Topic 15.2: Elliptic Partial-Differential Equations (HOWTO)

Contents Previous Chapter Start of Chapter Previous Topic Introduction Notes Theory HOWTO Examples Engineering Error Questions Matlab Maple No Next Topic Next Chapter

Problem

Approximate the solution of a Poisson equation

uxx(x, y) + uxx(x, y) = g

on a rectangular region R = [xa, xb] × [ya, yb] with Dirichlet boundary conditions given by a function ∂u(x, y).

Assumptions

The function ∂u(x, y) must be piecewise continuous. We will also assume that xb − xa = hn and yb − ya = hm for some integer values of n and m.

Tools

We will use the centred divided-difference formula for the partial derivatives and linear algebra.

Process

Divide the interval [xa, xb] into n sub-intervals by setting xi = xa + ih for i = 0, 1, 2, ..., n and yi = ya + jh for j = 0, 1, 2, ..., m. Let ui, j represent the approximation of the solution u(xi, yj).

At each interior point (xi, yj) where i = 1, 2, ..., n − 1 and j = 1, 2, ..., m − 1, evaluate the finite-difference equation:

ui + 1, j + ui − 1, j + ui, j + 1 + ui, j − 1 − 4 ui, j = gh2

This defines a system of (n − 1)(m − 1) linear equations and (n − 1)(m − 1) unknowns. We can now solve these for the approximations uij.

Copyright ©2005 by Douglas Wilhelm Harder. All rights reserved.