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

Sivaram Ambikasaran, Carlos Borges, Lise Marie Imbert-Gerard, Leslie Greengard

Research output: Contribution to journalArticle

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 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 low and high frequency regimes.

Original languageEnglish (US)
Pages (from-to)A1770-A1787
JournalSIAM Journal on Scientific Computing
Volume38
Issue number3
DOIs
StatePublished - 2016

Fingerprint

Quadtree
Exterior Domain
Compact Support
Scattering Problems
Tree Structure
Green's function
Partial differential equations
Integral equations
Data structures
Low Frequency
Mathematical operators
Data Structures
Integral Equations
Two Dimensions
Partial differential equation
Discretization
Degree of freedom
Scattering
Higher Order
kernel

Keywords

  • Acoustic scattering
  • Adaptivity
  • Electromagnetic scattering
  • Fast direct solver
  • High-order accuracy
  • Integral equation
  • Lippmann-schwinger equation
  • Penetrable media

ASJC Scopus subject areas

  • Applied Mathematics
  • Computational Mathematics

Cite this

Fast, adaptive, high-order accurate discretization of the Lippmann-Schwinger equation in two dimensions. / Ambikasaran, Sivaram; Borges, Carlos; Imbert-Gerard, Lise Marie; Greengard, Leslie.

In: SIAM Journal on Scientific Computing, Vol. 38, No. 3, 2016, p. A1770-A1787.

Research output: Contribution to journalArticle

Ambikasaran, Sivaram ; Borges, Carlos ; Imbert-Gerard, Lise Marie ; Greengard, Leslie. / Fast, adaptive, high-order accurate discretization of the Lippmann-Schwinger equation in two dimensions. In: SIAM Journal on Scientific Computing. 2016 ; Vol. 38, No. 3. pp. A1770-A1787.
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