Numerical Instability of the Direct Trefftz Method for Laplace Problems in a 2D Finite Domain

Abstract

abstract:This paper proposes a direct Trefftz boundary-type method to solve the Laplace problems in a two-dimensional finite domain. It is found that the ill-posed nature, i.e., the numerical instability, exists in the solver due to the regular formulation of the Trefftz method The Riemann-Lebesgue lemma is adopted to explain the inherent ill-posed nature. In order to deal with the numerical instability, the Tikhonov’s regularization method and the L-curve concept are suggested to overcome such a difficulty. In particular, how to choose an appropriate set of basis functions when the origin is placed inside or outside the finite domain is critically evaluated along with this scope. Based on the argument, it is further explained that for a multiply connected domain of genus 1, to place the origin inside the hole is the only selection in the numerical sense. Several numerical examples are demonstrated to show the validity of the proposed approach.