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Is there a technique where a square grid is used to as a mesh for time dependent partial differential equations (PDEs) but which the points are permuted in such a way as to minimize error?

e.g., instead of using the uniform coordinates (i,j,k) we would use (x[i,j,k], y[i,j,k], z[i,j,k]), where x, y, and z are lookup tables. As we evolve the PDE through time the points (x,y,z) are adjusted to minimize the total error.

This way we can have a constant number of points (say 10*10*10) but the grid points are adapted to maximize the approximations.

sc. Surely there are some meshes that are better than others in approximating the solution of a PDE. Changes are as we evolve the PDE through time the optimal mesh itself will evolve in a similar manner. In fact it at some points the direction to move a grid point may not be distinct in which case we could insert a new grid point.

Anyway, just curious about this concept. AFAIK finite element method (FEM) tend to be static in that they are based on the geometry. But maybe they do adapt the meshes based on the error produced as the PDE is evolved or maybe doing so is not very effective in the first place?

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  • $\begingroup$ Sorry for the ignorance: What are PDE and FEM? $\endgroup$ – Hsien-Chih Chang 張顯之 Mar 14 '11 at 15:56
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    $\begingroup$ @Hsien-Chih Chang: Partial Differential Equation. Finite Element Method. $\endgroup$ – Dave Clarke Mar 14 '11 at 15:59
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What follows is not quite an answer but some terminology to start the search ...

While the basic finite element methods are indeed static, many of the mature implementations are adaptive which means that the mesh adapts in order to minimize the error. The kind of adaptation you describe, where the mesh is evolving in tandem with the solution, is called moving mesh or r-adaptive finite elements.

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