A posteriori error estimation for the dual mixed finite element method for the \(p\)-Laplacian in a polygonal domain. (English) Zbl 1173.65345
Summary: For the discrete solution of the dual mixed formulation for the \(p\)-Laplace equation, we define two residues \(R\) and \(r\). Then we bound the norm of the errors on the two unknowns in terms of the norms of these two residues. Afterwards, we bound the norms of these two residues by functions of two error estimators whose expressions involve at the very most the datum and the computed quantities. We next explain how the discretized dual mixed formulation is hybridized and solved. We close our paper by numerical tests for \(p=1.8\) and \(p=3\) firstly to corroborate the orders of convergence established by M. Farhloul and H. Manouzi [Can. Appl. Math. Q. 8, No. 1, 67–78 (2000; Zbl 0982.65126)], and secondly to experimentally verify the reliability of our a posteriori error estimates.
MSC:
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |
| 35J60 | Nonlinear elliptic equations |
Citations:
Zbl 0982.65126References:
| [1] | Farhloul, M.; Manouzi, H., On a mixed finite element method for the \(p\)-Laplacian, Canad. Appl. Math. Quarterly, 8, 67-78 (2000) · Zbl 0982.65126 |
| [2] | J.M. Thomas, Sur l’analyse numérique des méthodes d’Éléments finis hybrides et mixtes, Thèse de l’université Pierre et Marie Curie, Paris, 1977.; J.M. Thomas, Sur l’analyse numérique des méthodes d’Éléments finis hybrides et mixtes, Thèse de l’université Pierre et Marie Curie, Paris, 1977. |
| [3] | Raviart, P. A.; Thomas, J. M., A mixed finite element method for second order elliptic problems, (Galligani, I.; Magenes, E., Mathematical Aspects of Finite Element Methods. Mathematical Aspects of Finite Element Methods, Lecture Notes in Mathematics, vol. 606 (1977), Springer-Verlag), 292-315 · Zbl 0362.65089 |
| [4] | Ciarlet, P. G., The Finite Element Method for Elliptic Problems (1978), North-Holland: North-Holland Amsterdam · Zbl 0445.73043 |
| [5] | Sandri, D., Sur l’approximation numérique des écoulements quasi-Newtoniens dont la viscosité suit la loi puissance ou la loi de Carreau, \(M^2\) AN, 27, 131-155 (1993) · Zbl 0764.76039 |
| [6] | Farhloul, M., A mixed finite element method for a nonlinear Dirichlet problem, IMA Journal of Numerical Analysis, 18, 121-132 (1998) · Zbl 0909.65086 |
| [7] | Grisvard, P., Elliptic Problems in Nonsmooth Domains. Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, vol. 24 (1985), Pitman · Zbl 0695.35060 |
| [8] | Girault, V.; Raviart, P. A., Finite Element Methods for Navier-Stokes Equations. Theory and algorithms, SCM5 (1986), Springer-Verlag · Zbl 0585.65077 |
| [9] | Dautray, R.; Lions, J.-L., Analyse mathématique et calcul numérique pour les sciences et les techniques, Transformations, Sobolev, Opérateurs, CEA, INSTN, Collection Enseignement, vol. 3 (1988), Masson |
| [10] | Z. Mghazli, Une méthode mixte pour les problémes d’hydrodynamique, Thése de l’université de Montréal, 1987.; Z. Mghazli, Une méthode mixte pour les problémes d’hydrodynamique, Thése de l’université de Montréal, 1987. |
| [11] | El Sossa, H.; Paquet, L., Refined mixed finite element method for the Poisson problem on a polygonal domain with a reentrant corner, Adv. Math. Sci. Appl., Gakkotosho, Tokyo, 12, 607-645 (2002) · Zbl 1039.65080 |
| [12] | Verfürth, R., A Review of a posteriori Error Estimation and Adaptive Mesh-Refinement Techniques (1996), Wiley and Teubner · Zbl 0853.65108 |
| [13] | J. Dieudonné, Eléments d’Analyse, tome II, Gauthier-Villars, Paris, Cahiers Scientifiques, Fascicule XXXI, 1969.; J. Dieudonné, Eléments d’Analyse, tome II, Gauthier-Villars, Paris, Cahiers Scientifiques, Fascicule XXXI, 1969. |
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