## On the sharpness of certain local estimates for $\text {\textit {\r {H}}}^1$ projections into finite element spaces: influence of a re-entrant corner

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- by Lars B. Wahlbin PDF
- Math. Comp.
**42**(1984), 1-8 Request permission

## 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 $|x - {v_M}{|^\beta }$ 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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## Additional Information

- © Copyright 1984 American Mathematical Society
- Journal: Math. Comp.
**42**(1984), 1-8 - MSC: Primary 65N30; Secondary 65N15
- DOI: https://doi.org/10.1090/S0025-5718-1984-0725981-7
- MathSciNet review: 725981