Convergence analysis of a coupled method for Helmholtz equation

Abstract

Based on a hypersingular integral method, we analyze the use of a coupled Nyström and finite-element procedure for approximating the Helmholtz equation in an exterior domain. The coupling technique uses a smooth artificial boundary and hypersingular integral equation and finite-element computations. We prove convergence of the method. One highlight of this analysis is that we prove Sobolev space error estimates for the Nyström scheme (which is usually analyzed in Hölder spaces). Finally, we discuss some details of implementing the coupled method and prove the discrete Dirichlet–Neumann alternation iterative method solving the problem more rapidly.

Keywords:

• coupled method; Nyström method; hypersingular integral equation; Helmholtz equation;

AMS Subject Classifications:

• 65N30
• 65F30
• 42C40

Acknowledgments

We thank professor Wei Lin for his helpful comments and insightful remarks. This work is supported partly by NSF of China (No. 10771218), the Hunan Provincial Natural Science Foundation of China (No. 11JJ6003), the Scientific Research Fund of Hunan Provincial Education Department (No. 10B040) and the Postdoctoral Science Foundation of Central South University.