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Numerical analysis of nonlinear elliptic-parabolic equations

Published online by Cambridge University Press:  15 April 2002

Emmanuel Maitre*
Affiliation:
Laboratoire de Mathématiques et Application, Université de Haute-Alsace, 4 rue des frères Lumière, 68093 Mulhouse Cedex, France. E.Maitre@univ-mulhouse.fr.
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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).

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2002

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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).

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