Nystrom discretizations of boundary integral equations for the solution of 2D elastic scattering problems

TL;DR

This paper develops high-order Nystrom discretization methods for boundary integral equations solving 2D elastic scattering problems, demonstrating their accuracy and efficiency for smooth and Lipschitz boundaries, and integrating them with Convolution Quadrature for time-domain solutions.

Contribution

It introduces novel high-order Nystrom discretization strategies for elastic scattering boundary integral equations, including approaches for non-standard operators and their integration with time-domain methods.

Findings

Achieves high-order accuracy for smooth and Lipschitz boundaries.

Provides extensive numerical comparisons of solver performance at high frequencies.

Demonstrates seamless integration of discretizations with Convolution Quadrature for time-domain problems.

Abstract

We present three high-order Nystrom discretization strategies of various boundary integral equation formulations of the impenetrable time-harmonic Navier equations in two dimensions. One class of such formulations is based on the four classical Boundary Integral Operators (BIOs) associated with the Green's function of the Navier operator. We consider two types of Nystrom discretizations of these operators, one that relies on Kussmaul-Martensen logarithmic splittings and the other on Alpert quadratures. In addition, we consider an alternative formulation of Navier scattering problems based on Helmholtz decompositions of the elastic fields, which can be solved via a system of boundary integral equations that feature integral operators associated with the Helmholtz equation. Owing to the fact that some of the BIOs that are featured in those formulations are non-standard, we use Quadrature by Expansion (QBX) methods for their high order Nystrom discretization. Alternatively, we use Maue integration by parts techniques to recast those non-standard operators in terms of single and double layer Helmholtz BIOs whose Nystrom discretizations is amenable to the Kussmaul-Martensen methodology. We present a variety of numerical results concerning the high order accuracy that our Nystrom discretization elastic scattering solvers achieve for both smooth and Lipschitz boundaries. We also present extensive comparisons regarding the iterative behavior of solvers based on different integral equations in the high frequency regime. Finally, we illustrate how some of the Nystrom discretizations we considered can be incorporated seamlessly into the Convolution Quadrature (CQ) methodology to deliver high-order solutions of the time domain elastic scattering problems.