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Finite element approximation for degenerate parabolic equations. an application of nonlinear semigroup theory

Published online by Cambridge University Press:  15 August 2005

Akira Mizutani ,
Norikazu Saito  and
Takashi Suzuki
Akira Mizutani
Affiliation:
Department of Mathematics, Faculty of Science, Gakushuin University, 1-5-1 Mejiro, Toshima-ku, Tokyo 171-8588, Japan.
Norikazu Saito
Affiliation:
Faculty of Education, Toyama University, 3190 Gofuku, Toyama 930-8555, Japan. saito@edu.toyama-u.ac.jp
Takashi Suzuki
Affiliation:
Division of Mathematical Science, Department of System Innovation, Graduate School of Engineering Science, Osaka University, 1-1 Machikaneyama, Toyonaka, 560-0043, Japan.
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Abstract

Finite element approximation for degenerate parabolic equations is considered. We propose a semidiscrete scheme provided with order-preserving and L1 contraction properties, making use of piecewise linear trial functions and the lumping mass technique. Those properties allow us to apply nonlinear semigroup theory, and the wellposedness and stability in L1 and L∞, respectively, of the scheme are established. Under certain hypotheses on the data, we also derive L1 convergence without any convergence rate. The validity of theoretical results is confirmed by numerical examples.

Keywords

Finite element methoddegenerate parabolic equationnonlinear semigroup.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 39 , Issue 4 , July 2005 , pp. 755 - 780
Copyright
© EDP Sciences, SMAI, 2005

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References

R.A. Adams, Sobolev Spaces. Academic Press, New York, London (1975).
Bénilan, P., Crandall, M.G. and Sacks, P., Some L 1 existence and dependence results for semilinear elliptic equations under nonlinear boundary conditions. Appl. Math. Optim. 17 (1988) 203–224. CrossRef
Berger, A.E., Brezis, H. and Rogers, J.C.W, A numerical method for solving the problem ut - Δƒ(u) = 0. RAIRO Anal. Numer. 13 (1979) 297–312. CrossRef
S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer (1994).
Brezis, H. and Pazy, A., Convergence and approximation of semigroups of nonlinear operators in Banach spaces. J. Funct. Anal. 9 (1972) 63–74. CrossRef
Brezis, H. and Strauss, W., Semi-linear second-order elliptic equations in L 1. J. Math. Soc. Japan 25 (1973) 565–590. CrossRef
P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North Holland, Amsterdam (1978).
P.G. Ciarlet, Basic Error Estimates for Elliptic Problems, in Finite Element Methods (Part 1), P.G. Ciarlet and J.L. Lions Eds., Handbook of Numerical Analysis, 17–351, Elsevier Science Publishers B.V., Amsterdam (1991).
Ciarlet, P.G. and Raviart, P.A., Maximum principle and uniform convergence for the finite element method. Comput. Methods Appl. Mech. Engrg. 2 (1973) 17–31. CrossRef
Ciavaldini, J.F., Analyse numérique d'un problème de Stefan à deux phases par une méthode d'éléments finis. SIAM J. Numer. Anal. 12 (1975) 464–487. CrossRef
Cockburn, B. and Gripenberg, G., Continuous dependence on the nonlinearities of solutions of degenerate parabolic equations. J. Differential Equations 151 (1999) 231–251. CrossRef
Crandall, M.G. and Liggett, T., Generation of semi-groups of nonlinear transformations on general Banach spaces. Amer. J. Math. 93 (1971) 265–293. CrossRef
Elliott, C.M., Error analysis of the enthalpy method for the Stefan problem. IMA J. Numer. Anal. 7 (1987) 61–71. CrossRef
C.M. Elliott and J.R. Ockendon, Weak and Variational Methods for Moving Boundary Problems. Pitman, Boston. Res. Notes Math. 59 (1982).
A. Friedman, Variational Principles and Free-Boundary Problems. Wiley, New York (1982).
H. Fujii, Some remarks on finite element analysis of time-dependent field problems, in Theory and Practice in Finite Element Structural Analysis, University of Tokyo Press, Tokyo (1973) 91–106.
H. Fujita, N. Saito and T. Suzuki, Operator Theory and Numerical Methods. North-Holland, Amsterdam (2001).
Gilding, B.H. and Peletier, L.A., On a class of similarity solutions of the porous media equation. J. Math. Anal. Appl. 55 (1976) 351–364. CrossRef
P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985).
Jäger, W. and Kačur, J., Solution of doubly nonlinear and degenerate parabolic problems by relaxation schemes. RAIRO Modél. Math. Anal. Numér. 29 (1995) 605–627. CrossRef
J. Kačur, A. Handlovicová and M. Kacurová, Solution of nonlinear diffusion problems by linear approximation schemes. SIAM J. Numer. Anal. 30 1703-1722 (1993).
Kato, T., Schrödinger operators with singular potentials. Israel J. Math. 13 (1972) 135–148. CrossRef
Le Roux, M.N., Semi-discretization in time for a fast diffusion equation. J. Math. Anal. Appl. 137 (1989) 354–370. CrossRef
Le Roux, M.N. and Mainge, P.E., Numerical solution of a fast diffusion equation. Math. Comp. 68 (1999) 461–485. CrossRef
Lesaint, P. and Pousin, J., Error estimates for a nonlinear degenerate parabolic equation. Math. Comp. 59 (1992) 339–358. CrossRef
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. Magenes, C. Verdi and A. Visintin, Semigroup approach to the Stefan problem with non-linear flux. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 75 (1983) 24–33.
Magenes, E., Verdi, C., and Visintin, A., Theoretical and numerical results on the two-phase Stefan problem. SIAM J. Numer. Anal. 26 (1989) 1425–1438. CrossRef
I. Miyadera, Nonlinear Semigroups. Amer. Math. Soc. Colloq. Publ. (1992).
Nochetto, R.H., Error estimates for two-phase Stefan problems in several space variables. I. Linear boundary conditions. Calcolo 22 (1985) 457–499. CrossRef
Nochetto, P.H., and Verdi, C., Approximation of degenerate parabolic problems using numerical integration. SIAM J. Numer. Anal. 25 (1988) 784–814. CrossRef
L.A. Peletier, The porous media equation, in Applications of Nonlinear Analysis in the Physical Sciences (Bielefeld, 1979), Surveys Reference Works Math., 6, Pitman, Boston, Mass.-London (1981) 229–241.
Rannacher, R. and Scott, R., Some optimal error estimates for piecewise linear finite element approximation. Math. Comp. 38 (1982) 437–445. CrossRef
Rose, M., Numerical methods for flows through porous media, I. Math. Comp. 40 (1983) 435–467. CrossRef
Scott, L.R. and Zhang, S., Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54 (1990) 483–493. CrossRef
White, R.E., An enthalpy formulation of the Stefan problem. SIAM J. Numer. Anal. 19 (1982) 1129–1157. CrossRef