WELL-POSED BOUNDARY INTEGRAL EQUATION FORMULATIONS AND NYSTROM DISCRETIZATIONS FOR THE SOLUTION OF HELMHOLTZ TRANSMISSION PROBLEMS IN TWO-DIMENSIONAL LIPSCHITZ DOMAINS

摘要

We present a comparison among the performance of solvers based on Nystrom discretizations of several well-posed boundary integral equation formulations of Helmholtz transmission problems in two-dimensional Lipschitz domains. Specifically, we focus on the following four classes of boundary integral formulations of Helmholtz transmission problems: (1) the classical first kind integral equations for transmission problems [1 3], (2) the classical second kind integral equations for transmission problems [2 5], (3) the single integral equation formulations [2 1], and (4) certain direct counterparts of recently introduced generalized combined source integral equations [4, 5]. The former two formulations were the only formulations whose wellposedness in Lipschitz domains was rigorously established [1 3, 3 6]. We establish the well-posedness of the latter two formulations in appropriate functional spaces of boundary traces of solutions of transmission Helmholtz problems in Lipschitz domains. We give ample numerical evidence that Nystrom solvers based on formulations (3) and (4) are computationally more advantageous than solvers based on the classical formulations (1) and (2), especially in the case of high-contrast transmission problems at high frequencies.