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Numerical solution of partial differential equations

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Partial differential equations are considerably more complex to solve than ordinary differential equations, and there are a substantial number of special techniques developed to handle them. All involve some sort of "grid" that subdivides space into small packets. The different ways in which the packets are used gives the different methods.


Multivariate calculus, basic numerical analysis.

Example 1: Galerkin

The most common (especially from a mathematical point of view) is the Galerkin method. The basic idea is that we set up the weak formulation of the partial differential equation. For a (linear) operator equation Ax=b, where X\to X^* is an operator (X^* is the dual space to X), we take a basis of a finite dimensional subspace \phi_1,\,\phi_2,\,\ldots,\,\phi_N, with X_N={\mathrm{span}}\{\phi_1,\,\phi_2,\,\ldots,\,\phi_N\}.

Then we choose x=\sum_{i=1}^N x_i\phi_i\in X_N where

   \langle \phi_i,\,Ax\rangle = \langle \phi_i,\,b\rangle

for i=1,\,2,\,\ldots,\,N. If A is an elliptic partial differential operator, then the linear system generated by the Galerkin method is positive definite, and the linear system can be solved.

This method is also related to the Rayleigh-Ritz method, and the more general Petrov-Galerkin method.

The Galerkin method is commonly known as the finite element method (FEM), at least under common choices of the basis functions.

Example 2: Finite difference

General discussion