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

Droniou, Jérôme

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 40 (2006), p. 1069-1100 / Harvested from Numdam

Access to full text
Full (PDF)
Full (DJVU)

Résumé

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: $- div (| \nabla u |^{p - 2} \nabla u) = f$ (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.

Publié le : 2006-01-01

MR 2297105 | Zbl 1117.65154

DOI : https://doi.org/10.1051/m2an:2007001
Classification: 65N12, 35J65, 65N30

@article{M2AN_2006__40_6_1069_0,
     author = {Droniou, J\'er\^ome},
     title = {Finite volume schemes for fully non-linear elliptic equations in divergence form},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {40},
     year = {2006},
     pages = {1069-1100},
     doi = {10.1051/m2an:2007001},
     mrnumber = {2297105},
     zbl = {1117.65154},
     language = {en},
     url = {http://dml.mathdoc.fr/item/M2AN_2006__40_6_1069_0}
}

Droniou, Jérôme. Finite volume schemes for fully non-linear elliptic equations in divergence form. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 40 (2006) pp. 1069-1100. doi : 10.1051/m2an:2007001. http://gdmltest.u-ga.fr/item/M2AN_2006__40_6_1069_0/

Bibliographie

[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 0093.10401

[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 1081.65105

[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 1106.65098

[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 1111.65101

[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 0799.35085

[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 0789.35056

[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 0643.65058

[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 0937.65116

[10] K. Deimling, Nonlinear functional analysis. Springer (1985). | MR 787404 | Zbl 0559.47040

[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 0808.35066

[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 1109.65099

[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 1070.65566

[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 0981.65095

[16] M. Feistauer and A. Ženíšek, Finite element solution of nonlinear elliptic problems. Numer. Math. 50 (1987) 451-475. | Zbl 0637.65107

[17] M. Feistauer and A. Ženíšek, Compactness method in the finite element theory of nonlinear elliptic problems. Numer. Math. 52 (1988) 147-163. | Zbl 0642.65075

[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 0712.65097

[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 0536.65054

[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 1046.76002

[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 0132.10502

[23] E.M. Stein, Singular Integrals and Differentiability Properties of Functions. Princetown University Press (1970). | MR 290095 | Zbl 0207.13501

[24] A. Ženíšek, The finite element method for nonlinear elliptic equations with discontinuous coefficients. Numer. Math. 58 (1990) 51-77. | Zbl 0709.65081