Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-30T21:18:28.475Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical analysis of nonlinear elliptic-parabolic equations

Published online by Cambridge University Press:  15 April 2002

Emmanuel Maitre
Emmanuel Maitre*
Affiliation:
Laboratoire de Mathématiques et Application, Université de Haute-Alsace, 4 rue des frères Lumière, 68093 Mulhouse Cedex, France. [email protected].
Get access

Abstract

This paper deals with the numerical approximation of mild solutions of elliptic-parabolic equations, relying on the existence results of Bénilan and Wittbold (1996). We introduce a new and simple algorithm based on Halpern's iteration for nonexpansive operators (Bauschke, 1996; Halpern, 1967; Lions, 1977), which is shown to be convergent in the degenerate case, and compare it with existing schemes (Jäger and Kačur, 1995; Kačur, 1999).

Keywords

Elliptic-parabolicnumericaliterative method.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 36 , Issue 1 , January 2002 , pp. 143 - 153
Copyright
© EDP Sciences, SMAI, 2002

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

Alt, H.W. and Luckhaus, S., Quasilinear Elliptic-Parabolic Differential Equations. Math. Z. 183 (1983) 311-341.
Bauschke, H., The approximation of fixed points of composition of nonexpansive mappings in Hilbert space. J. Math. Anal. Appl. 202 (1996) 150-159. CrossRef
Bénilan, Ph. and Ha, K., Equation d'évolution du type $({\rm d}u/{\rm d}t)+\beta\partial\varphi(u)\ni 0$ dans L ∞(Ω). C.R. Acad. Sci. Paris Sér. A 281 (1975) 947-950.
Berger, A., Brézis, H. and Rogers, J., A numerical method for solving the problem $u_t-\Delta f(u)=0$ . RAIRO Anal. Numér. 13 (1979) 297-312. CrossRef
Bénilan, Ph. and Wittbold, P., On mild and weak solutions of elliptic-parabolic problems. Adv. Differential Equations 1 (1996) 1053-1073.
Bénilan, Ph. and Wittbold, P., Sur un problème parabolique-elliptique. ESAIM: M2AN 33 (1999) 121-127 . CrossRef
P. Colli, On Some Doubly Nonlinear Evolution Equations in Banach Spaces. Technical Report 775, Università di Pavia, Istituto di Analisi Numerica (1991).
Colli, P. and Visintin, A., On a class of doubly nonlinear evolution equations. Comm. Partial Differential Equations 15 (1990) 737-756. CrossRef
Halpern, B., Fixed points of nonexpansive mappings. Bull. Amer. Math. Soc. 73 (1967) 957-961. CrossRef
Jäger, W. and Kacur, J., Solution of Porous Medium Type Systems by Linear Approximation Schemes. Numer. Math. 60 (1991) 407-427. CrossRef
Jäger, W. and Kacur, J., Solution of Doubly Nonlinear and Degenerate Parabolic Problems by Relaxation Schemes. RAIRO Modél. Math. Anal. Numér. 29 (1995) 605-627. CrossRef
Kacur, J., Solution of Some Free Boundary Problems by Relaxation Schemes. SIAM J. Numer. Anal. 36 (1999) 290-316. CrossRef
Kacur, J., Handlovicová, A. and Kacurová, M., Solution of Nonlinear Diffusion Problems by Linear Approximation Schemes. SIAM J. Numer. Anal. 30 (1993) 1703-1722. CrossRef
J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Dunod (1969).
Lions, P.-L., Approximation de points fixes de contractions. C.R. Acad. Sci. Paris Sér. A. 284 (1977) 1357-1359.
Magenes, E., Nochetto, R.H. and Verdi, C., Energy Error Estimates for a Linear Scheme to Approximate Nonlinear Parabolic Problems. RAIRO Modél. Math. Anal. Numér. 21 (1987) 655-678. CrossRef
E. Maitre, Sur une classe d'équations à double non linéarité : application à la simulation numérique d'un écoulement visqueux compressible. Thèse, Université Grenoble I (1997).
E. Maitre and P. Witomski, A pseudomonotonicity adapted to doubly nonlinear elliptic-parabolic equations. Nonlinear Anal. TMA (to appear).
Otto, F., L 1-Contraction and Uniqueness for Quasilinear Elliptic-Parabolic Equations. J. Differential Equations 131 (1996) 20-38. CrossRef
F. Simondon, Sur l'équation $b(u)_t-a(u,\nabla u)=0$ par la méthode des semi-groupes dans L 1. Séminaire d'analyse non linéaire, Laboratoire de Mathématiques de Besançon (1984).