Graded mesh approximation in weighted Sobolev spaces and elliptic equations in 2D

Adler, James; Nistor, Victor

doi:10.1090/S0025-5718-2015-02934-2

Graded mesh approximation in weighted Sobolev spaces and elliptic equations in 2D
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by James H. Adler and Victor Nistor;

Math. Comp. 84 (2015), 2191-2220

DOI: https://doi.org/10.1090/S0025-5718-2015-02934-2

Published electronically: February 26, 2015

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Abstract:

We study the approximation properties of some general finite-element spaces constructed using improved graded meshes. In our results, either the approximating function or the function to be approximated (or both) are in a weighted Sobolev space. We consider also the $L^p$-version of these spaces. The finite-element spaces that we define are obtained from conformally invariant families of finite elements (no affine invariance is used), stressing the use of elements that lead to higher regularity finite-element spaces. We prove that for a suitable grading of the meshes, one obtains the usual optimal approximation results. We provide a construction of these spaces that does not lead to long, “skinny” triangles. Our results are then used to obtain $L^2$-error estimates and $h^m$-quasi-optimal rates of convergence for the FEM approximation of solutions of strongly elliptic interface/boundary value problems.

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