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

Akira Mizutani; Norikazu Saito; Takashi Suzuki

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 39, Issue: 4, page 755-780
- ISSN: 0764-583X

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topMizutani, Akira, Saito, Norikazu, and Suzuki, Takashi. "Finite element approximation for degenerate parabolic equations. an application of nonlinear semigroup theory." ESAIM: Mathematical Modelling and Numerical Analysis 39.4 (2010): 755-780. <http://eudml.org/doc/194285>.

@article{Mizutani2010,

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.
},

author = {Mizutani, Akira, Saito, Norikazu, Suzuki, Takashi},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Finite element method;
degenerate parabolic equation; nonlinear semigroup.; order-preserving; contraction properties; rate of convergence},

language = {eng},

month = {3},

number = {4},

pages = {755-780},

publisher = {EDP Sciences},

title = {Finite element approximation for degenerate parabolic equations. an application of nonlinear semigroup theory},

url = {http://eudml.org/doc/194285},

volume = {39},

year = {2010},

}

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: Mathematical Modelling and Numerical Analysis

DA - 2010/3//

PB - EDP Sciences

VL - 39

IS - 4

SP - 755

EP - 780

AB -
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.

LA - eng

KW - Finite element method;
degenerate parabolic equation; nonlinear semigroup.; order-preserving; contraction properties; rate of convergence

UR - http://eudml.org/doc/194285

ER -

## References

top- R.A. Adams, Sobolev Spaces. Academic Press, New York, London (1975).
- P. Bénilan, M.G. Crandall and P. Sacks, Some L1 existence and dependence results for semilinear elliptic equations under nonlinear boundary conditions. Appl. Math. Optim.17 (1988) 203–224. Zbl0652.35043
- A.E. Berger, H. Brezis and J.C.W Rogers, A numerical method for solving the problem ut - Δƒ(u) = 0. RAIRO Anal. Numer.13 (1979) 297–312.
- S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer (1994). Zbl0804.65101
- H. Brezis and A. Pazy, Convergence and approximation of semigroups of nonlinear operators in Banach spaces. J. Funct. Anal.9 (1972) 63–74. Zbl0231.47036
- H. Brezis and W. Strauss, Semi-linear second-order elliptic equations in L1. J. Math. Soc. Japan25 (1973) 565–590. Zbl0278.35041
- P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North Holland, Amsterdam (1978). Zbl0383.65058
- 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).
- 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. Zbl0251.65069
- 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. Zbl0272.65101
- B. Cockburn and G. Gripenberg, Continuous dependence on the nonlinearities of solutions of degenerate parabolic equations. J. Differential Equations151 (1999) 231–251. Zbl0921.35017
- M.G. Crandall and T. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces. Amer. J. Math.93 (1971) 265–293. Zbl0226.47038
- C.M. Elliott, Error analysis of the enthalpy method for the Stefan problem. IMA J. Numer. Anal.7 (1987) 61–71. Zbl0638.65088
- C.M. Elliott and J.R. Ockendon, Weak and Variational Methods for Moving Boundary Problems. Pitman, Boston. Res. Notes Math.59 (1982). Zbl0476.35080
- A. Friedman, Variational Principles and Free-Boundary Problems. Wiley, New York (1982). Zbl0564.49002
- 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). Zbl0976.65098
- 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. Zbl0356.35049
- P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985). Zbl0695.35060
- 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. Zbl0837.65103
- J. Kačur, A. Handlovicová and M. Kacurová, Solution of nonlinear diffusion problems by linear approximation schemes. SIAM J. Numer. Anal.30 1703-1722 (1993). Zbl0792.65070
- T. Kato, Schrödinger operators with singular potentials. Israel J. Math.13 (1972) 135–148.
- M.N. Le Roux, Semi-discretization in time for a fast diffusion equation. J. Math. Anal. Appl.137 (1989) 354–370. Zbl0693.65085
- M.N. Le Roux and P.E. Mainge, Numerical solution of a fast diffusion equation. Math. Comp.68 (1999) 461–485. Zbl1020.65053
- P. Lesaint and J. Pousin, Error estimates for a nonlinear degenerate parabolic equation. Math. Comp.59 (1992) 339–358. Zbl0767.65071
- 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. Zbl0635.65123
- 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. Zbl0562.35089
- 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. Zbl0738.65092
- I. Miyadera, Nonlinear Semigroups. Amer. Math. Soc. Colloq. Publ. (1992).
- R.H. Nochetto, Error estimates for two-phase Stefan problems in several space variables. I. Linear boundary conditions. Calcolo22 (1985) 457–499. Zbl0606.65084
- P.H. Nochetto, and C. Verdi, Approximation of degenerate parabolic problems using numerical integration. SIAM J. Numer. Anal.25 (1988) 784–814. Zbl0655.65131
- 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.
- R. Rannacher and R. Scott, Some optimal error estimates for piecewise linear finite element approximation. Math. Comp.38 (1982) 437–445. Zbl0483.65007
- M. Rose, Numerical methods for flows through porous media, I. Math. Comp.40 (1983) 435–467. Zbl0518.76078
- L.R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp.54 (1990) 483–493. Zbl0696.65007
- R.E. White, An enthalpy formulation of the Stefan problem. SIAM J. Numer. Anal.19 (1982) 1129–1157. Zbl0501.65058

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