no. 6
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

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 .

DOI : 10.1051/m2an/2018038
Classification : 65N30, 65N85, 58J05
Keywords: Surface PDE, Laplace-Beltrami operator, cut finite element method, stabilization, condition number, a priori error estimates, arbitrary codimension

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

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},
     year = {2018},
     publisher = {EDP Sciences},
     volume = {52},
     number = {6},
     doi = {10.1051/m2an/2018038},
     zbl = {1417.65199},
     mrnumber = {3905189},
     language = {en},
     url = {https://www.numdam.org/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  - https://www.numdam.org/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 https://www.numdam.org/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

[1] G.E. Bredon, Topology and Geometry. In vol. 136 of Springer Science & Business Media (1993). | Zbl | MR

[2] E. Burman, Ghost penalty. C.R. Math. 348 (2010) 1217–1220. | Zbl | MR | DOI

[3] E. Burman, S. Claus, P. Hansbo, M.G. Larson, A. Massing, CutFEM: discretizing geometry and partial differential equations. Int. J. Numer. Meth. Eng. 104 (2015) 472–501. | Zbl | MR | DOI

[4] E. Burman, P. Hansbo, M.G. Larson, A stabilized cut finite element method for partial differential equations on surfaces: the Laplace-Beltrami operator. Comput. Methods Appl. Mech. Eng. 285 (2015) 188–207. | Zbl | MR | DOI

[5] E. Burman, P. Hansbo, M.G. Larson, S. Zahedi, Stabilized CutFEM for the convection problem on surfaces. Preprint (2015). | arXiv | MR

[6] E. Burman, P. Hansbo, M.G. Larson, A. Massing, A cut discontinuous Galerkin method for the Laplace-Beltrami operator. IMA J. Numer. Anal. 37 (2016) 138–169. | Zbl | MR | DOI

[7] E. Burman, P. Hansbo, M.G. Larson, A. Massing, S. Zahedi, Full gradient stabilized cut finite element methods for surface partial differential equations. Comput. Methods Appl. Mech. Eng. 310 (2016) 278–296. | Zbl | MR | DOI

[8] E. Burman, P. Hansbo, M.G. Larson, S. Zahedi, Cut finite element methods for coupled bulk-surface problems. Numer. Math. 133 (2016) 203–231. | Zbl | MR | DOI

[9] A.Y. Chernyshenko, M.A. Olshanskii, An adaptive octree finite element method for pdes posed on surfaces. Comput. Methods Appl. Mech. Eng. 291 (2015) 146–172. | Zbl | MR | DOI

[10] K. Deckelnick, C.M. Elliott, T. Ranner, Unfitted finite element methods using bulk meshes for surface partial differential equations. SIAM J. Numer. Anal. 52 (2014) 2137–2162. | Zbl | MR | DOI

[11] A. Demlow, Higher-order finite element methods and pointwise error estimates for elliptic problems on surfaces. SIAM J. Numer. Anal. 47 (2009) 805–827. | Zbl | MR | DOI

[12] A. Demlow, G. Dziuk, An adaptive finite element method for the Laplace-Beltrami operator on implicitly defined surfaces. SIAM J. Numer. Anal. 45 (2007) 421–442. | Zbl | MR | DOI

[13] A. Demlow, M.A. Olshanskii, An adaptive surface finite element method based on volume meshes. SIAM J. Numer. Anal. 50 (2012) 1624–1647. | Zbl | MR | DOI

[14] G. Dziuk, Finite elements for the Beltrami operator on arbitrary surfaces. In: Partial Differential Equations and Calculus of Variations. In vol. 1357 of Lecture Notes in Mathematics. Springer, Berlin (1988) 142–155. | Zbl | MR

[15] G. Dziuk, C.M. Elliott, Finite element methods for surface PDEs. Acta Numer. 22 (2013) 289–396. | Zbl | MR | DOI

[16] A. Ern, J.-L. Guermond, Evaluation of the condition number in linear systems arising in finite element approximations. ESAIM: M2AN 40 (2006) 29–48. | Zbl | MR | Numdam | DOI

[17] J. Grande, A. Reusken, A higher order finite element method for partial differential equations on surfaces. SIAM J. Numer. Anal. 54 (2016) 388–414. | Zbl | MR | DOI

[18] J. Grande, C. Lehrenfeld, A. Reusken, Analysis of a high order Trace Finite Element Method for PDEs on level set surfaces. SIAM J. Numer. Anal. 56 228–255. | Zbl | MR | DOI

[19] A. Gray, Tubes. Progress in Mathematics, vol 221, 2nd ed. Birkhauser, Basel (2004). | Zbl | MR

[20] P. Grisvard, Elliptic problems in nonsmooth domains. In: Vol. 24 of Monographs and Studies in Mathematics. Pitman Advanced Publishing Program, Boston, MA (1985). | Zbl | MR

[21] S. Groß, M.A. Olshanskii, A. Reusken, A trace finite element method for a class of coupled bulk-interface transport problems. ESAIM: M2AN 49 (2015) 1303–1330. | Zbl | MR | Numdam | DOI

[22] A. Hansbo, P. Hansbo, M.G. Larson, A finite element method on composite grids based on Nitsche’s method. ESAIM: M2AN 37 (2003) 495–514. | Zbl | MR | Numdam | DOI

[23] P. Hansbo, M.G. Larson, S. Zahedi, Characteristic cut finite element methods for convection-diffusion problems on time dependent surfaces. Comput. Methods Appl. Mech. Eng. 293 (2015) 431–461. | Zbl | MR | DOI

[24] P. Hansbo, M.G. Larson, A. Massing, A stabilized cut finite element method for the Darcy problem on surfaces. Comput. Methods Appl. Mech. Eng. 326 (2017) 298–318. | Zbl | MR | DOI

[25] M.G. Larson, S. Zahedi, Stabilization of Higher Order Cut Finite Element Methods on Surfaces. Preprint arXiv:1710.03343 (2017). | MR

[26] M.A. Olshanskii, A. Reusken, A finite element method for surface PDEs: matrix properties. Numer. Math. 114 (2010) 491–520. | Zbl | MR | DOI

[27] M.A. Olshanskii, A. Reusken, Error analysis of a space-time finite element method for solving PDEs on evolving surfaces. SIAM J. Numer. Anal. 52 (2014) 2092–2120. | Zbl | MR | DOI

[28] M.A. Olshanskii, A. Reusken, J. Grande, A finite element method for elliptic equations on surfaces. SIAM J. Numer. Anal. 47 (2009) 3339–3358. | Zbl | MR | DOI

[29] M.A. Olshanskii, A. Reusken, X. Xu, A stabilized finite element method for advection-diffusion equations on surfaces. IMA J. Numer. Anal. 34 (2014) 732–758. | Zbl | MR | DOI

[30] M.A. Olshanskii, A. Reusken, X. Xu, An Eulerian space-time finite element method for diffusion problems on evolving surfaces. SIAM J. Numer. Anal. 52 (2014) 1354–1377. | Zbl | MR | DOI

[31] A. Reusken, Analysis of trace finite element methods for surface partial differential equations. IMA J. Numer. Anal. 35 (2014) 1568–1590. | Zbl | MR | DOI

[32] H. Weyl, On the volume of tubes. Am. J. Math. 61 (1989) 461–472. | JFM | Zbl | MR | DOI

Cité par Sources :