Author’s summary: 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, \(0<h\leq 1\). 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 \(O(h^{2\beta}d^{-\beta})\).