Estimations a posteriori d’un schéma de volumes finis pour un problème non linéaire

Summary.RésuméDans cet article, nous développons des estimations a posteriori de l’erreur commise en approchant un problème elliptique non linéaire, par une Méthode de Volumes Finis (MVF) de type ‚‚Vertex-Centered’’. Les estimateurs obtenus mesurent, le résidu de l’équation forte et l’irrégularité de la solution discrète, qui se traduit par des sauts à travers les inter-élélments. Une condition sur la fonction flux numérique sera imposée pour traiter la non conformité du problème, au cours de l’évaluation de l’estimation. Summary.We derive a posteriori error estimates for vertex-centered finite volume discretizations of a class of non-linear elliptic pdes. The error estimates are of residual type and incorporate residuals of the pde in its strong form on the control volumes and jumps of the pde’s prinicipal part across the element boundaries. The non-conformity of the discretization is taken into account by a condition on the numerical flux function.

[1]  E. Süli,et al.  A dual graph-norm refinement indicator for finite volume approximations of the Euler equations , 1998 .

[2]  R. Verfürth.  A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations , 1994 .

[3]  Abdellatif Agouzal,et al.  Un nouveau résultat d'estimation d'erreur pour les éléments finis mixtes rectangulaires avec intégration numérique. Application à l'analyse de schémas de type volumes finis , 1996 .

[4]  Rüdiger Verfürth.  A Posteriori Error Estimators and Adaptive Mesh-Refinement for a Mixed Finite Element Discretization of the Navier-Stokes Equations , 1990 .

[5]  Ricardo G. Durán,et al.  On the asymptotic exactness of error estimators for linear triangular finite elements , 1991 .

[6]  Abdellatif Agouzal,et al.  Estimateur d'erreur a posteriori hiérarchique. Application aux éléments finis mixtes , 1998, Numerische Mathematik.

[7]  M. Ohlberger,et al.  A posteriori error estimate for finite volume approximations to singularly perturbed nonlinear convection-diffusion equations , 2001, Numerische Mathematik.

[8]  Randolph E. Bank,et al.  A posteriori error estimates based on hierarchical bases , 1993 .

[9]  R. Bank,et al.  Some a posteriori error estimators for elliptic partial differential equations , 1985 .

[10]  R. Verfürth.  A posteriori error estimators for the Stokes equations , 1989 .

[11]  Barbara I. Wohlmuth,et al.  A comparison of a posteriori error estimators for mixed finite element discretizations by Raviart-Thomas elements , 1999, Math. Comput..

[12]  W. Rheinboldt,et al.  Error Estimates for Adaptive Finite Element Computations , 1978 .

[13]  Ivo Babuška,et al.  Analysis of the efficiency of an a posteriori error estimator for linear triangular finite elements , 1992 .

[14]  Randolph E. Bank,et al.  A posteriori error estimates for the Stokes equations: a comparison , 1990 .

[15]  R. Verfürth.  A posteriori error estimates for nonlinear problems: finite element discretizations of elliptic equations , 1994 .

[16]  P. Raviart,et al.  Numerical Approximation of Hyperbolic Systems of Conservation Laws , 1996, Applied Mathematical Sciences.

[17]  J. Craggs.  Applied Mathematical Sciences , 1973 .

[18]  Ricardo G. Durán,et al.  On the asymptotic exactness of Bank-Weiser's estimator , 1992 .

[19]  Randolph E. Bank,et al.  A posteriori error estimates for the Stokes problem , 1991 .

[20]  I. Babuska,et al.  A‐posteriori error estimates for the finite element method , 1978 .

[21]  P. Clément.  Approximation by finite element functions using local regularization , 1975 .

[22]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[23]  Jacques Rappaz,et al.  Error estimates and adaptive finite elements for nonlinear diffusion-convection problems , 1996 .

[24]  Rüdiger Verfürth,et al.  A posteriori error estimation and adaptive mesh-refinement techniques , 1994 .

[25]  R. Eymard,et al.  Finite Volume Methods , 2019, Computational Methods for Fluid Dynamics.

[26]  Z. Mghazli,et al.  An Adaptive Method for Characteristics-Finite Element Method for Solute Transport Equation in Unsaturated Porous Media , 2000 .