Adaptive error estimation of the Trefftz method for solving the Cauchy problem

Abstract

In this paper, the Laplace problem with overspecified boundary conditions is investigated by using the Trefftz method. The main difficulty will appear an obvious divergent result when the boundary condition on an overspecified boundary contaminates artificial errors. The occurring mechanism of the unreasonable result originates from an ill-posed influence matrix. The accompanied ill-posed problem is remedied by using the Tikhonov regularization technique and the linear regularization method respectively, to reconstruct the influence matrix. The optimal parameters of the Tikhonov technique and linear regularization method are determined by adopting the adaptive error estimation technique. The numerical evidence of the Trefftz method is given to verify the accuracy of the solutions after comparison with the results of analytical solution and to demonstrate the validity and instructions of the proposed adaptive error estimation technique. The comparison of the Tikhonov regularization technique and the linear regularization method was also discussed in the example.

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

“…The main idea of this method is to extend the numerical solution in terms of T-complete functions that satisfy the governing equation. The TM is less popular than other numerical methods such as Finite Diff erence Method (FDM), Finite Element Method (FEM), Boundary Element Method (BEM), etc., because the system of linear equations resulting from the Treff tz method is an ill-posed problem, even for a well-posed boundary value problem, also for multi-connected domains, the conventional TM fails [15,16].…”