Isoparametric finite element analysis of a generalized Robin boundary value problem on curved domains
The SMAI Journal of computational mathematics, Volume 7 (2021), pp. 57-73.

We study the discretization of an elliptic partial differential equation, posed on a two- or three-dimensional domain with smooth boundary, endowed with a generalized Robin boundary condition which involves the Laplace–Beltrami operator on the boundary surface. The boundary is approximated with piecewise polynomial faces and we use isoparametric finite elements of arbitrary order for the discretization. We derive optimal-order error bounds for this non-conforming finite element method in both L 2 - and H 1 -norm. Numerical examples illustrate the theoretical results.

Published online:
DOI: 10.5802/smai-jcm.71
Keywords: generalized Robin boundary conditions, Laplace–Beltrami operator, isoparametric finite elements, finite element method, error analysis

Dominik Edelmann 1

1 Mathematisches Institut, Universität Tübingen, Germany
License: CC-BY-NC-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Dominik Edelmann. Isoparametric finite element analysis of a generalized  Robin boundary value problem on curved domains. The SMAI Journal of computational mathematics, Volume 7 (2021), pp. 57-73. doi : 10.5802/smai-jcm.71. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.71/

[1] S. Bartels; C. Carstensen; A. Hecht P2Q2Iso2D=2D Isoparametric FEM in Matlab, J. Comput. Appl. Math., Volume 192 (2006) no. 2, pp. 219-250 | DOI | MR | Zbl

[2] C. Bernardi Optimal Finite-Element Interpolation on Curved Domains, SIAM J. Numer. Anal., Volume 26 (1989) no. 5, pp. 1212-1240 | DOI | MR | Zbl

[3] A. Demlow Higher-Order Finite Element Methods and Pointwise Error Estimates for Elliptic Problems on Surfaces, SIAM J. Numer. Anal., Volume 47 (2009) no. 2, pp. 805-827 | DOI | MR | Zbl

[4] F. Dubois Discrete Vector Potential Representation of a Divergence-Free Vector Field in Three-Dimensional Domains: Numerical Analysis of a Model Problem, SIAM J. Numer. Anal., Volume 27 (1990) no. 5, pp. 1103-1141 | DOI | MR | Zbl

[5] G. Dziuk Finite Elements for the Beltrami Operator on arbitrary Surfaces, Partial Differential Equations and Calculus of Variations, Springer, 1988, pp. 142-155 | Zbl

[6] G. Dziuk; C. M. Elliott Finite Elements on Evolving Surfaces, IMA J. Numer. Anal., Volume 27 (2007) no. 2, pp. 262-292 | DOI | MR | Zbl

[7] G. Dziuk; C. M. Elliott Finite element methods for surface PDEs, Acta Numer., Volume 22 (2013), p. 289 | DOI | MR | Zbl

[8] C. M. Elliott; T. Ranner Finite element analysis for a coupled bulk–surface partial differential equation, IMA J. Numer. Anal., Volume 33 (2013) no. 2, pp. 377-402 | DOI | MR | Zbl

[9] C. M. Elliott; T. Ranner A unified theory for continuous in time evolving finite element space approximations to partial differential equations in evolving domains (2017) (https://arxiv.org/abs/1703.04679)

[10] M. J. Gander; L. Halpern Optimized Schwarz waveform relaxation methods for advection reaction diffusion problems, SIAM J. Numer. Anal., Volume 45 (2007) no. 2, pp. 666-697 | DOI | MR | Zbl

[11] L. Gerardo-Giorda; F. Nobile; C. Vergara Analysis and optimization of Robin–Robin partitioned procedures in fluid-structure interaction problems, SIAM J. Numer. Anal., Volume 48 (2010) no. 6, pp. 2091-2116 | DOI | MR | Zbl

[12] F. Gesztesy; M. Mitrea Generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains (2008) (https://arxiv.org/abs/0803.3179) | DOI | Zbl

[13] G. R. Goldstein Derivation and physical interpretation of general boundary conditions, Adv. Differ. Equ., Volume 11 (2006) no. 4, pp. 457-480 | MR | Zbl

[14] L. Halpern Optimized Schwarz waveform relaxation: roots, blossoms and fruits, Domain Decomposition Methods in Science and Engineering XVIII, Springer, 2009, pp. 225-232 | DOI | MR | Zbl

[15] T. Kashiwabara; C. M. Colciago; L. Dedè; A. Quarteroni Well-posedness, regularity, and convergence analysis of the finite element approximation of a generalized Robin boundary value problem, SIAM J. Numer. Anal., Volume 53 (2015) no. 1, pp. 105-126 | DOI | MR | Zbl

[16] M. Lenoir Optimal isoparametric finite elements and error estimates for domains involving curved boundaries, SIAM J. Numer. Anal., Volume 23 (1986) no. 3, pp. 562-580 | DOI | MR | Zbl

[17] P. Persson; G. Strang A simple mesh generator in MATLAB, SIAM Rev., Volume 46 (2004) no. 2, pp. 329-345 | DOI | MR | Zbl

[18] A. Quarteroni; A. Valli Domain decomposition methods for partial differential equations, Oxford University Press, 1999 | Zbl

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