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Keywords:
finite element method; error estimates; maximum angle condition
finite element method; error estimates; maximum angle condition
Summary:
In this short note we provide an optimal analysis of finite element convergence on meshes containing a so-called band of caps. These structures consist of a zig-zag arrangement of `degenerating' triangles which violate the maximum angle condition. A necessary condition on the geometry of such a structure for various $H^1$-convergence rates was previously given by Kučera. Here we prove that the condition is also sufficient, providing an optimal analysis of this special case of meshes. In the special case of optimal $O(h)$-convergence of finite elements, the analysis states that such optimal convergence is possible if and only if the height of the band of caps is at least $Ch^2$ for some constant $C$. Numerical experiments confirm this result.
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