A finite element method for domain decomposition with non-matching grids

Roland Becker[1]; Peter Hansbo; Rolf Stenberg

  • [1] Heidelberg University, Germany

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2003)

  • Volume: 37, Issue: 2, page 209-225
  • ISSN: 0764-583X

Abstract

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In this note, we propose and analyse a method for handling interfaces between non-matching grids based on an approach suggested by Nitsche (1971) for the approximation of Dirichlet boundary conditions. The exposition is limited to self-adjoint elliptic problems, using Poisson’s equation as a model. A priori and a posteriori error estimates are given. Some numerical results are included.

How to cite

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Becker, Roland, Hansbo, Peter, and Stenberg, Rolf. "A finite element method for domain decomposition with non-matching grids." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 37.2 (2003): 209-225. <http://eudml.org/doc/245245>.

@article{Becker2003,
abstract = {In this note, we propose and analyse a method for handling interfaces between non-matching grids based on an approach suggested by Nitsche (1971) for the approximation of Dirichlet boundary conditions. The exposition is limited to self-adjoint elliptic problems, using Poisson’s equation as a model. A priori and a posteriori error estimates are given. Some numerical results are included.},
affiliation = {Heidelberg University, Germany},
author = {Becker, Roland, Hansbo, Peter, Stenberg, Rolf},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {Nitsche’s method; domain decomposition; non-matching grids; Nitsche's method; Poisson problem; error estimates; numerical results; finite element method},
language = {eng},
number = {2},
pages = {209-225},
publisher = {EDP-Sciences},
title = {A finite element method for domain decomposition with non-matching grids},
url = {http://eudml.org/doc/245245},
volume = {37},
year = {2003},
}

TY - JOUR
AU - Becker, Roland
AU - Hansbo, Peter
AU - Stenberg, Rolf
TI - A finite element method for domain decomposition with non-matching grids
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2003
PB - EDP-Sciences
VL - 37
IS - 2
SP - 209
EP - 225
AB - In this note, we propose and analyse a method for handling interfaces between non-matching grids based on an approach suggested by Nitsche (1971) for the approximation of Dirichlet boundary conditions. The exposition is limited to self-adjoint elliptic problems, using Poisson’s equation as a model. A priori and a posteriori error estimates are given. Some numerical results are included.
LA - eng
KW - Nitsche’s method; domain decomposition; non-matching grids; Nitsche's method; Poisson problem; error estimates; numerical results; finite element method
UR - http://eudml.org/doc/245245
ER -

References

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  1. [1] J.-P. Aubin, Approximation of Elliptic Boundary-Value Problem. Wiley (1972). Zbl0248.65063MR478662
  2. [2] D. Arnold, An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19 (1982) 742–760. Zbl0482.65060
  3. [3] C. Baiocchi, F. Brezzi and L.D. Marini, Stabilization of Galerkin methods and applications to domain decomposition, in Future Tendencies in Computer Science, Control and Applied Mathematics, A. Bensoussan and J.-P. Verjus Eds., Springer (1992) 345–355. 
  4. [4] J.C. Barbosa and T.J.R. Hughes, Boundary Lagrange multipliers in finite element methods: error analysis in natural norms. Numer. Math. 62 (1992) 1–15. Zbl0765.65102
  5. [5] J.W. Barrett and C.M. Elliot, Finite element approximation of the Dirichlet problem using the boundary penalty method. Numer. Math. 49 (1986) 343–366. Zbl0614.65116
  6. [6] R. Becker and P. Hansbo, Discontinuous Galerkin methods for convection-diffusion problems with arbitrary Péclet number, in Numerical Mathematics and Advanced Applications: Proceedings of the 3rd European Conference, P. Neittaanmäki, T. Tiihonen and P. Tarvainen Eds., World Scientific (2000) 100–109. Zbl0968.65084
  7. [7] R. Becker and R. Rannacher, A feed-back approach to error control in finite element methods: basic analysis and examples. East-West J. Numer. Math. 4 (1996) 237–264. Zbl0868.65076
  8. [8] C. Bernadi, Y. Maday and A. Patera, A new nonconforming approach to domain decomposition: the mortar element method, in Nonlinear Partial Differential Equations and Their Application, H. Brezis and J.L. Lions Eds., Pitman (1989). Zbl0797.65094
  9. [9] F. Brezzi, L.P. Franca, D. Marini and A. Russo, Stabilization techniques for domain decomposition methods with non-matching grids, IAN-CNR Report N. 1037, Istituto di Analisi Numerica Pavia. 
  10. [10] J. Freund and R. Stenberg, On weakly imposed boundary conditions for second order problems, in Proceedings of the Ninth Int. Conf. Finite Elements in Fluids, M. Morandi Cecchi et al. Eds., Venice (1995) 327–336. 
  11. [11] J. Freund, Space-time finite element methods for second order problems: an algorithmic approach. Acta Polytech. Scand. Math. Comput. Manage. Eng. Ser. 79 (1996). Zbl0861.65083MR1422305
  12. [12] B. Heinrich and S. Nicaise, Nitsche mortar finite element method for transmission problems with singularities. SFB393-Preprint 2001-10, Technische Universität Chemnitz (2001). Zbl1027.65149MR1975269
  13. [13] B. Heinrich and K. Pietsch, Nitsche type mortaring for some elliptic problem with corner singularities. Computing 68 (2002) 217–238. Zbl1002.65124
  14. [14] C. Johnson and P. Hansbo, Adaptive finite element methods in computational mechanics. Comput. Methods Appl. Mech. Engrg. 101 (1992) 143–181. Zbl0778.73071
  15. [15] P. Le Tallec and T. Sassi, Domain decomposition with nonmatching grids: augmented Lagrangian approach. Math. Comp. 64 (1995) 1367–1396. Zbl0849.65087
  16. [16] P.L. Lions, On the Schwarz alternating method III: a variant for nonoverlapping subdomains, in Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, T.F. Chan, R. Glowinski, J. Periaux and O.B. Widlund Eds., SIAM (1989) 202–223. Zbl0704.65090
  17. [17] J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg 36 (1971) 9–15. Zbl0229.65079
  18. [18] R. Stenberg, On some techniques for approximating boundary conditions in the finite element method. J. Comput. Appl. Math. 63 (1995) 139–148. Zbl0856.65130
  19. [19] R. Stenberg, Mortaring by a method of J.A. Nitsche, in Computational Mechanics: New Trends and Applications, S. Idelsohn, E. Onate and E. Dvorkin Eds., CIMNE, Barcelona (1998). MR1839048
  20. [20] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. Springer (1997). Zbl0884.65097MR1479170
  21. [21] B.I. Wohlmuth, A residual based error estimator for mortar finite element discretizations. Numer. Math. 84 (1999) 143–171. Zbl0962.65090

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