Direct Adjoint Method for Inverse Boundary Value Problems of the Laplace Equation

Abstract

We present a unified treatment of the Laplace equation in two-dimensional domain enclosed by a smooth curve. The Dirichlet data can be prescribed on some part of the boundary, while the Neumann data can be prescribed on some other part of the boundary. This problem is reformulated as a variational problem, and it is recast into primary and adjoint boundary value problems of the Laplace equation. A non-iterative numerical method of solution using the BEM is presented. A numerical example is demonstrated for the Cauchy problem, showing viability of the treatment.