Penalty combinations of the Ritz-Galerkin and finite difference methods for singularity problems (Q1362344)

It is well known that incorporation of singular functions in the basis of the approximating subspace may greatly improve the accuracy of the finite element method [see results and references in the reviewer's book: Optimization in solving elliptic problems (1996; Zbl 0852.65087)]. In this paper, special attention is paid to a similar combination of difference methods (in \(S_1\)) and projective methods (in \(S_2\)), where \(\overline S_1\) and \(\overline S_2\) form a partition of the original region \(\overline S\) and \(\overline S_2\) contains a singular point. The coupling is done via a special penalty method on the cutting line \(\partial S_1\cap\partial S_2\). Estimates of type \(O(h^{3/2})\) are proved for a grid norm. Numerical experiments for a model problem are presented but the accuracy of approximations is not very high.