A simple Trefftz method for solving the Cauchy problems of three-dimensional Helmholtz equation

In this paper we numerically solve the direct as well as the inverse Cauchy problems of the three-dimensional Helmholtz equation in an arbitrary bounded region by using a multiple/scale/direction Trefftz method (MSDTM), which is a simple extension of the two-dimensional Trefftz method to a Trefftz method for the three-dimensional Helmholtz equation. In the MSDTM the directions are uniformly distributed unit vectors in the plane (y,z)(y,z), and the scales are determined by the boundary collocation points. The regularization of the MSDTM roots in a small number of unknown coefficients, a simple three-dimensional Trefftz basis and a simple multiple-scale post-conditioner, which are very easy to be numerically implemented. The three-dimensional numerical examples of direct and the inverse Cauchy problems in irregular domains guarantee the efficiency of the MSDTM. Although under a large noise, the solutions of inverse Cauchy problems are quite accurate.