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We develop a theoretical framework for the analysis of stabilized cut finite element methods for the Laplace-Beltrami operator on a manifold embedded in of arbitrary codimension. The method is based on using continuous piecewise linears on a background mesh in the embedding space for approximation together with a stabilizing form that ensures that the resulting problem is stable. The discrete manifold is represented using a triangulation which does not match the background mesh and does not need to be shape-regular, which includes level set descriptions of codimension one manifolds and the non-matching embedding of independently triangulated manifolds as special cases. We identify abstract key assumptions on the stabilizing form which allow us to prove a bound on the condition number of the stiffness matrix and optimal order estimates. The key assumptions are verified for three different realizations of the stabilizing form including a novel stabilization approach based on penalizing the surface normal gradient on the background mesh. Finally, we present numerical results illustrating our results for a curve and a surface embedded in .
Burman, Erik 1 ; Hansbo, Peter 1 ; Larson, Mats G. 1 ; Massing, André 1
@article{M2AN_2018__52_6_2247_0,
author = {Burman, Erik and Hansbo, Peter and Larson, Mats G. and Massing, Andr\'e},
title = {Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {2247--2282},
publisher = {EDP-Sciences},
volume = {52},
number = {6},
year = {2018},
doi = {10.1051/m2an/2018038},
zbl = {1417.65199},
mrnumber = {3905189},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2018038/}
}
TY - JOUR AU - Burman, Erik AU - Hansbo, Peter AU - Larson, Mats G. AU - Massing, André TI - Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 2247 EP - 2282 VL - 52 IS - 6 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2018038/ DO - 10.1051/m2an/2018038 LA - en ID - M2AN_2018__52_6_2247_0 ER -
%0 Journal Article %A Burman, Erik %A Hansbo, Peter %A Larson, Mats G. %A Massing, André %T Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 2247-2282 %V 52 %N 6 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2018038/ %R 10.1051/m2an/2018038 %G en %F M2AN_2018__52_6_2247_0
Burman, Erik; Hansbo, Peter; Larson, Mats G.; Massing, André. Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 6, pp. 2247-2282. doi: 10.1051/m2an/2018038
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