Numerical solution of partial differential equations

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Numerical solution of partial differential equations

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[QUICK DESCRIPTION]
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.

[PREREQUISITES]
Multivariate calculus, basic numerical analysis.

[EXAMPLE|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 $A:X\to X^*$ 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 
[maths]
  \langle \phi_i,\,Ax\rangle = \langle \phi_i,\,b\rangle
[/maths]
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|Finite difference]

[GENERAL DISCUSSION]

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Numerical solution of partial differential equations

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