On the sharpness of certain local estimates for 𝐻¹ projections into finite element spaces: influence of a re-entrant corner
Lars B. Wahlbin · Mathematics of Computation · 1984
In a plane polygonal domain with a reentrant corner, consider a homogeneous Dirichlet problem for Poisson’s equation − Δ u = f - \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 ⩽ 1 0 > h \leqslant 1 . Assuming that u has a singularity of the type | x − v M | β |x - {v_M}{|^\beta } at the vertex v M {v_M} of maximal angle π / β \pi /\beta , we show: (i) For any subdomain A and any s , the error measured in H − s ( A ) {H^{ - s}}(A) is not better than O ( h 2 β ) O({h^{2\beta }}) . (ii)On annular strips of points of distance of order d from v M {v_M} , the pointwise error is not better than