The numerical solution of partial differential equations ( PDEs ) has been dominated by either finite difference methods ( FDM ), finite element methods ( FEM ), and finite volume methods ( FVM ). These methods can be derived from the assumptions of the local interpolation schemes. These methods require a mesh to support the localized approximations; the construction of a mesh in three or more dimensions is a non-trivial problem. Typically with these methods only the function is continuous across meshes, but not its partial derivatives.

In practice, only low order approximations are used because of the notorious polynomial snaking problem. While higher order schemes are necessary for more accurate approximations of the spatial derivatives, they are not sufficient without monotonicity constraints. Because of the low order schemes typically employed, the spatial truncation errors can only be controlled by using progressively smaller meshes. The mesh spacing, h, must be sufficiently fine to capture the functions partial derivative behavior and to avoid unnecessarily large amounts of numerical artifacts contaminating the solution. Spectral methods while offering very high order spatial schemes typically depend upon tensor product grids in higher dimensions . . . . .

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