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In numerical mathematics, the relaxation method is a method for obtaining numerical approximations to the solutions of systems of equations, including certain types of elliptic partial differential equations, in particular Laplace's equation and its generalization, Poisson's equation. The function is assumed to be given on the boundary of a shape, and has to be computed on its interior.## Sketch

When φ is a smooth real-valued function on the real numbers, its second derivative can be approximated by:
## Convergence and acceleration

## See also

This relaxation method should not be confused with the unrelated relaxation technique in mathematical optimization.

- $frac\{d^2\}\{\{dx\}^2\}varphi(x)\; =\; h^\{-2\}left(varphi(x\{-\}h)-2varphi(x)+varphi(x\{+\}h)right),+,mathcal\{O\}(h^2),.$

- $varphi(x,\; y)\; =\; tfrac\{1\}\{4\}left(varphi(x\{+\}h,y)+varphi(x,y\{+\}h)+varphi(x\{-\}h,y)+varphi(x,y\{-\}h)$

- $\{nabla\}^2\; varphi\; =\; f,$

- $varphi^*(x,\; y)\; =\; tfrac\{1\}\{4\}left(varphi(x\{+\}h,y)+varphi(x,y\{+\}h)+varphi(x\{-\}h,y)+varphi(x,y\{-\}h)$

The method, sketched here for two dimensions, is readily generalized to other numbers of dimensions.

While the method always converges, it does so very slowly. To speed up the computation, one can first compute an approximation on a coarser grid – usually the double spacing 2h – and use that solution with interpolated values for the other grid points as the initial assignment. This can then also be done recursively for the coarser computation.

- The Jacobi method is a simple relaxation method.
- The Gauss–Seidel method is an improvement upon the Jacobi method.
- Successive over-relaxation can be applied to either of the Jacobi and Gauss–Seidel methods to speed convergence.

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Last updated on Friday August 08, 2008 at 16:54:06 PDT (GMT -0700)

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Last updated on Friday August 08, 2008 at 16:54:06 PDT (GMT -0700)

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