Well-posed boundary integral equation formulations and Nystr'om discretizations for the solution of Helmholtz transmission problems in two-dimensional Lipschitz domains

摘要

We present a comparison between the performance of solvers based on Nystr\"omdiscretizations of several well-posed boundary integral equation formulationsof Helmholtz transmission problems in two-dimensional Lipschitz domains.Specifically, we focus on the following four classes of boundary integralformulations of Helmholtz transmission problems (1) the classical first kindintegral equations for transmission problems, (2) the classical second kindintegral equations for transmission problems, (3) the {\em single} integralequation formulations, and (4) certain direct counterparts of recentlyintroduced Generalized Combined Source Integral Equations. The former twoformulations were the only formulations whose well-posedness in Lipschitzdomains was rigorously established. We establish the well-posedness of thelatter two formulations in appropriate functional spaces of boundary traces ofsolutions of transmission Helmholtz problems in Lipschitz domains. We giveample numerical evidence that Nystr\"om solvers based on formulations (3) and(4) are computationally more advantageous than solvers based on the classicalformulations (1) and (2), especially in the case of high-contrast transmissionproblems at high frequencies.