This paper develops a framework to include Dirichlet boundary conditions on a subset of the boundary which depends on time. In this model, the boundary conditions are weakly enforced with the help of a Lagrange multiplier method. In order to avoid that the ansatz space of the Lagrange multiplier depends on time, a bi-Lipschitz transformation, which maps a fixed interval onto the Dirichlet boundary, is introduced. An inf-sup condition as well as existence results are presented for a class of second order initial-boundary value problems. For the semi-discretization in space, a finite element scheme is presented which satisfies a discrete stability condition. Because of the saddle point structure of the underlying PDE, the resulting system is a DAE of index 3.
@article{M2AN_2014__48_6_1859_0,
author = {Altmann, Robert},
title = {Moving Dirichlet boundary conditions},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {48},
year = {2014},
pages = {1859-1876},
doi = {10.1051/m2an/2014022},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2014__48_6_1859_0}
}
Altmann, Robert. Moving Dirichlet boundary conditions. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 48 (2014) pp. 1859-1876. doi : 10.1051/m2an/2014022. http://gdmltest.u-ga.fr/item/M2AN_2014__48_6_1859_0/
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