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

Faker Ben Belgacem; Padmanabhan Seshaiyer; Manil Suri

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 34, Issue: 3, page 591-608
  • ISSN: 0764-583X

Abstract

top
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.

How to cite

top

Ben Belgacem, Faker, Seshaiyer, Padmanabhan, and Suri, Manil. "Optimal convergence rates of hp mortar finite element methods for second-order elliptic problems." ESAIM: Mathematical Modelling and Numerical Analysis 34.3 (2010): 591-608. <http://eudml.org/doc/197446>.

@article{BenBelgacem2010,
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. },
author = {Ben Belgacem, Faker, Seshaiyer, Padmanabhan, Suri, Manil},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Non conforming; mortar method; hp finite elements; optimal convergence; mortar finite element methods; second-order elliptic problems; error estimate; non-conforming finite element method},
language = {eng},
month = {3},
number = {3},
pages = {591-608},
publisher = {EDP Sciences},
title = {Optimal convergence rates of hp mortar finite element methods for second-order elliptic problems},
url = {http://eudml.org/doc/197446},
volume = {34},
year = {2010},
}

TY - JOUR
AU - Ben Belgacem, Faker
AU - Seshaiyer, Padmanabhan
AU - Suri, Manil
TI - Optimal convergence rates of hp mortar finite element methods for second-order elliptic problems
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 34
IS - 3
SP - 591
EP - 608
AB - 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.
LA - eng
KW - Non conforming; mortar method; hp finite elements; optimal convergence; mortar finite element methods; second-order elliptic problems; error estimate; non-conforming finite element method
UR - http://eudml.org/doc/197446
ER -

References

top
  1. 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.  
  2. Y. Achdou, Y. Maday and O.B. Widlund, Iterative substructuring preconditioners for the mortar finite element method in two dimensions. SIAM J. Num. Anal.36 (1999) 551-580.  
  3. Y. Achdou and O. Pironneau, 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.  
  4. I. Babuska and M. Suri, The h-p-version of the finite element method with quasi-uniform meshes. Modél. Math. et Anal. Numér.21 (1987) 199-238.  
  5. I. Babuska and M. Suri, The p and h-p-versions of the finite element method: basic principles and properties. SIAM Review36 (1984) 578-632.  
  6. I. Babuska and M. Suri, The optimal convergence rate of the p-Version of the finite element method. SIAM J. Num. Anal.24 (1987) 750-776.  
  7. 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).  
  8. F. Ben Belgacem, The mortar finite element method with Lagrange multipliers. Num. Mathematik (to appear).  
  9. F. Ben Belgacem and Y. Maday, Non conforming spectral element methodology tuned to parallel implementation. Compu. Meth. Appl. Mech. Eng.116 (1994) 59-67.  
  10. C. Bernardi, N. Débit and Y. Maday, Coupling finite element and spectral methods: first results. Math. Compu.54 (1990), 21-39.  
  11. 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 Mathematics167 Dekker (1994) 27-46.  
  12. 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.  
  13. C. Bernardi and Y. Maday, Relèvement de traces polynomiales et applications. RAIRO Modél. Math. Anal. Numér.24 (1990) 557-611.  
  14. 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).  
  15. C. Bernardi, Y. Maday and G. Sacchi-Landriani, Non conforming matching conditions for coupling spectral and finite element methods. Appl. Numer. Math.54 (1989) 64-84.  
  16. A. Berger, R. Scott and G. Strang, Approximate boundary conditions in the finite element method. Symposia Mathematica10 (1972) 295-313.  
  17. S. Brenner, A non-standard finite element interpolation estimate. Research Report 1998:07, Department of Mathematics, University of South Carolina (1998).  
  18. P.-G. Ciarlet, The finite element Method for Elliptic Problems. North Holland (1978).  
  19. 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).  
  20. 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).  
  21. V. Girault and P.-A. Raviart, Finite element methods for Navier-Stokes equations. Springer Verlag (1986).  
  22. P. Grisvard, Elliptic problems in nonsmooth domains. Monographs and Studies in Mathematics24 (Pitman, 1985).  
  23. W. Gui and I. Babuska, The h-p-version of the finite element method in one dimension. Num. Mathematik3 (1986) 577-657.  
  24. 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.  
  25. P. Seshaiyer, Non-Conforming h-p finite element methods. Doctoral Thesis, University of Maryland Baltimore County (1998).  
  26. 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).  
  27. 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.  
  28. P. Seshaiyer and M. Suri, h-p submeshing via non-conforming finite element methods. Submitted to Compu. Meth. Appl. Mech. Eng. (1998).  
  29. G. Strang and G. J. Fix, An analysis of the finite element method. Wellesly, Cambridge Press Masson (1973).  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.