Backward Euler discretization of fully nonlinear parabolic problems

González, C.; Ostermann, A.; Palencia, C.; Thalhammer, M.

doi:10.1090/S0025-5718-01-01330-8

Backward Euler discretization of fully nonlinear parabolic problems
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by C. González, A. Ostermann, C. Palencia and M. Thalhammer;

Math. Comp. 71 (2002), 125-145

DOI: https://doi.org/10.1090/S0025-5718-01-01330-8

Published electronically: July 22, 2001

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Abstract:

This paper is concerned with the time discretization of nonlinear evolution equations. We work in an abstract Banach space setting of analytic semigroups that covers fully nonlinear parabolic initial-boundary value problems with smooth coefficients. We prove convergence of variable stepsize backward Euler discretizations under various smoothness assumptions on the exact solution. We further show that the geometric properties near a hyperbolic equilibrium are well captured by the discretization. A numerical example is given.

References

Georgios Akrivis, Michel Crouzeix, and Charalambos Makridakis, Implicit-explicit multistep methods for quasilinear parabolic equations, Numer. Math. 82 (1999), no. 4, 521–541. MR 1701828, DOI 10.1007/s002110050429
Nikolai Yu. Bakaev, On variable stepsize Runge-Kutta approximations of a Cauchy problem for the evolution equation, BIT 38 (1998), no. 3, 462–485. MR 1652765, DOI 10.1007/BF02510254
A. Belleni-Morante and A. C. McBride, Applied nonlinear semigroups, Wiley Series in Mathematical Methods in Practice, vol. 3, John Wiley & Sons, Ltd., Chichester, 1998. An introduction. MR 1654473
Kenneth Eriksson, Claes Johnson, and Stig Larsson, Adaptive finite element methods for parabolic problems. VI. Analytic semigroups, SIAM J. Numer. Anal. 35 (1998), no. 4, 1315–1325. MR 1620144, DOI 10.1137/S0036142996310216
C. González and C. Palencia, Stability of Runge-Kutta methods for quasilinear parabolic problems, Math. Comp. 69 (2000), no. 230, 609–628. MR 1659851, DOI 10.1090/S0025-5718-99-01156-4
E. Hairer and M. Zennaro, On error growth functions of Runge-Kutta methods, Appl. Numer. Math. 22 (1996), no. 1-3, 205–216. Special issue celebrating the centenary of Runge-Kutta methods. MR 1424297, DOI 10.1016/S0168-9274(96)00032-3
Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244
Marie-Noëlle Le Roux, Méthodes multipas pour des équations paraboliques non linéaires, Numer. Math. 35 (1980), no. 2, 143–162 (French, with English summary). MR 585243, DOI 10.1007/BF01396312
Ch. Lubich and A. Ostermann, Runge-Kutta methods for parabolic equations and convolution quadrature, Math. Comp. 60 (1993), no. 201, 105–131. MR 1153166, DOI 10.1090/S0025-5718-1993-1153166-7
Christian Lubich and Alexander Ostermann, Runge-Kutta approximation of quasi-linear parabolic equations, Math. Comp. 64 (1995), no. 210, 601–627. MR 1284670, DOI 10.1090/S0025-5718-1995-1284670-0
Ch. Lubich and A. Ostermann, Linearly implicit time discretization of non-linear parabolic equations, IMA J. Numer. Anal. 15 (1995), no. 4, 555–583. MR 1355637, DOI 10.1093/imanum/15.4.555
Alessandra Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Progress in Nonlinear Differential Equations and their Applications, vol. 16, Birkhäuser Verlag, Basel, 1995. MR 1329547, DOI 10.1007/978-3-0348-9234-6
Etsushi Nakaguchi and Atsushi Yagi, Error estimates of implicit Runge-Kutta methods for quasilinear abstract equations of parabolic type in Banach spaces, Japan. J. Math. (N.S.) 25 (1999), no. 1, 181–226. MR 1698356, DOI 10.4099/math1924.25.181
C. Palencia, On the stability of variable stepsize rational approximations of holomorphic semigroups, Math. Comp. 62 (1994), no. 205, 93–103. MR 1201070, DOI 10.1090/S0025-5718-1994-1201070-9
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486, DOI 10.1007/978-1-4612-5561-1
Michael E. Taylor, Partial differential equations. III, Applied Mathematical Sciences, vol. 117, Springer-Verlag, New York, 1997. Nonlinear equations; Corrected reprint of the 1996 original. MR 1477408
M. Thalhammer, Runge-Kutta time discretization of nonlinear parabolic equations. Thesis, Universität Innsbruck, 2000.