Semi-Discrete Galerkin Approximations for the Single-Layer Equation on Lipschitz Curves
Ian H. Sloan, KENDALL E. ATKINSON · Journal of Integral Equations and Applications · 1997
We study a semi-discrete Galerkin method for solving the singlelayer equation Vu = f with an approximating subspace of piecewise constant functions. Error bounds in Sobolev norms k\\Deltak s with \\Gamma1 s ! 1 2 are proven and are of the same order as for the original Galerkin method. The distinctive features of the present work are that we handle irregular meshes and do not rely on Fourier methods. The main assumptions are that the quadrature rule used to approximate the inner product is a composite rule and that the underlying quadrature rule that is mapped to each subinterval has a sufficiently small Peano constant. 1 Introduction The single-layer equation Vu = f (1) is an important boundary integral equation. It arises, for example, in the solution of the Laplace equation on interior or exterior domains. If\\Omega is a School of Mathematics, University of New South Wales, Sydney 2052, Australia. Supported by the Australian Research Council. y Dept of Mathematics, Univ. of ...