A Regularized Trefftz Method for Solving an Inverse Problem for the Laplace Equation

Abstract

The present paper proposes a new regularised Trefftz method to recover the boundary value on a non-accessible boundary of an annulus from overdetermined data on the accessible boundary of that annulus. Considering that the available data have a Fourier expansion, we consider the Tikhonov damping factor in constructing our regularized scheme, which satisfies the boundary value problem. At the same time, the convergence estimates for the regularised solution will be established under an assumption for the exact solution. The finite term truncation of the series expansion allows us to match the boundary condition as accurately as desired. By the collocation method, we derive a system of linear equations that can be uniquely solved to obtain the coefficients and preset the regularisation parameter and the damping factor to ensure adequate stability. The numerical efficiency of the proposed method is investigated with a high truncation number in comparison with the modified collocation Trefftz method.