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A Fast Solver for Boundary Integral Equations of the Modified Helmholtz Equation

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Abstract

The main purpose of this paper is to develop a fast fully discrete Fourier–Galerkin method for solving the boundary integral equations reformulated from the modified Helmholtz equation with boundary conditions. We consider both the nonlinear and the Robin boundary conditions. To tackle the difficulties caused by the two types of boundary conditions, we provide an improved version of the Galerkin method based on the Fourier basis. By employing a matrix compression strategy and efficient numerical quadrature schemes for oscillatory integrals, we obtain fully discrete nonlinear or linear system. Finally, we use the multilevel augmentation method to solve the resulting systems. We point out that the proposed method enjoys an optimal convergence order and a nearly linear computational complexity. The theoretical estimates are confirmed by the performance of this method on several numerical examples.

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Acknowledgments

This research is partially supported by Guangdong Provincial Government of China through the “Computational Science Innovative Research Team” program, by the Natural Science Foundation of China under Grants 11401207, 11301208, 11271370, by Hunan Provincial Natural Science Foundation of China under Grant 2015JJ6069.

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Authors and Affiliations

  1. School of Mathematics, Jilin University, Changchun, 130012, People’s Republic of China

    Rui Wang

  2. College of Mathematics and Computer Science, Hunan Normal University, Changsha, 410081, Hunan, People’s Republic of China

    Xiangling Chen

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  1. Rui Wang
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  2. Xiangling Chen
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Correspondence to Xiangling Chen.

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Wang, R., Chen, X. A Fast Solver for Boundary Integral Equations of the Modified Helmholtz Equation. J Sci Comput 65, 553–575 (2015). https://doi.org/10.1007/s10915-014-9975-1

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  • DOI: https://doi.org/10.1007/s10915-014-9975-1

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