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

We study polynomial approximations of vertex singularities of the type rλ log rβ on three-dimensional surfaces. The analysis focuses on the case when λ > -1/2 . 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. Furthermore, such strong vertex singularities may appear in solutions to boundary integral formulations of time-harmonic problems of electromagnetism in Lipschitz domains. Thus, the approximation results for such singularities are needed for the error analysis of boundary element methods in three dimensions. Moreover, to our knowledge, the approximation of strong singularities (-1/2 <λ = 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. © 2009 Rocky Mountain Mathematics Consortium.