Solving dirichlet and neumann problems with discontinuous coefficients on bounded simply and multiply connected regions

Abstract

Many problems in science and engineering require the solution of the Dirichlet problem and Neumann problem with discontinuous coefficients. In this thesis, a boundary integral equation method is developed for solving Laplace’s equation with Dirichlet condition and Neumann condition with discontinuous coefficients in both simply and multiply connected regions. The methods are based on a uniquely solvable boundary linear integral equation with the Neumann kernel. For numerical experiments, discretizing each integral equation leads to a system of linear equations. The system is then solved using the generalized minimum residual method (gmres) powered by the fast multipole method (FMM). After the boundary values of the solution of the Dirichlet problem and the Neumann problem with discontinuous coefficients in both simply and multiply connected regions are computed, the solution of the problem at the interior points are calculated by means of the Cauchy integral formula. The numerical examples presented have illustrated that the boundary integral equation methods developed yield high accuracy. Then a method by using these concepts is suggested for solving the mixed boundary value problem. The method is based on converting the mixed problem to a Riemann-Hilbert problem with discontinuous coefficients which is then reduced to two Dirichlet problems, one with discontinuous coefficients and one with unbounded coefficients.