Error Analysis of Trefftz Methods for Laplace's Equations and Its Applications

Abstract

For Laplace’s equation and other homogeneous elliptic equations, when the particular and fundamental solutions can be found, we may choose their linear combination as the admissible functions, and obtain the expansion coefficients by satisfying the boundary conditions only. This is known as the Trefftz method (TM) (or boundary approximation methods). Since the TM is a meshless method, it has drawn great attention of researchers in recent years, and Inter. Workshops of TM and MFS (i.e., the method of fundamental solutions). A number of efficient algorithms, such the collocation algorithms, Lagrange multiplier methods, etc., have been developed in computation. However, there still exists a gap of convergence and errors between computation and theory. In this paper, convergence analysis and error estimates are explored for Laplace’s equations with the solution u∈Hk(k > 1 2), to achieve polynomial convergence rates. Such a basic theory is important for TM and MFS and their further developments. Numerical experiments are provided to support the analysis and to display the significance of its applications.