The scheme of finite element method with multiplicative separation of singularities for boundary value problems in domains with angles (Q1909824)

This paper is concerned with the finite element solution of boundary value problems in domains with angles. The author considers the homogeneous Dirichlet problem for the Poisson equation in a two-dimensional polygonal domain. The solution of the problem has singularities at the vertices of the domain depending on the values of the corresponding inner angles. The author proposes a finite element scheme on a regular triangulation of the domain based on a new multiplicative representation of the singularity of the exact solution of the problem. The scheme has an error of order \(O(h \ln^\theta (1/h))\) in the grid norm \(W^1_2\) where \(h\) is the characteristic size of a cell, \(\theta = 0.5\) if there exist angles equaling \(2\pi\), and \(\theta = 0\) otherwise. The results are illustrated by a numerical example.