Finite element approximation for degenerate parabolic equations. An application of nonlinear semigroup theory
ESAIM: Modélisation mathématique et analyse numérique, Tome 39 (2005) no. 4, pp. 755-780.

Finite element approximation for degenerate parabolic equations is considered. We propose a semidiscrete scheme provided with order-preserving and L 1 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 L 1 and L , respectively, of the scheme are established. Under certain hypotheses on the data, we also derive L 1 convergence without any convergence rate. The validity of theoretical results is confirmed by numerical examples.

DOI : 10.1051/m2an:2005033
Classification : 35K65, 47H20, 65M60
Mots clés : finite element method, degenerate parabolic equation, nonlinear semigroup
@article{M2AN_2005__39_4_755_0,
     author = {Mizutani, Akira and Saito, Norikazu and Suzuki, Takashi},
     title = {Finite element approximation for degenerate parabolic equations. {An} application of nonlinear semigroup theory},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {755--780},
     publisher = {EDP-Sciences},
     volume = {39},
     number = {4},
     year = {2005},
     doi = {10.1051/m2an:2005033},
     mrnumber = {2165678},
     zbl = {1078.35009},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an:2005033/}
}
TY  - JOUR
AU  - Mizutani, Akira
AU  - Saito, Norikazu
AU  - Suzuki, Takashi
TI  - Finite element approximation for degenerate parabolic equations. An application of nonlinear semigroup theory
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2005
SP  - 755
EP  - 780
VL  - 39
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an:2005033/
DO  - 10.1051/m2an:2005033
LA  - en
ID  - M2AN_2005__39_4_755_0
ER  - 
%0 Journal Article
%A Mizutani, Akira
%A Saito, Norikazu
%A Suzuki, Takashi
%T Finite element approximation for degenerate parabolic equations. An application of nonlinear semigroup theory
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2005
%P 755-780
%V 39
%N 4
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an:2005033/
%R 10.1051/m2an:2005033
%G en
%F M2AN_2005__39_4_755_0
Mizutani, Akira; Saito, Norikazu; Suzuki, Takashi. Finite element approximation for degenerate parabolic equations. An application of nonlinear semigroup theory. ESAIM: Modélisation mathématique et analyse numérique, Tome 39 (2005) no. 4, pp. 755-780. doi : 10.1051/m2an:2005033. http://www.numdam.org/articles/10.1051/m2an:2005033/

[1] R.A. Adams, Sobolev Spaces. Academic Press, New York, London (1975). | MR | Zbl

[2] P. Bénilan, M.G. Crandall and P. Sacks, Some L 1 existence and dependence results for semilinear elliptic equations under nonlinear boundary conditions. Appl. Math. Optim. 17 (1988) 203-224. | Zbl

[3] A.E. Berger, H. Brezis and J.C.W Rogers, A numerical method for solving the problem u t -Δf(u)=0. RAIRO Anal. Numer. 13 (1979) 297-312. | Numdam | Zbl

[4] S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer (1994). | MR | Zbl

[5] H. Brezis and A. Pazy, Convergence and approximation of semigroups of nonlinear operators in Banach spaces. J. Funct. Anal. 9 (1972) 63-74. | Zbl

[6] H. Brezis and W. Strauss, Semi-linear second-order elliptic equations in L 1 . J. Math. Soc. Japan 25 (1973) 565-590. | Zbl

[7] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North Holland, Amsterdam (1978). | MR | Zbl

[8] 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). | Zbl

[9] P.G. Ciarlet and P.A. Raviart, Maximum principle and uniform convergence for the finite element method. Comput. Methods Appl. Mech. Engrg. 2 (1973) 17-31. | Zbl

[10] J.F. Ciavaldini, Analyse numérique d'un problème de Stefan à deux phases par une méthode d'éléments finis. SIAM J. Numer. Anal. 12 (1975) 464-487. | Zbl

[11] B. Cockburn and G. Gripenberg, Continuous dependence on the nonlinearities of solutions of degenerate parabolic equations. J. Differential Equations 151 (1999) 231-251. | Zbl

[12] M.G. Crandall and T. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces. Amer. J. Math. 93 (1971) 265-293. | Zbl

[13] C.M. Elliott, Error analysis of the enthalpy method for the Stefan problem. IMA J. Numer. Anal. 7 (1987) 61-71. | Zbl

[14] C.M. Elliott and J.R. Ockendon, Weak and Variational Methods for Moving Boundary Problems. Pitman, Boston. Res. Notes Math. 59 (1982). | MR | Zbl

[15] A. Friedman, Variational Principles and Free-Boundary Problems. Wiley, New York (1982). | MR | Zbl

[16] 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. | Zbl

[17] H. Fujita, N. Saito and T. Suzuki, Operator Theory and Numerical Methods. North-Holland, Amsterdam (2001). | MR | Zbl

[18] B.H. Gilding and L.A. Peletier, On a class of similarity solutions of the porous media equation. J. Math. Anal. Appl. 55 (1976) 351-364. | Zbl

[19] P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985). | MR | Zbl

[20] W. Jäger and J. Kačur, Solution of doubly nonlinear and degenerate parabolic problems by relaxation schemes. RAIRO Modél. Math. Anal. Numér. 29 (1995) 605-627. | EuDML | Numdam | Zbl

[21] J. Kačur, A. Handlovicová and M. Kacurová, Solution of nonlinear diffusion problems by linear approximation schemes. SIAM J. Numer. Anal. 30 1703-1722 (1993). | MR | Zbl

[22] T. Kato, Schrödinger operators with singular potentials. Israel J. Math. 13 (1972) 135-148. | Zbl

[23] M.N. Le Roux, Semi-discretization in time for a fast diffusion equation. J. Math. Anal. Appl. 137 (1989) 354-370. | Zbl

[24] M.N. Le Roux and P.E. Mainge, Numerical solution of a fast diffusion equation. Math. Comp. 68 (1999) 461-485. | Zbl

[25] P. Lesaint and J. Pousin, Error estimates for a nonlinear degenerate parabolic equation. Math. Comp. 59 (1992) 339-358. | Zbl

[26] E. Magenes, R.H. Nochetto and C. Verdi, Energy error estimates for a linear scheme to approximate nonlinear parabolic problems. RAIRO Modél. Math. Anal. Numér. 21 (1987) 655-678. | EuDML | Numdam | Zbl

[27] 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. | EuDML | Zbl

[28] E. Magenes, C. Verdi, and A. Visintin, Theoretical and numerical results on the two-phase Stefan problem. SIAM J. Numer. Anal. 26 (1989) 1425-1438. | Zbl

[29] I. Miyadera, Nonlinear Semigroups. Amer. Math. Soc. Colloq. Publ. (1992). | Zbl

[30] R.H. Nochetto, Error estimates for two-phase Stefan problems in several space variables. I. Linear boundary conditions. Calcolo 22 (1985) 457-499. | Zbl

[31] P.H. Nochetto, and C. Verdi, Approximation of degenerate parabolic problems using numerical integration. SIAM J. Numer. Anal. 25 (1988) 784-814. | Zbl

[32] 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. | Zbl

[33] R. Rannacher and R. Scott, Some optimal error estimates for piecewise linear finite element approximation. Math. Comp. 38 (1982) 437-445. | Zbl

[34] M. Rose, Numerical methods for flows through porous media, I. Math. Comp. 40 (1983) 435-467. | Zbl

[35] L.R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54 (1990) 483-493. | Zbl

[36] R.E. White, An enthalpy formulation of the Stefan problem. SIAM J. Numer. Anal. 19 (1982) 1129-1157. | Zbl

Cité par Sources :