A modified collocation Trefftz method for the inverse Cauchy problem of Laplace equation

Abstract

We consider an inverse problem for Laplace equation by recovering the boundary value on an inaccessible part of a circle from an overdetermined data on an accessible part of that circle. The available data are assumed to have a Fourier expansion, and thus the finite terms truncation plays a role of regularization to perturb the ill-posedness of this inverse problem into a well-posed one. Hence, we can apply a modified indirect Trefftz method to solve this problem and then a simple collocation technique is used to determine the unknown coefficients, which is named a modified collocation Trefftz method. The results may be useful to detect the corrosion inside a pipe through the measurements on a partial boundary. Numerical examples show the effectiveness of the new method in providing an excellent estimate of unknown data from the given data under noise.