Complex Riemannian metric and absorbing boundary conditions
type: Item
instance of: scholarly article
title: Complex Riemannian metric and absorbing boundary conditions
zbMATH Open document ID: 1031.35141
DOI: 10.1016/S0021-7824(01)01214-4
author: M. Lassas
author: Jukka Liukkonen
author: Erkki Somersalo
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2002-07-24T00:00:00Z
review text: The scattering of electromagnetic or acoustic waves from a given obstacle takes place in an unbounded space which causes difficulties in numerical simulations. One possible approach is to surround the scatterer by an absorbing boundary layer. The common PLM (perfectly matched layer) method consists in embedding of the original space \(\mathbb{R}^3\) into \(\mathbb{C}^3\), the complex stretching of coordinates. The authors propose a different and more geometrical approach by using complex Riemannian metric. This provides a new kind of absorbing boundary layers for Maxwell and Helmholtz equations. A pseudo-Riemannian manifold \((M,g)\) is called an absorbing manifold if \(g\) is flat and nondegenerate complex metric such that there is an isometry \(M\to\mathbb{C}^3\) on a totally real submanifold of \(\mathbb{C}^3\) but \(g\) is the common Euclidean metric on a relatively compact open subset \(D\subset M\). If we place a source or scatterer inside \(D\), the waves within \(D\) coincide with the scattering solution while outside \(D\), the complex metric causes reflectionless attenuation of exponential type. The authors carefully introduce all geometrical prerequisites and thoroughly deal with the existence proofs of solutions on absorbing manifolds.
reviewed by: Jan Chrastina
Mathematics Subject Classification ID: 35Q60
Mathematics Subject Classification ID: 78A45
Mathematics Subject Classification ID: 76Q05
zbMATH DE Number: 1770845
zbMATH Keywords: absorbing boundary conditions
zbMATH Keywords: complex Riemannian metrics
zbMATH Keywords: scattering poles
MaRDI profile type: Publication
Wikidata QID: Q115340566
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description: scientific article; zbMATH DE number 1770845