A finite volume scheme for a noncoercive elliptic equation with measure data

SIAM Journal on Numerical Analysis

Volumn 41, Issue 6, 2003, Pages 1997-2031

  Droniou, Jérôme ^a   Gallouët, Thierry ^b   Herein, Raphaèle ^b  

^a UNIV MONTPELLIER   (France)

^b CNRS   (France)

Author keywords

Finite volume; Measure data; Noncoercive elliptic equations

Indexed keywords

DATA REDUCTION; DIFFUSION; MATHEMATICAL OPERATORS; NONLINEAR EQUATIONS; PROBLEM SOLVING; SET THEORY;

FINITE VOLUME; MEASURE DATA; NONCOERCIVE ELLIPTIC EQUATIONS; REGULARITY ASSUMPTION;

FINITE VOLUME METHOD;

EID: 0742320132     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142902405205     Document Type: Article

References (29)

1. R.A. ADAMS, Sobolev Spaces, Academic Press, New York, 1975.
2. L. BOCCARDO AND T. GALLOUËT, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87 (1989), pp. 241-273.
3. N without growth restrictions on the data, J. Differential Equations, 105 (1993), pp. 334-363.
4. S. BOIVIN, F. CAYRÉ, AND J.M. HÉRARD, A finite volume method to solve the Navier-Stokes equations for incompressible flows on unstructured meshes, Int. J. Therm. Sci., 39 (2000), pp. 806-825.
5. S.H. CHOU AND P.S. VASSILEVSKI, A general mixed covolume framework for constructing conservative schemes for elliptic problems, Math. Comp., 68 (1999), pp. 991-1011.
6. p error estimates for approximate finite volume solutions of convection diffusion equations, M2AN Math. Model. Numer. Anal., 35 (2001), pp. 767-778.
7. R. EYMARD, T. GALLOUËT, AND R. HERBIN, Finite volume approximation of elliptic problems and convergence of an approximate gradient, Appl. Numer. Math., 37 (2001), pp. 31-53.
8. Y. COUDIÈRE, J.P. VILA, AND P. VILLEDIEU, Convergence rate of a finite volume scheme for a two dimensional convection diffusion problem, M2AN Math. Model. Numer. Anal., 33 (1999), pp. 493-516.
9. J. DRONIOU, Noncoercive linear elliptic problems, Potential Anal., to appear.
10. J. DRONIOU, Etude de Certaines Equations aux Dérivées Partielles, Ph.D. thesis, Centre de Mathématiques et Informatique, Université de Provence, 2001.
11. -1, in Finite Volume for Complex Applications III, R. Herbin and D. Kröner, eds., Hermes Penton Science, London, 2002, pp. 195-202.
12. -1, M2AN Math. Model. Numer. Anal., 36 (2002), pp. 705-724.
13. J. DRONIOU AND T. GALLOUËT, A uniqueness result for quasilinear elliptic equations with measures as data, Rend. Mat. Appl. (7), 21 (2001), pp. 57-86.
14. n. Part 3, Techniques of Scientific Computing, Handb. Numer. Anal. 7, P.G. Ciarlet and J.L. Lions, eds., North-Holland, Amsterdam, pp. 713-1020.
15. R. EYMARD, T. GALLOUËT, AND R. HERBIN, Convergence of finite volume approximations to the solutions of semilinear convection diffusion reaction equations, Numer. Math., 82 (1999), pp. 91-116.
16. R. EYMARD, T. GALLOUËT, R. HERBIN, AND A. MICHEL, Convergence of a finite volume scheme for nonlinear degenerate parabolic equations, Numer. Math., 92 (2002), pp. 41-82.
17. 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. Methods Appl. Mech. Engrg., 115 (1994), pp. 315-338.
18. P.A. FORSYTH AND P.H. SAMMON, Quadratic convergence for cell-centered grids, Appl. Numer. Math., 4 (1988), pp. 377-394.
19. T. GALLOUËT AND R. HERBIN, Finite volume methods for diffusion problems and irregular data, in Finite Volumes for Complex Applications, Problems and Perspectives, II, F. Benkhaldoun, M. Hänel, and R. Vilsmeier, eds., Hermes, Paris, 1999, pp. 155-162.
20. T. GALLOUËT, R. HERBIN, AND M.H. VIGNAL, Error estimates for the approximate finite volume solution of convection diffusion equations with general boundary conditions, SIAM J. Numer. Anal., 37 (2000), pp. 1935-1972.
21. T. GALLOUËT AND A. MONIER, On the regularity of solutions to elliptic equations, Rend. Mat. Appl. (7), 19 (1999), pp. 471-488.
22. P. GRISVARD, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985.
23. R. HERBIN, An error estimate for a finite volume scheme for a diffusion-convection problem on a triangular mesh, Numer. Methods Partial Differential Equations, 11 (1995), pp. 165-173.
24. R. HERBIN, Finite volume approximation of elliptic problems with irregular data, in Finite Volumes for Complex Applications, Problems and Perspectives, F. Benkhaldoun, D. Hanel, and R. Vilsmeier, eds., Hermes, Paris, 1999, pp. 153-160.
25. T. GALLOUËT AND R. HERBIN, Finite volume methods for diffusion convection equations on general meshes, in Finite Volumes for Complex Applications, Problems and Perspectives, F. Benkhaldoun and R. Vilsmeier, eds., Hermes, Paris, 1996, pp. 153-160.
26. R.D. LAZAROV, I.D. MISHEV, AND P.S. VASSILEVSKI, Finite volume methods for convection-diffusion problems, SIAM J. Numer. Anal., 33 (1996), pp. 31-55.
27. T.A. MANTEUFFEL AND A.B. WHITE, The numerical solution of second-order boundary value problems on nonuniform meshes, Math. Comp., 47 (1986), pp. 511-535.
28. p estimate for the gradient of solutions of second order divergence equations, Ann. Scuola Norm. Sup. Pisa, 17 (1963), pp. 189-206.
29. I.D. MISHEV, Finite volume methods on Voronoï meshes, Numer. Methods Partial Differential Equations, 14 (1998), pp. 193-212.