Finite volume schemes for fully non-linear elliptic equations in divergence form

Droniou, Jérôme

ESAIM: Modélisation mathématique et analyse numérique, Tome 40 (2006) no. 6, pp. 1069-1100.

We construct finite volume schemes, on unstructured and irregular grids and in any space dimension, for non-linear elliptic equations of the $p$-laplacian kind: $-div\left(\right|\nabla u{|}^{p-2}\nabla u)=f$ (with $1<p<\infty$). 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.

[1] S. Agmon, A. Douglis and L. Niremberg, 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. | Zbl

[2] B. Andreianov, F. Boyer and F. Hubert, Finite-volume schemes for the $p$-laplacian on cartesian meshes. ESAIM: M2AN 38 (2004) 931-960. | Numdam | Zbl

[3] B. Andreianov, F. Boyer and F. Hubert, Besov regularity and new error estimates for finite volume approximation of the $p$-Laplacian. Numer. Math. 100 (2005) 565-592. | Zbl

[4] B. Andreianov, F. Boyer and F. Hubert, Discrete duality finite volume schemes for Leray-Lions type elliptic problems on general $2$D meshes. Numer. Methods Partial Differ. Equ. 23 (2007) 145-195. | Zbl

[5] J.W. Barrett and W.B. Liu, 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. | Zbl

[6] L. Boccardo, T. Gallouët and F. Murat, Unicité de la solution de certaines équations elliptiques non linéaires. C.R. Acad. Sci. Paris 315 (1992) 1159-1164. | Zbl

[7] 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.

[8] S. Chow, Finite element error estimates for non-linear elliptic equations of monotone type. Numer. Math. 54 (1989) 373-393. | Zbl

[9] Y. Coudiere, J.-P. Vila and P. Villedieu, Convergence rate of a finite volume scheme for a two dimensional convection-diffusion problem. ESAIM: M2AN 33 (1999) 493-516. | Numdam | Zbl

[10] K. Deimling, Nonlinear functional analysis. Springer (1985). | MR | Zbl

[11] J.I. Diaz and F. De Thelin, On a nonlinear parabolic problem arising in some models related to turbulent flows. SIAM J. Math. Anal. 25 (1994) 1085-1111. | Zbl

[12] J. Droniou and R. Eymard, A mixed finite volume scheme for anisotropic diffusion problems on any grid. Num. Math. 105 (2006) 35-71. | Zbl

[13] 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.

[14] J. Droniou and T. Gallouët, Finite volume methods for convection-diffusion equations with right-hand side in $H^{-1}$. ESAIM: M2AN 36 (2002) 705-724. | Numdam | Zbl

[15] 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). | Zbl

[16] M. Feistauer and A. Ženíšek, Finite element solution of nonlinear elliptic problems. Numer. Math. 50 (1987) 451-475. | Zbl

[17] M. Feistauer and A. Ženíšek, Compactness method in the finite element theory of nonlinear elliptic problems. Numer. Math. 52 (1988) 147-163. | Zbl

[18] M. Feistauer and V. Sobotíková, Finite element approximation of nonlinear elliptic problems with discontinuous coefficients. RAIRO Modél. Math. Anal. Numér. 24 (1990) 457-500. | Numdam | Zbl

[19] J.M. Fiard and R. Herbin, 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.

[20] R. Glowinski, Numerical methods for nonlinear variational problems. Springer (1984). | Zbl

[21] R. Glowinski and J. Rappaz, Approximation of a nonlinear elliptic problem arising in a non-newtonian fluid flow model in glaciology. ESAIM: M2AN 37 (2003) 175-186. | Numdam | Zbl

[22] J. Leray and J.L. Lions, 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. | Numdam | Zbl

[23] E.M. Stein, Singular Integrals and Differentiability Properties of Functions. Princetown University Press (1970). | MR | Zbl

[24] A. Ženíšek, The finite element method for nonlinear elliptic equations with discontinuous coefficients. Numer. Math. 58 (1990) 51-77. | Zbl