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Nitsche Type Mortaring for Singularly Perturbed Reaction-diffusion Problems

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Abstract

The paper is concerned with the Nitsche mortaring in the framework of domain decomposition where non-matching meshes and weak continuity of the finite element approximation at the interface are admitted. The approach is applied to singularly perturbed reaction-diffusion problems in 2D. Non-matching meshes of triangles being anisotropic in the boundary layers are applied. Some properties as well as error estimates of the Nitsche mortar finite element schemes are proved. In particular, using a suitable degree of anisotropy of triangles in the boundary layers of a rectangle, convergence rates as known for the conforming finite element method are derived. Numerical examples illustrate the approach and the results.

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References

  • Apel, T.: Anisotropic finite elements: local estimates and applications. Adv. Numer. Math. Stuttgart: Teubner 1999.

  • T. Apel G. Lube (1998) ArticleTitleAnisotropic mesh refinement for a singularly perturbed reaction diffusion model problem Appl. Numer. Math. 26 415–433 Occurrence Handle10.1016/S0168-9274(97)00106-2

    Article  Google Scholar 

  • D. N. Arnold (1982) ArticleTitleAn interior penalty finite element method with discontinuous elements SIAM J. Numer. Anal. 19 742–760 Occurrence Handle10.1137/0719052

    Article  Google Scholar 

  • Becker, R., Hansbo, P.: A finite element method for domain decomposition with non-matching grids. Technical Report INRIA 3613, 1999.

  • R. Becker P. Hansbo R. Stenberg (2003) ArticleTitleA finite element method for domain decomposition with non-matching grids M2AN Math. Model. Numer. Anal. 37 209–225 Occurrence Handle10.1051/m2an:2003023

    Article  Google Scholar 

  • F. B. Belgacem (1999) ArticleTitleThe Mortar finite element method with Lagrange multipliers Numer. Math. 84 173–197 Occurrence Handle10.1007/s002110050468

    Article  Google Scholar 

  • C. Bernardi Y. Maday A. T. Patera (1994) A new non-conforming approch to domain decomposition: The mortar element method H. Brezis (Eds) et al. Nonlinear partial differential equations and their applications Longman Paris 13–51

    Google Scholar 

  • D. Braess W. Dahmen C. Wieners (1999) ArticleTitleA multigrid algorithm for the Mortar finite element method SIAM J. Numer. Anal. 37 48–69 Occurrence Handle10.1137/S0036142998335431

    Article  Google Scholar 

  • F. Brezzi M. Fortin (1991) Mixed and hybrid finite element methods Springer New York

    Google Scholar 

  • Fritz, A., Hüeber, S., Wohlmuth, B. I.: A comparison of Mortar and Nitsche techniques for linear elasticity. Preprint 008, IANS, Universitaet Stuttgart 2003.

  • P. Grisvard (1985) Elliptic problems in nonsmooth domains Pitman Boston

    Google Scholar 

  • H. Han R. B. Kellogg (1990) ArticleTitleDifferentiability properties of solutions of the equation -ɛ2 ▵u+ru = f in a square SIAM J. Math. Anal. 21 394–408 Occurrence Handle10.1137/0521022

    Article  Google Scholar 

  • B. Heinrich S. Nicaise (2003) ArticleTitleNitsche mortar finite element method for transmission problems with singularities IMA J. Numer. Anal. 23 331–358 Occurrence Handle10.1093/imanum/23.2.331

    Article  Google Scholar 

  • B. Heinrich K. Pietsch (2002) ArticleTitleNitsche type mortaring for some elliptic problem with corner singularities Computing 68 217–238 Occurrence Handle10.1007/s00607-002-1446-0

    Article  Google Scholar 

  • G. Kunert (2001) ArticleTitleRobust a posteriori error estimation for a singularly perturbed reaction-diffusion equation on anisotropic tetrahedral meshes Adv. Comput. Math. 15 237–259 Occurrence Handle10.1023/A:1014248711347

    Article  Google Scholar 

  • Melenk, J. M.: hp-finite element methods for singular perturbations. Lecture Notes in Mathematics., vol. 1796. Berlin: Springer 2002

  • Nitsche, J.: Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abhandlung aus dem Mathematischen Seminar der Universität Hamburg 36, 9–15 (1970/1971).

  • Stenberg, R.: Mortaring by a method of J. A. Nitsche. In: (Idelsohn, S., Onate, E., Dvorkin, E., eds.) Computational mechanics, new trends and applications, Barcelona 1998.

  • V. Thomeé (1997) Galerkin finite element methods for parabolic problems Springer Berlin New York

    Google Scholar 

  • B. I. Wohlmuth (2000) ArticleTitleA mortar finite element method using dual spaces for the Lagrange multiplier SIAM J. Numer. Anal. 38 989–1012 Occurrence Handle10.1137/S0036142999350929

    Article  Google Scholar 

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Authors and Affiliations

  1. Fakultät fär Mathematik, Technische Universität Chemnitz, 09107, Chemnitz, Germany

    B. Heinrich

  2. Fakultät fär Mathematik, Technische Universität Chemnitz, 09107, Chemnitz, Germany

    K. Pönitz

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  1. B. Heinrich
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  2. K. Pönitz
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Correspondence to B. Heinrich.

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Heinrich, B., Pönitz, K. Nitsche Type Mortaring for Singularly Perturbed Reaction-diffusion Problems. Computing 75, 257–279 (2005). https://doi.org/10.1007/s00607-005-0123-5

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  • DOI: https://doi.org/10.1007/s00607-005-0123-5

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