Norm-preserving discretization of integral equations for elliptic PDEs with internal layers I: the one-dimensional case

TL;DR

This paper studies how to discretize integral equations for 1D elliptic PDEs with internal layers, ensuring well-conditioned systems by preserving norms and adaptively refining discretizations.

Contribution

It demonstrates that norm-preserving Nyström discretizations, combined with adaptive refinement, yield well-conditioned linear systems for 1D elliptic PDEs with internal layers.

Findings

High-order Nyström discretization preserves norms in suitable L_p spaces.

Adaptive refinement improves the conditioning of discretized systems.

Analytical solutions and condition number estimates are provided for 1D cases.

Abstract

We investigate the behavior of integral formulations of variable coefficient elliptic partial differential equations (PDEs) in the presence of steep internal layers. In one dimension, the equations that arise can be solved analytically and the condition numbers estimated in various L_p norms. We show that high-order accurate Nystr\"{o}m discretization leads to well-conditioned finite dimensional linear systems if and only if the discretization is both norm-preserving in a correctly chosen L_p space and adaptively refined in the internal layer.