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On the sharpness of certain local estimates for H$ ^1$ projections into finite element spaces: influence of a re-entrant corner

Author: Lars B. Wahlbin
Journal: Math. Comp. 42 (1984), 1-8
MSC: Primary 65N30; Secondary 65N15
MathSciNet review: 725981
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Abstract: In a plane polygonal domain with a reentrant corner, consider a homogeneous Dirichlet problem for Poisson's equation $ - \Delta u = f$ with f smooth and the corresponding Galerkin finite element solutions in a family of piecewise polynomial spaces based on quasi-uniform (uniformly regular) triangulations with the diameter of each element comparable to h, $ 0 < h \leqslant 1$. Assuming that u has a singularity of the type $ \vert x - {v_M}{\vert^\beta }$ at the vertex $ {v_M}$ of maximal angle $ \pi /\beta $, we show: (i) For any subdomain A and any s, the error measured in $ {H^{ - s}}(A)$ is not better than $ O({h^{2\beta }})$. (ii)On annular strips of points of distance of order d from $ {v_M}$, the pointwise error is not better than $ O({h^{2\beta }}{d^{ - \beta }})$.

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Article copyright: © Copyright 1984 American Mathematical Society

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