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The finite elements (FEM) method is particularly useful when a robust approximation is sought to solve partial differential equations on an inhomogeneous mesh. Solid mathematical foundations and a great deal of generality allow for different implementations that are only sketched in the first section: an example given for the advection-diffusion equation, for example, shows how the Crank-Nicholson method described in the previous section is only a particular case of a finite elements scheme.

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