An analytical method for linear elliptic PDEs and its numerical implementation

Abstract

A new numerical method for solving linear elliptic boundary value problems with constant coefficients in a polygonal domain is introduced. This method produces a generalized Dirichlet–Neumann map: given the derivative of the solution along a direction of an arbitrary angle to the boundary, the derivative of the solution perpendicular to this direction is computed without solving on the interior of the domain. If desired, the solution on the interior can then be computed via an integral representation. The key to the method is a “global condition” which couples known and unknown components of the derivative on the boundary and which is valid for all values of a complex parameter k. This condition has been solved recently analytically for several equations on simple domains. In this paper, first the previous analytical result is strengthened, and then a numerical method is introduced for solving the global condition for the Laplace equation on an arbitrary bounded convex polygon. Numerical results demonstrate the applicability and convergence of the method; however, a rigorous proof of convergence remains open. Extensions to other problems are also discussed.