Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-23T20:02:13.237Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite volume schemes for fully non-linear elliptic equationsin divergence form

Published online by Cambridge University Press:  15 February 2007

Jérôme Droniou
Jérôme Droniou*
Affiliation:
Département de Mathématiques, UMR CNRS 5149, CC 051, Université Montpellier II, Place Eugène Bataillon, 34095 Montpellier cedex 5, France. [email protected]
Get access

Abstract

We construct finite volume schemes, on unstructured and irregular grids andin any space dimension, for non-linear elliptic equations of the p-Laplacian kind: -div(|∇u|p-2 ∇u) = ƒ(with 1 < p < ∞). We prove the existence and uniqueness of the approximate solutions,as well as their strong convergence towards the solution of the PDE.The outcome of some numerical tests are also provided.

Keywords

Finite volume schemesirregular gridsnon-linear elliptic equationsLeray-Lions operators.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 40 , Issue 6 , November 2006 , pp. 1069 - 1100
Copyright
© EDP Sciences, SMAI, 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Agmon, S., Douglis, A. and Niremberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, Part I and Part II. Comm. Pure. Appl. Math. 12 (1959) 623–727 and 17 (1964) 35–92. CrossRef
Andreianov, B., Boyer, F. and Hubert, F., Finite-volume schemes for the p-laplacian on cartesian meshes. ESAIM: M2AN 38 (2004) 931–960. CrossRef
Andreianov, B., Boyer, F. and Hubert, F., Besov regularity and new error estimates for finite volume approximation of the p-Laplacian. Numer. Math. 100 (2005) 565–592. CrossRef
Andreianov, B., Boyer, F. and Hubert, F., Discrete duality finite volume schemes for Leray-Lions type elliptic problems on general 2D meshes. Numer. Methods Partial Differ. Equ. 23 (2007) 145–195. CrossRef
Barrett, J.W. and Liu, W.B., A remark on the regularity of the solutions of the p-Laplacian and its application to the finite element approximation. J. Math. Anal. Appl. 178 (1993) 470–487.
Boccardo, L., Gallouët, T. and Murat, F., Unicité de la solution de certaines équations elliptiques non linéaires. C.R. Acad. Sci. Paris 315 (1992) 1159–1164.
C. Chainais and J. Droniou, Convergence analysis of a mixed finite volume scheme for an elliptic-parabolic system modeling miscible fluid flows in porous media, submitted. Available at http://hal.ccsd.cnrs.fr/ccsd-00022910.
Chow, S., Finite element error estimates for non-linear elliptic equations of monotone type. Numer. Math. 54 (1989) 373–393. CrossRef
Coudiere, Y., Vila, J.-P. and Villedieu, P., Convergence rate of a finite volume scheme for a two dimensional convection-diffusion problem. ESAIM: M2AN 33 (1999) 493–516. CrossRef
K. Deimling, Nonlinear functional analysis. Springer (1985).
Diaz, J.I. and de Thelin, F., On a nonlinear parabolic problem arising in some models related to turbulent flows. SIAM J. Math. Anal. 25 (1994) 1085–1111. CrossRef
Droniou, J. and Eymard, R., A mixed finite volume scheme for anisotropic diffusion problems on any grid. Num. Math. 105 (2006) 35–71. CrossRef
J. Droniou and R. Eymard, Study of the mixed finite volume method for Stokes and Navier-Stokes equations, submitted. Available at http://hal.archives-ouvertes.fr/hal-00110911.
Droniou, J. and Gallouët, T., Finite volume methods for convection-diffusion equations with right-hand side in H -1. ESAIM: M2AN 36 (2002) 705–724. CrossRef
R. Eymard, T. Gallouët and R. Herbin, Finite Volume Methods, Handbook of Numerical Analysis, P.G. Ciarlet and J.L. Lions Eds., Vol. VII, 713–1020 (North Holland).
Feistauer, M. and A. Ženíšek, Finite element solution of nonlinear elliptic problems. Numer. Math. 50 (1987) 451–475. CrossRef
Feistauer, M. and A. Ženíšek, Compactness method in the finite element theory of nonlinear elliptic problems. Numer. Math. 52 (1988) 147–163. CrossRef
Feistauer, M. and Sobotíková, V., Finite element approximation of nonlinear elliptic problems with discontinuous coefficients. RAIRO Modél. Math. Anal. Numér. 24 (1990) 457–500. CrossRef
Fiard, J.M. and Herbin, R., Comparison between finite volume finite element methods for the numerical simulation of an elliptic problem arising in electrochemical engineering. Comput. Meth. Appl. Mech. Engin. 115 (1994) 315–338. CrossRef
R. Glowinski, Numerical methods for nonlinear variational problems. Springer (1984).
Glowinski, R. and Rappaz, J., Approximation of a nonlinear elliptic problem arising in a non-newtonian fluid flow model in glaciology. ESAIM: M2AN 37 (2003) 175–186. CrossRef
Leray, J. and Lions, J.L., Quelques résultats de Višik sur les problèmes elliptiques semi-linéaires par les méthodes de Minty et Browder. Bull. Soc. Math. France 93 (1965) 97–107. CrossRef
E.M. Stein, Singular Integrals and Differentiability Properties of Functions. Princetown University Press (1970).
Ženíšek, A., The finite element method for nonlinear elliptic equations with discontinuous coefficients. Numer. Math. 58 (1990) 51–77. CrossRef