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My current work requires using (Adaptive Mesh Refinement) AMR to resolve multi scale physics. I have a general question whether finite element is better than finite difference in this aspect or not.

I know that finite element can solve equations on unstructured grid. Since AMR is nonuniform and can be structured or unstructured, does finite element do better than finite difference when AMR is applied?

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    $\begingroup$ Doing AMR with finite differences is not a good idea in general. While hanging nodes are a problem for FEMs, they are a bigger problem for FDM. Also, under AMR, you may have weird meshes where there may not be any stable FDMs on that mesh. $\endgroup$ Dec 4, 2021 at 21:19
  • $\begingroup$ In how many dimensions? $\endgroup$
    – Bill Barth
    Dec 5, 2021 at 1:52
  • $\begingroup$ @BillBarth 3 dimension. Will the answer be different for lower dimension? $\endgroup$
    – jchan192
    Dec 5, 2021 at 3:39
  • $\begingroup$ In 1D for many PDEs, many finite element methods have equivalent finite different methods (probably congruent is a better word), hence if a AMR-FEM is stable in 1D, then it is likely that there is a stable AMR-FDM too. The claim breaks in 2 or more dimensions. But Bill Barth may be thinking of something else. $\endgroup$ Dec 5, 2021 at 3:45
  • $\begingroup$ @jchan192, I think 1D would be a lot easier in both general and in an FDM with both having about the same difficulty. May depend on your refinement strategy. $\endgroup$
    – Bill Barth
    Dec 5, 2021 at 5:25

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Finite element methods are generally easier to deal with when using adaptive mesh refinement because higher order finite difference methods have stencils that extend for several mesh sizes away from a point, and one has to deal with all of the possible hanging nodes one might encounter in this large neighborhood even if the mesh is rectangular and is derived from a uniform mesh. The finite element method does not have this drawback: What happens on a cell is restricted to this cell and its immediate neighbors.

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