Volume 59, pp. 319-341, 2023.

Domain truncation, absorbing boundary conditions, Schur complements, and Padé approximation

Martin J. Gander, Lukáš Jakabčin, and Michal Outrata

Abstract

We show for a model problem that the truncation of an unbounded domain by an artificial Dirichlet boundary condition placed far away from the domain of interest is equivalent to a specific absorbing boundary condition placed closer to the domain of interest. This specific absorbing boundary condition can be implemented as a truncation layer terminated by a Dirichlet condition. We prove that the absorbing boundary condition thus obtained is a spectral Padé approximation about infinity of the transparent boundary condition. We also study numerically two improvements for this boundary condition, the truncation with an artificial Robin condition placed at the end of the truncation layer and a Padé approximation about a different point than infinity. Both of these give new and substantially better results compared to using the artificial Dirichlet boundary condition at the end of the truncation layer. We prove our results in the context of linear algebra, using spectral analysis of finite and infinite Schur complements, which we relate to continued fractions. We illustrate our results with numerical experiments.

Full Text (PDF) [465 KB], BibTeX

Key words

domain truncation, ABC, Schur complement, continued fractions, Padé approximants

AMS subject classifications

65N85, 65N06, 65E05, 41A21

Links to the cited ETNA articles

[10]	Vol. 31 (2008), pp. 228-255 Martin J. Gander: Schwarz methods over the course of time

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