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arXiv · 1505.07157

Fast, adaptive, high order accurate discretization of the Lippmann-Schwinger equation in two dimension

Sivaram Ambikasaran·Carlos Borges·Lise-Marie Imbert-Gerard·Leslie Greengard
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Abstract

We present a fast direct solver for two dimensional scattering problems, where an incident wave impinges on a penetrable medium with compact support. We represent the scattered field using a volume potential whose kernel is the outgoing Green's function for the exterior domain. Inserting this representation into the governing partial differential equation, we obtain an integral equation of the Lippmann-Schwinger type. The principal contribution here is the development of an automatically adaptive, high-order accurate discretization based on a quad tree data structure which provides rapid access to arbitrary elements of the discretized system matrix. This permits the straightforward application of state-of-the-art algorithms for constructing compressed versions of the solution operator. These solvers typically require $O(N^{3/2})$ work, where $N$ denotes the number of degrees of freedom. We demonstrate the performance of the method for a variety of problems in both the low and high frequency regimes.

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Sivaram Ambikasaran, Carlos Borges, Lise-Marie Imbert-Gerard, Leslie Greengard. 2015-05-26. Fast, adaptive, high order accurate discretization of the Lippmann-Schwinger equation in two dimension. https://arxiv.org/abs/1505.07157

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