A Highly Accurate MCTM for Inverse Cauchy Problems of Laplace Equation in Arbitrary Plane Domains

Abstract

We consider the inverse Cauchy problems for Laplace equation in simply and doubly connected plane domains by recoverning the unknown bound- ary value on an inaccessible part of a noncircular contour from overspecified data. A modified Trefftz method is used directly to solve those problems with a simple collocation technique to determine unknown coefficients, which is named a mod- ified collocation Trefftz method (MCTM). Because the condition number is small for the MCTM, we can apply it to numerically solve the inverse Cauchy problems without needing of an extra regularization, as that used in the solutions of direct problems for Laplace equation. So, the computational cost of MCTM is very sav- ing. Numerical examples show the effectiveness of the new method in providing an excellent estimate of unknown boundary data, even by subjecting the given data to a large noise. Keyword: Inverse Cauchy problem, Modified Trefftz method, Laplace equation, Modified collocation Trefftz method (MCTM)