In this paper, we present a general approach to obtain numerical schemes with good mesh properties for problems with moving boundaries, that is for evolving submanifolds with boundaries. This includes moving domains and surfaces with boundaries. Our approach is based on a variant of the so-called the DeTurck trick. By reparametrizing the evolution of the submanifold via solutions to the harmonic map heat flow of manifolds with boundary, we obtain a new velocity field for the motion of the submanifold. Moving the vertices of the computational mesh according to this velocity field automatically leads to computational meshes of high quality both for the submanifold and its boundary. Using the ALE-method in [16], this idea can be easily built into algorithms for the computation of physical problems with moving boundaries.
DOI: 10.5802/smai-jcm.12
Keywords: Moving boundary, surface finite elements, mesh improvement, harmonic map heat flow, DeTurck trick
@article{SMAI-JCM_2016__2__141_0,
author = {Charles M. Elliott and Hans Fritz},
title = {On algorithms with good mesh properties for problems with moving boundaries based on the {Harmonic} {Map} {Heat} {Flow} and the {DeTurck} trick},
journal = {The SMAI Journal of computational mathematics},
pages = {141--176},
publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
volume = {2},
year = {2016},
doi = {10.5802/smai-jcm.12},
zbl = {1416.65324},
mrnumber = {3633548},
language = {en},
url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.12/}
}
TY - JOUR AU - Charles M. Elliott AU - Hans Fritz TI - On algorithms with good mesh properties for problems with moving boundaries based on the Harmonic Map Heat Flow and the DeTurck trick JO - The SMAI Journal of computational mathematics PY - 2016 SP - 141 EP - 176 VL - 2 PB - Société de Mathématiques Appliquées et Industrielles UR - https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.12/ DO - 10.5802/smai-jcm.12 LA - en ID - SMAI-JCM_2016__2__141_0 ER -
%0 Journal Article %A Charles M. Elliott %A Hans Fritz %T On algorithms with good mesh properties for problems with moving boundaries based on the Harmonic Map Heat Flow and the DeTurck trick %J The SMAI Journal of computational mathematics %D 2016 %P 141-176 %V 2 %I Société de Mathématiques Appliquées et Industrielles %U https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.12/ %R 10.5802/smai-jcm.12 %G en %F SMAI-JCM_2016__2__141_0
Charles M. Elliott; Hans Fritz. On algorithms with good mesh properties for problems with moving boundaries based on the Harmonic Map Heat Flow and the DeTurck trick. The SMAI Journal of computational mathematics, Volume 2 (2016), pp. 141-176. doi : 10.5802/smai-jcm.12. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.12/
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