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arXiv · 1501.03003

On optimal $L^2$- and surface flux convergence in FEM (extended version)

T. Horger·J. M. Melenk·B. Wohlmuth
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Abstract

We show that optimal $L^2$-convergence in the finite element method on quasi-uniform meshes can be achieved if, for some $s_0 > 1/2$, the boundary value problem has the mapping property $H^{-1+s} \rightarrow H^{1+s}$ for $s \in [0,s_0]$. The lack of full elliptic regularity in the dual problem has to be compensated by additional regularity of the exact solution. Furthermore, we analyze for a Dirichlet problem the approximation of the normal derivative on the boundary without convexity assumption on the domain. We show that (up to logarithmic factors) the optimal rate is obtained.

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T. Horger, J. M. Melenk, B. Wohlmuth. 2015-01-13. On optimal $L^2$- and surface flux convergence in FEM (extended version). https://doi.org/10.1007/s00791-015-0237-z

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