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Unformatted text preview: online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 7.6 Positive Definite Matrices http://www.amazon.com/exec/obidos/ASIN/0898714540 573 It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] Discretize the problem by overlaying the square with a regular mesh containing n2 interior points at equally spaced intervals of length h as explained in Example 7.6.2 (p. 563). Let fij = f (xi , yj ), and define f to be the vector f = (f11 , f12 , . . . , f1n |f21 , f22 . . . , f2n | · · · |fn1 , fn2 , . . . , fnn )T . Show that the discretization of Poisson’s equation produces a system of linear equations of the form Lu = g − h2 f , where L is the discrete Laplacian and where u and g are as described in Example 7.6.2. 7.6.10. As defined in Exercise 5.8.15 (p. 380) and discussed in Exercise 7.8.11 (p. 597) the Kronecker product (sometimes called tensor product , or direct product ) of matrices Am×n and Bp×q is the mp × nq matrix ⎛ ⎞ a1n B a2n B ⎟ . ⎟. .⎠ . a11 B ⎜ a21 B A⊗B=⎜ . ⎝. . a12 B a22 B . . . ··· ··· .. . am1 B am2 B · · · amn B D E T H Verify that if In is the n × n identity matrix, and if ⎛ 2 ⎜ −1 ⎜ An = ⎜ ⎜ ⎝ −1 2 .. . R Y IG −1 .. . −1 .. . 2 −1 ⎞ ⎟ ⎟ ⎟ ⎟ −1 ⎠ 2 n×n P is the nth -order finite difference matrix of Example 1.4.1 (p. 19), then the discrete Laplacian is given by O C Ln2 ×n2 = (In ⊗ An ) + (An ⊗ In ). Thus we have an elegant matrix connection between the finite difference approximations of the one-dimensional and two-dimensional Laplacians. This formula leads to a simple alternate derivation of (7.6.8)—see Exercise 7.8.12 (p. 598). As you might guess, the discrete three-dimensional Laplacian is Ln3 ×n3 = (In ⊗ In ⊗ An ) + (In ⊗ An ⊗ In ) + (An ⊗ In ⊗ In ). Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html It is illegal to print, duplicate, or distribu...
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This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

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