Perfectly Matched Layers on Cubic Domains forPauli's Equations

TL;DR

This paper proves the well-posedness of perfectly matched layers (PML) applied to Pauli's equations on cubic domains, addressing complex boundary conditions and variable coefficients with a novel stability proof for non-smooth boundaries.

Contribution

It provides the first stability proof for PML on bounded domains with non-smooth boundaries and x-dependent absorptions in the context of Pauli's equations.

Findings

Established well-posedness of PML for Pauli's equations on cubic domains.

Analyzed the Laplace transform and complex stretching for variable coefficient problems.

Proved existence using boundary value problems on smoothed domains.

Abstract

This article proves the well posedness of the boundary value problemthat arises when PML algorithms are applied to Pauli's equationswith a three dimensional rectangle as computational domain. The absorptionsare positive near the boundary and zero far from the boundary so are always x-dependent. At the flat parts of the boundary of the rectangle, the natural absorbing boundary conditions are imposed.The difficulty addressed is the analysis of the resulting variable coeffi-cient problem on the rectanglar solid with its edges and corners. TheLaplace transform is analysed. It turns on the analysis of a boundaryvalue problem formally obtained by complex stretching. Existence isproved by deriving a boundary value problems for a complex stretchedHelmholtz equation on smoothed domains. This is the first stabilityproof with x-dependent absorptions on a bounded domain whoseboundary is not smooth.