Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions

Burman, Erik; Hansbo, Peter; Larson, Mats G.; Massing, André

doi:10.1051/m2an/2018038

Burman, Erik ¹ ; Hansbo, Peter ¹ ; Larson, Mats G.¹ ; Massing, André ¹

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 6, pp. 2247-2282.

Résumé

We develop a theoretical framework for the analysis of stabilized cut finite element methods for the Laplace-Beltrami operator on a manifold embedded in $ℝ^{d}$ 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 $𝑎 𝑝𝑟𝑖𝑜𝑟𝑖$ 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 $ℝ^{3}$ .

MR Zbl

DOI : 10.1051/m2an/2018038

Classification : 65N30, 65N85, 58J05
Mots clés : Surface PDE, Laplace-Beltrami operator, cut finite element method, stabilization, condition number, a priori error estimates, arbitrary codimension

Affiliations des auteurs :

Burman, Erik ¹ ; Hansbo, Peter ¹ ; Larson, Mats G. ¹ ; Massing, André ¹

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     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://www.numdam.org/articles/10.1051/m2an/2018038/}
}

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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. http://www.numdam.org/articles/10.1051/m2an/2018038/

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