Fast, adaptive, high order accurate discretization of the Lippmann-Schwinger equation in two dimensions
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Author
Ambikasaran, Sivaram
Borges, Carlos
Imbert-Gerard, Lise-Marie
Greengard, Leslie
Date
2015-05-26Metadata
Show full item recordAbstract
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 eld using a volume potential
whose kernel is the outgoing Green's function for the exterior domain. Inserting this representation into the governing
partial di erential 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(N3=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.