A posteriori error estimates for nonlinear problems. 𝐿^{𝑟}(0,𝑇;𝐿^{𝜌}(Ω))-error estimates for finite element discretizations of parabolic equations

Verfürth, R.

doi:10.1090/S0025-5718-98-01011-4

A posteriori error estimates for nonlinear problems. $L^r(0,T;L^\rho (\Omega ))$-error estimates for finite element discretizations of parabolic equations
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Math. Comp. 67 (1998), 1335-1360 Request permission

Abstract:

Using the abstract framework of [R. Verfürth: A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations. Math. Comput. 62 (206), 445 – 475 (1994)] we analyze a residual a posteriori error estimator for space-time finite element discretizations of quasilinear parabolic pdes. The estimator gives global upper and local lower bounds on the error of the numerical solution. The finite element discretizations in particular cover the so-called $\theta$-scheme, which includes the implicit and explicit Euler methods and the Crank-Nicholson scheme.

References

Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
Herbert Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function spaces, differential operators and nonlinear analysis (Friedrichroda, 1992) Teubner-Texte Math., vol. 133, Teubner, Stuttgart, 1993, pp. 9–126. MR 1242579, DOI 10.1007/978-3-663-11336-2_{1}
Jacques Baranger and Hassan El Amri, Estimateurs a posteriori d’erreur pour le calcul adaptatif d’écoulements quasi-newtoniens, RAIRO Modél. Math. Anal. Numér. 25 (1991), no. 1, 31–47 (French, with English summary). MR 1086839, DOI 10.1051/m2an/1991250100311
Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
Ph. Clément, Approximation by finite element functions using local regularization, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. 9 (1975), no. R-2, 77–84 (English, with Loose French summary). MR 0400739
Robert Dautray and Jacques-Louis Lions, Mathematical analysis and numerical methods for science and technology. Vol. 5, Springer-Verlag, Berlin, 1992. Evolution problems. I; With the collaboration of Michel Artola, Michel Cessenat and Hélène Lanchon; Translated from the French by Alan Craig. MR 1156075, DOI 10.1007/978-3-642-58090-1
Kenneth Eriksson and Claes Johnson, Adaptive finite element methods for parabolic problems. IV. Nonlinear problems, SIAM J. Numer. Anal. 32 (1995), no. 6, 1729–1749. MR 1360457, DOI 10.1137/0732078
Kenneth Eriksson and Claes Johnson, Adaptive finite element methods for parabolic problems. V. Long-time integration, SIAM J. Numer. Anal. 32 (1995), no. 6, 1750–1763. MR 1360458, DOI 10.1137/0732079
R. Verfürth, A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations, Math. Comp. 62 (1994), no. 206, 445–475. MR 1213837, DOI 10.1090/S0025-5718-1994-1213837-1
—, A posteriori error estimates for nonlinear problems. Finite element discretizations of parabolic equations. Report 180, Ruhr-Universität Bochum (1995) http://www.num1.ruhr-uni-bochum.de/rv/preprints.html