Chapter 12 Spectral and Pseudo-Spectral Methods for Periodic Elliptic Equations In this chapter, we will investigate the solutions to periodic elliptic equations in one space dimension. Most everything can be generalized to two and three- dimensional problems as well. 12.1 Periodic Di↵erential Equations Definition 12.1.1. Suppose that <a<b< . We say that a function 1 1 f : R R is [a, b]-periodic i↵ f (x + n(b a)) = f (x) for all x R and any ! − 2 n Z.Form =0, 1, 2, , we define the set 2 ··· m m C (a, b):= f C (R) f is [a, b]-periodic . p { 2 | } Definition 12.1.2. Suppose that <a<b< , c 0,andf C (a, b). 1 1 ≥ 2 p We say that u : R R is a classical solution to the [a, b]-periodic Helmholtz ! problem, d 2u (x)+cu(x)=f (x),xR, (12.1.1) −dx2 2 i↵ u C2(a, b) and u satisfies (12.1.1) point-wise. We call this the [a, b]- 2 p periodic Poisson problem when c =0. Classical solutions are defined simi- larly. Remark 12.1.3. One interesting feature of the periodic problem is that there are, in some sense, no boundary conditions, since we demand that (12.1.1) 173 174 CHAPTER 12. SPECTRAL METHODS holds on the whole real number line. Still, we some times say the boundary conditions are of periodic type. In any case, for the periodic problem, one must be careful about the value of c. Proposition 12.1.4. Suppose that f C (a, b) and u C2(a, b) is a classical 2 p 2 p solution to the [a, b]-periodic Helmholtz problem with c 0,then ≥ b b c u(x) dx = f (x) dx. Za Za Consequently, a necessary condition for there to be a solution to the [a, b] b periodic Poisson problem (c =0) is that a f (x) dx =0. Proof. The proof follows using integration-by-parts,R Theorem 4.2.14, and is left as an exercise. Theorem 12.1.5. Suppose that <a<b< , c 0,andf C (a, b). 1 1 ≥ 2 p If c>0, there exists a unique classical solution u C2(a, b) to the [a, b]- 2 p periodic Helmholtz problem (12.1.1).Anecessaryandsufficient condition for the existence of a classical solution to the [a, b]-periodic Poisson prob- b b lem (12.1.1) (c =0) is that a f (x) dx =0.Ifc =0and a f (x) dx =0 there is a unique classical solution u to (12.1.1) in the function class R R b C˚2(a, b)= u C2(a, b) u(x) dx =0 . p 2 p ⇢ Za Proof. The proof can be gotten using the periodic Green’s function, or by using Fourier series techniques. See [9, 12, 17]. 12.2 Finite Di↵erence Approximation Now, we wish to approximate the classical solutions to these problems using finite di↵erence methods. Many of the techniques that we describe here are similar to those developed in Chapter 10. We need a couple of preliminary definitions. Definition 12.2.1. Suppose that m N. Define the index set 2 V := 1, ,m , m { ··· } 1 and set h = and pi = ih,fori Z. Define the following sets m 2 m := w : Vm C , := w : Z C . V { ! } V { ! } 12.2. FINITE DIFFERENCE APPROXIMATION 175 Elements of these sets – which are clearly linear vector spaces of functions – are called grid functions. We use the notation wi = w(i),fori Vm or i Z. 2 2 We define subspace of periodic grid functions m,p := w wi+n m = wi , for all i,n Z . V { 2 V| · 2 } Finally, we define the discrete Laplacian operator ∆ : via h V ! V wi 1 2wi + wi+1 ∆hwi := − − ,iZ. h2 2 1 Definition 12.2.2. Suppose m N, h = , pi = ih, f Cp(0, 1), c 0,and 2 m 2 ≥ u C2(0, 1) is a classical solution to the one dimensional periodic Helmholtz 2 p problem d 2u (x)+cu = f (x),xR. (12.2.1) −dx2 2 We call w a finite di↵erence approximation to (12.2.1) i↵ 2 Vm,p ∆hwi + cwi = fi ,iZ, (12.2.2) − 2 where fi := f (pi ), for all i Z. The global error e m,p is the grid function 2 2 V defined via ei = ui wi ,iZ, − 2 where ui := u(pi ), for all i Z. The local truncation error ⌧[u] m,p is 2 2 V the grid function ⌧i [u]:=fi + ∆hui cui ,iZ. − 2 One of our first concerns should be to determine whether our finite di↵er- ence approximation is well-defined. To make this determination, we first state some simple facts. Proposition 12.2.3. Let m N. There is a one-to-one correspondence be- 2 m tween the space of grid functions m and C . Furthermore, there is a one- V to-one correspondence between and the space of periodic grid functions Vm . Therefore, dim( ) = dim( )=m. Vm,p Vm Vm,p m Proof. The (canonical) correspondence between m and C has already been V established, albeit for the real case. Regarding the second correspondence, if w , then we can define its unique periodic extension w ext via 2 Vm 2 Vm,p ext wi+n m := wi , for all i Vm,n Z. · 2 2 176 CHAPTER 12. SPECTRAL METHODS Conversely, for any w , there is a unique restriction w res defined 2 Vm,p 2 Vm via w res := w , for all i V . i i 2 m Remark 12.2.4. As before, we will write m w C w m w m,p. 2 $ 2 V $ 2 V If we know that the components of the vector are real, we will write bf w Rm 2 instead. We use the same convention as before, denoting a grid function by a greek or roman character and its corresponding canonical vector representative by the boldface of the same character. 1 Theorem 12.2.5. Suppose m N, h = m , pi = ih, c 0, f Cp(0, 1),and 2 2 ≥ 2 u Cp (0, 1) is a classical solution to the one dimensional periodic Helmholtz 2 m m problem (12.2.1). Define A R ⇥ , called the sti↵ness matrix, via 2 2 10 1 − . ··· − . 2 12.. 3 − .. .. A = 6 0 . 107 . (12.2.3) 6 − 7 6 . 7 6 . 121 7 6 − − 7 6 1 0 127 6 − ··· − 7 4 5 m Define u m,p u R component-wise via ui = u(pi ), for all i Z, 2 V $ 2 m 2 and likewise, f m,p f R , via fi = f (pi ), for all i Z. Then 2 Vm $ 2 2 w m,p w R is a solution to the finite di↵erence problem (12.2.2) i↵ 2 V $ 2 it is a solution to the problem (A + ch2I)w = h2f . (12.2.4) m Likewise, the global and local truncation errors, e m,p e R and m 2 V $ 2 ⌧[u] m,p ⌧[u] R , respectively, satisfy 2 V $ 2 (A + ch2I)e = h2⌧[u]. (12.2.5) − Proof. The proof is an exercise. We have proven that (12.2.2) is equivalent to (12.2.4). As before, this equivalence is useful, because it is easy analyze the coefficient matrix and give sufficient and necessary conditions for the solvability of (12.2.4). 12.3. CONVERGENCE OF FINITE DIFFERENCE APPROX. 177 5 4 3 2 1 0 -1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 12.1: The eigenvalues of the sti↵ness matrix A are show (open circles) for m = 10. Observe that 0 λ 4. ` 12.3 Well-Definedness and Convergence of the Finite Di↵erence Approximation As in Chapter 10, we will analyze the sti↵ness matrix A, define an appropriate `2-norm and prove the convergence of our method in that norm. m m Theorem 12.3.1. Let m N and suppose A R ⇥ is the sti↵ness matrix 2 2 m defined in (12.2.3). Consider the set of vectors S = v ` C ` Vm , { 2 | 2 } where the components of v ` are [v ] =: v = e`(2⇡i)kh,kV , ` k `,k 2 m with i = p 1. Then − 1. S is an orthonormal set of vectors in the sense that H (v k , v j )2 = v ` v j = δk,j . 2. S is a set of eigenvectors of A, and the eigenvalue λ` corresponding to the eigenvector v ` is given by 2 λ` = 4 sin (`⇡h) . Since 0 λ 4 (see Figure 12.2), for all ` V , A is a symmetric ` 2 m positive semi-definite matrix. 178 CHAPTER 12. SPECTRAL METHODS 3. A is not invertible, and, in particular, ker(A)=span( 1 ). { } 1 4. There is a constant, C1 > 0, independent of h, such that, if 0 <h< 2 , 1 1 1 2 = = 2 C1h− . λ1 λm 1 4 sin (h⇡) − Proof. The proofs are left for exercises. The details are similar those from Chapter 10. Corollary 12.3.2. Suppose that c>0.Foreverym N, there is a unique m 2 solution w m,p w R to the finite di↵erence problem (12.2.2). 2 V $ 2 Proof. This follows because, when c>0, the coefficient matrix A + ch2I for problem (12.2.4) is symmetric positive definite.