A note on the polynomial approximation of vertex singularities in the boundary element method in three dimensions

Abstract

We study polynomial approximations of vertex singularities of the type $r^\lambda |\log r|^\beta$ on three-dimensional surfaces. The analysis focuses on the case when $\lambda > -\frac 12$. This assumption is a minimum requirement to guarantee that the above singular function is in the energy space for boundary integral equations with hypersingular operators. Thus, the approximation results for such singularities are needed for the error analysis of boundary element methods on piecewise smooth surfaces. Moreover, to our knowledge, the approximation of strong singularities ($-\frac 12 < \lambda \le 0$) by high-order polynomials is missing in the existing literature. In this note we prove an estimate for the error of polynomial approximation of the above vertex singularities on quasi-uniform meshes discretising a polyhedral surface. The estimate gives an upper bound for the error in terms of the mesh size $h$ and the polynomial degree $p$.