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

TL;DR

This paper introduces a fast, adaptive, high-order accurate direct solver for 2D scattering problems based on the Lippmann-Schwinger equation, utilizing a quad tree structure for efficient discretization and matrix compression.

Contribution

The paper develops an automatically adaptive, high-order discretization method that enables rapid construction of compressed solution operators for 2D scattering problems.

Findings

Achieves high accuracy in both low and high frequency regimes.

Reduces computational complexity to approximately O(N^{3/2}) for large problems.

Demonstrates effectiveness through various numerical experiments.

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.