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Error indicators for the mortar finite element discretization of the Laplace equation. (English) Zbl 1012.65108

The paper deals with the numerical analysis of residual error indicators for mortar finite element discretizations for the Laplace equation. Optimal estimates, which allow a comparison with the error, are derived without any saturation assumption. Numerical tests, also involving a comparison to a conforming approach, are presented indicating the efficiency of the non-conforming method considered.

MSC:

65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65Y20 Complexity and performance of numerical algorithms

Software:

FreeFem++
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[1] M. Azaïez, C. Bernardi, Y. Maday – Some tools for adaptivity in the spectral element method, in Proc. of the third Int. Conf. On Spectral And High Order Methods, Houston J. of Math. (1996), 243-253.
[2] Faker Ben Belgacem, The mixed mortar finite element method for the incompressible Stokes problem: convergence analysis, SIAM J. Numer. Anal. 37 (2000), no. 4, 1085 – 1100. · Zbl 0959.65126
[3] F. Ben Belgacem, C. Bernardi, N. Chorfi, Y. Maday – Inf-sup conditions for the mortar spectral element discretization of the Stokes problem, Numer. Math. 85 (2000), 257-281.CMP 2000:12
[4] C. Bernardi and V. Girault, A local regularization operator for triangular and quadrilateral finite elements, SIAM J. Numer. Anal. 35 (1998), no. 5, 1893 – 1916. · Zbl 0913.65007
[5] C. Bernardi, Y. Maday – Mesh adaptivity in finite elements by the mortar method, Revue européenne des éléments finis 9 (2000), 451-465.
[6] C. Bernardi, Y. Maday, and A. T. Patera, A new nonconforming approach to domain decomposition: the mortar element method, Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. XI (Paris, 1989 – 1991) Pitman Res. Notes Math. Ser., vol. 299, Longman Sci. Tech., Harlow, 1994, pp. 13 – 51. · Zbl 0797.65094
[7] C. Bernardi, Y. Maday, and A. T. Patera, Domain decomposition by the mortar element method, Asymptotic and numerical methods for partial differential equations with critical parameters (Beaune, 1992) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 384, Kluwer Acad. Publ., Dordrecht, 1993, pp. 269 – 286. · Zbl 0799.65124
[8] C. Bernardi, R.G. Owens, J. Valenciano – An error indicator for mortar element solutions to the Stokes problem, Internal Report 99030, Laboratoire d’Analyse Numérique, Université Pierre et Marie Curie, Paris (1999).
[9] C. Bernardi and R. Verfürth, Adaptive finite element methods for elliptic equations with non-smooth coefficients, Numer. Math. 85 (2000), no. 4, 579 – 608 (English, with English and French summaries). · Zbl 0962.65096
[10] D. Braess and R. Verfürth, A posteriori error estimators for the Raviart-Thomas element, SIAM J. Numer. Anal. 33 (1996), no. 6, 2431 – 2444. · Zbl 0866.65071
[11] F. Bouillault, A. Buffa, Y. Maday, F. Rapetti – The mortar edge element method in three dimensions: application to magnetostatics, Internal Report, Laboratoire d’Analyse Numérique, Université Pierre et Marie Curie, Paris (2000).
[12] P.G. Ciarlet – Basic Error Estimates for Elliptic Problems, in the Handbook of Numerical Analysis, Vol. II, P.G. Ciarlet & J.-L. Lions eds., North-Holland (1991), 17-351. CMP 91:14
[13] Ph. Clément, Approximation by finite element functions using local regularization, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. \jname RAIRO Analyse Numérique 9 (1975), no. R-2, 77 – 84 (English, with Loose French summary). · Zbl 0368.65008
[14] M. Crouzeix – Personal communication.
[15] F. Hecht, O. Pironneau – Multiple meshes and the implementation of freefem+, I.N.R.I.A. Report, Rocquencourt (1999).
[16] P. Joly – Remise en forme, analyse numérique matricielle, Cours de D.E.A., Laboratoire d’Analyse Numérique, Université Pierre et Marie Curie, Paris (1999).
[17] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968 (French). · Zbl 0212.43801
[18] L. Ridgway Scott and Shangyou Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), no. 190, 483 – 493. · Zbl 0696.65007
[19] R. Verfürth, A posteriori error estimators for the Stokes equations. II. Nonconforming discretizations, Numer. Math. 60 (1991), no. 2, 235 – 249. · Zbl 0739.76035
[20] R. Verfürth – A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley & Teubner (1996). · Zbl 0853.65108
[21] R. Verfürth – Error estimates for some quasi-interpolation operators, Modél. Math. et Anal. Numér. 33 (1999), 695-713. · Zbl 0938.65125
[22] O.B. Widlund – An extension theorem for finite element spaces with three applications, in Numerical Techniques in Continuum Mechanics, Proceedings of the Second GAMM Seminar, W. Hackbush & K. Witsch eds., Kiel (1986).
[23] Barbara I. Wohlmuth, A residual based error estimator for mortar finite element discretizations, Numer. Math. 84 (1999), no. 1, 143 – 171. · Zbl 0962.65090
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