A Singularity Guided Nystr\"om Method for Elastostatics on Two Dimensional Domains with Corners

TL;DR

This paper introduces a new singularity-guided Nyström method for solving boundary integral equations of the 2D Lamé system on domains with corners, combining analytical insights with numerical refinement for improved accuracy.

Contribution

It develops a comprehensive analytical framework for cornered domains, derives corner exponents, and proposes a novel SGN numerical scheme utilizing these exponents for enhanced precision.

Findings

SGN outperforms uniform Nyström in accuracy

Error analysis confirms exponential convergence with refinement

Method effectively handles re-entrant angles

Abstract

We develop a comprehensive analytical and numerical framework for boundary integral equations (BIEs) of the 2D Lam\'e system on cornered domains. By applying local Mellin analysis on a wedge, we obtain a factorizable characteristic equation for the singular exponents of the boundary densities, and clarify their dependence on boundary conditions. The Fredholm well-posedness of the BIEs on cornered domains is proved in weighted Sobolev spaces. We further construct an explicit density-to-Taylor mapping for the BIE and show its invertibility for all but a countable set of angles. Based on these analytical results, we propose a singularity guided Nystr\"om (SGN) scheme for the numerical solution of BIEs on cornered domains. The SGN uses the computed corner exponents and a Legendre-tail indicator to drive panel refinement. An error analysis that combines this refinement strategy with an exponentially accurate far-field quadrature rule is provided. Numerical experiments across various cornered geometries demonstrate that SGN obtains higher order accuracy than uniform Nystr\"om method and reveal a crowding-limited regime for domains with re-entrant angles.