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Collocation and Galerkin Methods for the Approximate Solution of Linear and Nonlinear Differential Equations

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Collocation and Galerkin Methods for the Approximate Solution of Linear and Nonlinear Differential Equations 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<f>IIL2 --+ 0 for every </> E H 2 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<x<l with Dirichlet boundary conditions {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: Ci</>i 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.
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