Solving Fredholm second-kind integral equations with singular right-hand sides on non-smooth boundaries

TL;DR

This paper introduces a universal numerical scheme based on RCIP for solving Fredholm second-kind integral equations with singular right-hand sides on non-smooth boundaries, achieving high accuracy and stability.

Contribution

The paper presents a novel, boundary-agnostic scheme that handles strong singularities in right-hand sides of integral equations with adaptive refinement and recursive preconditioning.

Findings

Effective treatment of singular right-hand sides close to 1/|r| in full machine precision.

Adaptive refinement improves discretization efficiency and stability.

Numerical examples demonstrate the scheme's applicability to kinetic equations.

Abstract

A numerical scheme is presented for the solution of Fredholm second-kind boundary integral equations with right-hand sides that are singular at a finite set of boundary points. The boundaries themselves may be non-smooth. The scheme, which builds on recursively compressed inverse preconditioning (RCIP), is universal as it is independent of the nature of the singularities. Strong right-hand-side singularities, such as $1/|r|^\alpha$ with $\alpha$ close to $1$, can be treated in full machine precision. Adaptive refinement is used only in the recursive construction of the preconditioner, leading to an optimal number of discretization points and superior stability in the solve phase. The performance of the scheme is illustrated via several numerical examples, including an application to an integral equation derived from the linearized BGKW kinetic equation for the steady Couette flow.