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 mathcal{O}(Nlog (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.
Highlights
1 Introduction Laplace’s equation is of particular importance in applied mathematics because it is applicable to a wide range of different physical and mathematical phenomena, including electromagnetism, fluid and solid mechanics, conductivity
For discretization of the integral equation (2.1) with the given parametrization of Γ, a Nyström method can be applied by using the mid-point quadrature rule
The discretized form of the direct problem (2.2) is as follows: 1 A(p)u := I + D + Sp u = f, where p = diag(p), which can be solved by using a Gaussian elimination method (GE)
Summary
Laplace’s equation is of particular importance in applied mathematics because it is applicable to a wide range of different physical and mathematical phenomena, including electromagnetism, fluid and solid mechanics, conductivity. Jin solved the Robin inverse problem by using conjugate gradient (CG) methods [15], analyzing the convergence of different regular terms. The resulting integral equation can be discretized by using a boundary element method or numerical quadratures [14, 15, 22, 23].
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