = 0, 1) is linear and every bounded subset of L! has the closure of its image being compact. In (4.75), (4.76) set v(x) = u"(x) such that u(k)(x) = Gkv(x) k = 0, 1. Equa tion (4.76) can be rewritten as (4.84) Kv = f*(x,G0v,G1v). 73 As a preliminary we choose I to be the identity operator embedding C into L!(I : C --+ L!) and therefore the boundary value problem (4.73), (4.74) is equivalent to the nonlinear operator equation (4.85) v=IKv considered in the Banach space L!, Denote T = I K (T : L! --+ L!) and hence T is completely continuous in the ball llv - vollL} $ f. Suppose XN = SPo(IIN, 3, 2) C L!, YN = SPo(IIN, 1, 0) C L! are two se quences of subspaces with dimXN = (2N +2)-2 = 2N, dimYN = dimSPo(IIN, 1, 0) = 2N - 2 and let P;,°: L!--+ YN be a projection operator onto YN, Suppose also that VN E XN, then the projection method generated by XN andPN = pNG approximates the exact equation (4.86) v = Tv = f*(x,Gov,G1v) by the projected equation (4.87) This projection method is called convergent for T if there exists an index N0 such that for each f*(x, G0v, G1v)_ EL!, the approximating equation PNTVN = VN has a unique solution VN E YN for all N ~ N0 and if these solutions converge VN --+ v as N--+ oo to the unique solution v of Tv = v. The fixed point equation analogue (4.86) of the system (4.73), (4.74) 1s uniquely solvable by (4.88) u(x) = fo1 G(x, t)v(t)dt where G(x, t) is given explicitly by G(x, t) (4.89) and H(x - t) is the Heaviside unit function defined as H(x - t) - 1, - 0, X < t. 74 The properties of the Green's function G(x, t) and the continuity of f*(x, G0v, G1v) show that T is a nonlinear completely continuous (compact) operator which maps the Hilbert space L!(J) equipped with the L!-norm consisting of square integrable functions onto L!(J), where I= (0, 1]. In this non.linear framework the spline projector P;, : L!(I) -+ SPo(IIN, l, 0) defines an orthogonal projection of v onto a finite dimensional subspace of L! consisting of piecewise linear functions on a uniform partition IIN of the interval (0, 1]. For the cubic spline solution space, introduce the notation GoSPo(IIN, 1,0) = SPo(IIN, 3, 2) which is attained from UN( x) = GovN( x) with VN = u'f.,. Then for any real valued function Tv EL! we can express the spline projector as follows: 2N-2 {4.90) P;,Tv = L (Tv, t/Ji)t/Ji; dimSPo{IIN, 1, 0) = 2N - 2 i=l where { tp1 , tp2 , tp3 , ••• , t/J2N-2 } is an orthonormal basis for the approximation spline space YN = SPo(IIN, l, 0) and v = Tv E L!(J). Thus we are in a position to formulate the H 1-Galerkin method for the solution of (4.73), (4.74). The function VN E YN = SPo(IIN, l, 0) is an approximate solution to the non.linear equation v = Tv generated by XN and P;, if and only if it satisfies the projection equation ( 4.91) (TvN,9) = (vN,9) for all g E SPo(IIN, 1, 0), the set of test functions. If we write the residual as (4.92) we find that ( 4.93) P;,Tv = 0 in {4.90) if and only if the following condition holds (4.94) (Tv, -,Pi) = 0 i = 1, 2, 3, ... , 2N - 2 where ( ·, ·) is the usual L2 inner-product on L!. 75 Without loss of generality, we force the residual TN in equation ( 4.92) as in the linear case to be approximately zero in some sense in the expectation that the resulting function vN(x) will be a good approximation of the true solution v(x) of (4.86). To find VN, we must solve (4.91) obtaining a nonlinear algebraic system of 2N - 2 equations in 2N - 2 unknowns. This nonlinear system is solved by the Newton-Kantorovich iterative scheme and is expected to produce quadratic convergence provided its Jacobian matrix is invertible and a sufficiently accurate starting value is known. We label this as Galerkin's method for obtaining an approximate solution to (4.86) and hence to the original boundary value problem (4.73), (4.74). Equation (4.94) now becomes (4.95) i = 1, 2, 3, ... , 2N - 2 which is equivalent to the form ( 4.96) Expanding TN out in (4.96) and taking into account P;,vN = VN, leads to the Galerkin approximate equations: We now state Vainikko's Theorem as modified in (Lucas and Reddien [27, p342]) to derive our upper bound for the error, then show that the approximate solutions VN converge to v as N-+ oo. Theorem 4.6 Suppose { P;,} is a sequence of continuous projections converg ing pointwise to the identity operator on L!(I) and Ta nonlinear operator acting on L!(I). Let v be a solution to the nonlinear operator equation v = Tv with T completely continuous on an open set containing v, T Frechet differentiable at v and the equation v0 -'- T'(v)v0 = 0 possessing only the trivial solution in L!(I). Then vis unique in some sphere llv - vollLi ~ f, f > 0 and there exists an integer 76 N such that for k > N the equation v = PkTv has a unique solution VN in the same sphere. Furthermore there exists a constant C > 0 such that (4.97) Proof. llv - VNIIL~ - ll(v - P.vv) + (P.vv - VN)IIL~ < llv - PNvll~ + IIPNv - VNII~ - llv - P.vvllL~ + IIPNT(v - VN)IIL~ by using the exact and approximate equation definitions (4.86) and ( 4.87), re spectively. so that (4.98) < llv-P.vvllL~ 1 - ll~NTIIL~ llv - P.vvllL~ where O < IIPNTII < 1 and hence M2 > 1. In what follows, we endeavour to make the right side quantity (PNv - v) approach zero with respect to the L2 inner-product norm. Before we proceed to do this we shall make use of two facts in completing this proof. First of all that the set of all continuous functions on [O, 1] are dense in L!(O, 1). Also, that the orthogonal projector PN satisfies IIPNIIL~ = 1. Choose v E L!(O, 1) and let { vm} be a sequence of continuous functions which converge to v in L!(O, 1). We have, ll(v - Vm) + (vm - P.vvm) + (P.vVm - PNv)IIL~ < llv - VmllL~ + llvm - PNvmllL~ + IIPN(v - Vm)IIL~ < 2llv - VmllL~ + llvm - P.vvmllL~ 77 Given an E > 0 {i sma.11), fix m such that llv - vmllLj < 'i" This implies that for a.11 N {4.99) llv - PNvllYp < i + llvm - PNvmllYp Employing a feature of our Hilbert space structure into the right side of inequality {4.99), viz {2.7), shows that PNvm minimizes this error and hence can be made arbitrarily small as N --+ oo. We have, {4.100) Combining this result into {4.98) shows that Therefore VN converges to the exact solution v of equation ( 4.86) as N --+ oo. Because IIPN°IIL2 = 1, N ~ 1 are uniformly bounded then the estimate IIPNv - vllL2p converges to zero due to the fact that (4.101) lim {inf llv - VNIIL2 : VN E Spo(IIN, 1, O)} = 0 N-+oo P for all v E L!, implying that PN --+ I on L!. However, the latter is a natural hypothesis of the convergence theorem 4.6. We see this: IIPNv - vllpp = IIPN(v - vo) - (v - vo)IIL2p because PNvo belongs to the subspace. We have by theorem 4.6. Hence llv - vNIIL2p --+ 0 as required. We focus our attention briefly now to the discrete H 1-Galerkin method. The central motivation for introducing a bilinear form(·, ·)h defined by (4.103) for all g E SPo(IIN, 1,0), is basically that the matrix coefficients (TvN,9) which arise in the nonlinear system of algebraic equations are very difficult to compute exactly. 78 Thus it is necessary to implement a quadrature rule to numerically approx imate these integrals. In this framework define a linear discrete projection op erator ~ onto S11o(IIN, 1, 0) by RNg = g E S,11o(IIN, 1, 0) which correlates with the qualocation theory developed by Sloan (Sloan Tran, Fairweather (40]; Sloan and Grigorieff (41]), we have that the approximate solution VN sought in the trial space is actually of the same dimension as the test space S11o(IIN, 1, 0). The only 'ingredient' left to apply this theory is to specify the quadrature rule Qh. We judiciously choose the composite 2-point Gauss rule by 1 N (4.104) Qhg = 2 :E hk[g(xk,1) + g(xk,2)] k=l with discrete inner-product (w,v)h = Qh(wv) in the bilinear form (4.103) and where 1 1 Xk,1 Xk + 2(1 - vJ )hk 1 1 Xk,2 - XA:+ 2(1+ \1'3)hk are Gauss-Legendre points. We demonstrate that (4.105) is equivalent to ( 4.103). Using the definition of RN; (RNg,g)h = (g,g)hVg E SPo(IIN, 1,0) and (4.105) implies that as well as (TvN - VN,9)h = (0,g)h so that Since the discrete projection operator RN has its range in SPo(IIN, 1, 0) then this allows us to write R'F,g in terms of the piecewise linear chapeau functions { 2N-2 (4.107) I: (i, 1c)hCi = (g, 1c)h, k = 1, 2, 3, ... , 2N - 2 i=l or equivalently the banded system of nonlinear algebraic equations; for the coefficients Ci(i = 1, 2, 3, ... , 2N - 2) and z1c in z (k = 1, 2, 3, ... , 2N - 2) is the scaled variable 1 Zk = 2h (g, Suppose RNvN, VN E H 2 then a result from Sloan and Grigorieff [41] shows that RNvN converges to VN in the L2 norm II· IIL2 with order h2 accuracy, i.e, (4.108) We postulate that llv - vNIIL2 < CIIRNvN - vNIIL2 < Ch2 llv~IIL2 -+ 0 as N -+ oo. Thus we have shown the discrete H 1-Galerkin method converges as the mesh spac ing h-+ 0. It follows that the Galerkin approximation equations RNTVN = RNVN possess a unique solution VN in some sphere llv - voll0 ~ f containing v0 where v is the exact solution of equation ( 4.86). For comparison purposes of the two meth ods, if we consider collocation of the nonlinear boundary value problem ( 4. 73), ( 4. 74) using cubic B-splines UN E SPo(IIN, 3, 2) with u( x) a classical solution or 80 true solution to the same problem, then, the conditions Uf:,(x,) - f*(x,, UN(x,), UN(x,)) i = 1, 2, 3, ... , N AoUN(0) + µoUN(0) - o A1UN(l) + µ1UN(l) - /J define spline nodal collocation in S.Po(ITN, 1, 0). The expected order of convergence for the homogeneous case (..\0 = ..\1 = o = /3 = 0, µ0 = µ 1 = 1) with u E C''[0, 1] is summarised as llu - UNllv :5 Cllu - UNIILoo :5 Ch2 (see Lucas and Reddien, [27]) which moreover is a suboptimal estimate in the literature and cannot be improved. 81 References [1) Ahlberg, J., Nilson, E. and Walsh, J. 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