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Optimal convergence rates of hp mortar finite element methods for second-order elliptic problems

Published online by Cambridge University Press:  15 April 2002

Faker Ben Belgacem
Affiliation:
Mathématiques pour l'industrie et la physique, Unité mixte de recherche CNRS-UPS-INSAT-UT1 (UMR 5640), Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 04, France. (belgacem@mip.ups-tlse.fr)
Padmanabhan Seshaiyer
Affiliation:
Department of Biomedical Engineering, 233 Zachry Engineering Center, Texas A & M University, College Station, Texas 77843-3120, U.S.A. (padhu@math.umbc.edu)
Manil Suri
Affiliation:
Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, Maryland 21250, U.S.A. (suri@math.umbc.edu)
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Abstract

We present an improved, near-optimal hp error estimate for a non-conforming finite element method, called the mortar method (M0). We also present a new hp mortaring technique, called the mortar method (MP), and derive h, p and hp error estimates for it, in the presence of quasiuniform and non-quasiuniform meshes. Our theoretical results, augmented by the computational evidence we present, show that like (M0), (MP) is also a viable mortaring technique for the hp method.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2000

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References

Y. Achdou, Y. Maday and O.B. Widlund, Méthode itérative de sous-structuration pour les éléments avec joints. C.R. Acad. Sci. Paris Série I 322 (1996) 185-190.
Achdou, Y., Maday, Y. and Widlund, O.B., Iterative substructuring preconditioners for the mortar finite element method in two dimensions. SIAM J. Num. Anal. 36 (1999) 551-580. CrossRef
Achdou, Y. and Pironneau, O., A fast solver for Navier-Stokes equations in the laminar regime using mortar finite element and boundary element methods. SIAM J. Num. Anal. 32 (1995) 985-1016. CrossRef
Babuska, I. and Suri, M., The h-p-version of the finite element method with quasi-uniform meshes. Modél. Math. et Anal. Numér. 21 (1987) 199-238. CrossRef
Babuska, I. and Suri, M., The p and h-p-versions of the finite element method: basic principles and properties. SIAM Review 36 (1984) 578-632. CrossRef
Babuska, I. and Suri, M., The optimal convergence rate of the p-Version of the finite element method. SIAM J. Num. Anal. 24 (1987) 750-776. CrossRef
F. Ben Belgacem, Disrétisations 3D non conformes par la méthode de décomposition de domaine des éléments avec joints : Analyse mathématique et mise en œuvre pour le problème de Poisson. Thèse de l'Université Pierre et Marie Curie, Paris VI. Note technique EDF, ref. HI72/93017 (1993).
F. Ben Belgacem, The mortar finite element method with Lagrange multipliers. Num. Mathematik (to appear).
Ben Belgacem, F. and Maday, Y., Non conforming spectral element methodology tuned to parallel implementation. Compu. Meth. Appl. Mech. Eng. 116 (1994) 59-67. CrossRef
C. Bernardi, N. Débit and Y. Maday, Coupling finite element and spectral methods: first results. Math. Compu. 54 (1990), 21-39.
C. Bernardi, M. Dauge and Y. Maday, Interpolation of nullspaces for polynomial approximation of divergence-free functions in a cube. Proc. Conf. Boundary Value Problems and Integral Equations in Nonsmooth Domains, M. Costabel, M. Dauge and S. Nicaise Eds., Lecture Notes in Pure and Applied Mathematics 167 Dekker (1994) 27-46.
C. Bernardi and Y. Maday, Spectral, spectral element and mortar element methods. Technical report of the Laboratoire d'analyse numérique, Université Pierre et Marie Curie, Paris VI, 1998.
Bernardi, C. and Maday, Y., Relèvement de traces polynomiales et applications. RAIRO Modél. Math. Anal. Numér. 24 (1990) 557-611. CrossRef
C. Bernardi, Y. Maday and A. T. Patera, A new non conforming approach to domain decomposition: The mortar element method. Pitman, H. Brezis, J.-L. Lions Eds., Collège de France Seminar (1990).
Bernardi, C., Maday, Y. and Sacchi-Landriani, G., Non conforming matching conditions for coupling spectral and finite element methods. Appl. Numer. Math. 54 (1989) 64-84.
Berger, A., Scott, R. and Strang, G., Approximate boundary conditions in the finite element method. Symposia Mathematica 10 (1972) 295-313.
S. Brenner, A non-standard finite element interpolation estimate. Research Report 1998:07, Department of Mathematics, University of South Carolina (1998).
P.-G. Ciarlet, The finite element Method for Elliptic Problems. North Holland (1978).
N. Débit, La méthode des éléments avec joints dans le cas du couplage des méthodes spectrales et méthodes des éléments finis : Résolution des équations de Navier-Stokes. Thèse de l'Université Pierre et Marie Curie, Paris VI (1992).
M. Dorr, On the discretization of inter-domain coupling in elliptic boundary-value problems via the p -Version of the finite element method. T.F. Chan, R. Glowinski, J. Periaux. O.B. Widlund, Eds., SIAM (1989).
V. Girault and P.-A. Raviart, Finite element methods for Navier-Stokes equations. Springer Verlag (1986).
P. Grisvard, Elliptic problems in nonsmooth domains. Monographs and Studies in Mathematics 24 (Pitman, 1985).
Gui, W. and Babuska, I., The h-p-version of the finite element method in one dimension. Num. Mathematik 3 (1986) 577-657. CrossRef
B. Guo and I. Babuska, The h-p-version of the finite element method. Compu. Mech. 1 (1986), Part 1: 21-41, Part 2: 203-220.
P. Seshaiyer, Non-Conforming h - p finite element methods. Doctoral Thesis, University of Maryland Baltimore County (1998).
P. Seshaiyer and M. Suri,: Uniform h-p Convergence results for the mortar finite element method. Math. Compu. PII: S 0025-5718(99)01083-2 (to appear).
P. Seshaiyer and M. Suri, Convergence results for the non-Conforming h-p methods: The mortar finite element method. AMS, Cont. Math. 218 (1998) 467-473.
P. Seshaiyer and M. Suri, h-p submeshing via non-conforming finite element methods. Submitted to Compu. Meth. Appl. Mech. Eng. (1998).
G. Strang and G. J. Fix, An analysis of the finite element method. Wellesly, Cambridge Press Masson (1973).