The numerical solution of forward and inverse Robin problems for Laplace’s equation

Abstract

Abstract The Robin boundary value problem for Laplace’s equation in the elliptic region (which is a forward problem) and its related inverse problem can be used to reconstruct Robin coefficients from measurements on a partial boundary (inverse problem). We present a numerical solution of the forward problem that uses a boundary integral equation method, and we propose a fast solver based on one that reduces the computational complexity to O(Nlog(N)) $\mathcal{O}(N\log (N))$, where N is the size of the data. We compute the solution of the inverse problem using a preconditioned Krylov subspace method where the preconditioner is based on a block matrix decomposition. The structure of the matrix is then exploited to solve the direct problem. Numerical examples are presented to illustrate the effectiveness of the proposed approach.

Keywords

• Robin inverse problem
• Laplace equation
• Boundary integral equations
• Preconditioned Krylov subspace method