Perfectly Matched Layers on Cubic Domains for Pauli's Equations

Laurence Halpern (1) , Jeffrey Rauch (2)

1 LAGA - Laboratoire Analyse, Géométrie et Applications
2 Department of Mathematics - University of Michigan

This article proves the well posedness of the boundary value problem that arises when PML algorithms are applied to Pauli's equations with a three dimensional rectangle as computational domain. The absorptions are 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. The Laplace transform is analysed. It turns on the analysis of a boundary value problem formally obtained by complex stretching. Existence is proved by deriving a boundary value problems for a complex stretched Helmholtz equation on smoothed domains. This is the first stability proof with x-dependent absorptions on a bounded domain whose boundary is not smooth.

Mots clés: Hyperbolic boundary value problem; PML; trihedral corner; dissipative boundary conditions; Bérenger; Pauli system

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]

Laurence Halpern, Jeffrey Rauch. Perfectly Matched Layers on Cubic Domains for Pauli's Equations. 2022. ⟨hal-02872141v3⟩