Isotropic and anisotropic a posteriori error estimation of the mixed finite element method for second order operators in divergence form

Electronic Transactions on Numerical Analysis

Volumn 23, Issue , 2006, Pages 38-62

  Nicaise, Serge ^a   Creusé, Emmanuel ^a  

^a UNIVERSITÉ DE VALENCIENNES   (France)

Author keywords

Anisotropic solution; Error estimator; Mixed FEM; Stretched elements

Indexed keywords

ANISOTROPY; ESTIMATION; FINITE ELEMENT METHOD; MATHEMATICAL OPERATORS; PROBLEM SOLVING; RELIABILITY;

ANISOTROPIC SOLUTION; ERROR ESTIMATORS; MIXED FEM; STRETCHED ELEMENTS;

ERROR ANALYSIS;

EID: 33846230937     PISSN: None     EISSN: 10689613     Source Type: Journal    
DOI: None     Document Type: Article

References (32)

1. G. ACOSTA AND R. G. DURÁN, The maximum angle condition for mixed and non-conforming elements,Application to the Stokes equations, SIAM J. Numer. Anal, 37 (1999), pp. 18-36.
2. A. ALONSO, Error estimators for mixed methods, Numer. Math., 74 (1996), pp. 385-395.
3. T. APEL, Anisotropic Finite Elements: Local Estimates and Applications, Adv. Numer. Math, (1999).
4. T. APEL AND S. NICAISE, The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges, Math. Methods Appl. Sci., 21 (1998), pp. 519-549.
5. T. APEL, S. NICAISE AND J. SCHöBERL, Crouzeix-Raviart type finite elements on anisotropic meshes, Numer. Math., 89 (2001), pp. 193-223.
6. D. BRAESS AND R. VERFÜRTH, A posteriori error estimators for the Raviart-Thomas element, SIAM J, Numer, Anal., 33 (1996), pp. 2431-2444.
7. F. BREZZI AND M. FORTIN, Mixed and Hybrid Finite Element Methods, Springer, New York, 1991.
8. C. CARSTENSEN, A posteriori error estimate for the mixed finite element method, Math. Comp., 66 (1997), pp. 465-476.
9. C. CARSTENSEN AND G. DOLZMANN, A posteriori error estimates for mixed fern in elasticity, Numer. Math., 81 (1998), pp. 187-209.
10. P. G. CIARLET, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
11. M. COSTABEL, M. DAUGE AND S. NICAISE, Singularities of Maxwell interface problems, RAIRO Model. Math. Anal. Numér., 33 (1999), pp. 627-649.
12. E. CREUSÉ, G. KUNERT AND S. NICAISE, A posteriori error estimation for the Stokes problem: Anisotropic and Isotropic discretizations, Math. Models Methods Appl. Sci., 14 (2004), pp. 1297-1341.
13. M. DAUGE, Elliptic boundary value problems on corner domains - smoothness and asymptotics of solutions, Lecture Notes in Math., Vol. 1341, Springer, Berlin, 1988.
14. M. DOBROWOLSKl, Numerical approximation of elliptic interface and corner problems, Bonn, Habilitationsscrift, 1981.
15. M. FARHLOUL, S. NICAISE AND L. PAQUET, Some mixed finite element methods on anisotropic meshes, RAIRO Modél. Math. Anal. Numér., 35 (2001), pp. 907-920.
16. L. FORMAGGIA AND S. PEROTTO, New anisotropic a priori error estimates, Numer. Math., 89 (2001), pp. 641-667.
17. V. GIRAULT AND P.-A. RAVIART, Finite element methods for Navier-Stokes equations, Springer Ser. Comput. Math. Vol. 5, Springer, Berlin, 1986.
18. G. KUNERT, A posteriori error estimation for anisotropic tetrahedral and triangular finite element meshes, Logos Verlag, Berlin, 1999. Also PhD thesis, TU Chemnitz, http://archiv.tu-chemnitz.de/pub/1999/0012/ index.html.
19. G. KUNERT, Towards anisotropic mesh construction and error estimation in the finite element method, Numer. Methods Partial Differential Equations, 18 (2002), pp. 625-648.
20. G. KUNERT AND R. VERFÜRTH, Edge residuals dominate a posteriori error estimates for linear finite element methods on anisotropic triangular and tetrahedral meshes, Numer. Math., 86 (2000), pp. 283-303.
21. M. KŘÍŽEK, On the maximum angle condition for linear tetrahedral elements, SIAM J. Numer. Anal., 29 (1992), pp. 513-520.
22. D. LEGUILLON AND E. SANCHEZ-PALENCIA, Computation of singular solutions in elliptic problems and elasticity, RMA 5. Masson, Paris, 1991.
23. J.-L. LIONS, Perturbations singulières dans les problèmes aux limites et en contrôle optimal, Lecture Notes in Math., Vol. 323, Springer, Berlin, 1973.
24. J. M. MELENK, hp-Finite Element Methods for Singular Perturbations, Lecture Notes in Math., Vol. 1796, Springer, Berlin, 2002.
25. S. NICAISE, Polygonal Interface Problems, Peter Lang Verlag, Berlin, 1993.
26. S. NICAISE AND A.-M. SÄNDIG, General interface problems I, II, Math. Methods Appl. Sci., 17 (1994), pp. 395-450.
27. P. A. RAVIART AND J. M. THOMAS, A mixed finite element method for second order elliptic problems, In I. Galligani and E. Magenes, eds., Mathematical Aspects of Finite Element Methods, Lecture Notes in Math., Vol. 606, pp. 292-315, Springer, Berlin, 1977.
28. J. E. ROBERTS AND J. M. THOMAS, Mixed and Hybrid Methods, pp. 523-639, North-Holland, Amsterdam, 1991.
29. K. G. SIEBERT, An a posteriori error estimator for anisotropic refinement, Numer. Math., 73 (1996), pp. 373398.
30. H. E. SOSSA AND L. PAQUET, Refined mixed finite element method of the Dirichlet problem for the Laplace equation in a polygonal domain, Adv. Math. Sci. Appl., 12 (2002), pp. 607-643.
31. J. M. THOMAS, Sur l'analyse numérique des méthodes d'élements finis mixtes et hybrides, PhD thesis (Thèse d'Etat), Université Pierre et Marie Curie, Paris, 1977.
32. R. VERFÜRTH, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley and Teubner, Chichester and Stuttgart, 1996.