Nochetto, Ricardo H.; Savaré, Giuseppe; Verdi, Claudio A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations. (English) Zbl 1021.65047 Commun. Pure Appl. Math. 53, No. 5, 525-589 (2000). Summary: We study the backward Euler method with variable time steps for abstract evolution equations in Hilbert spaces. Exploiting convexity of the underlying potential or the angle-bounded condition, thereby assuming no further regularity, we derive novel a posteriori estimates of the discretization error in terms of computable quantities related to the amount of energy dissipation or monotonicity residual. These estimators depend solely on the discrete solution and data and impose no constraints between consecutive time steps. We also prove that they converge to zero with an optimal rate with respect to the regularity of the solution. We apply the abstract results to a number of concrete, strongly nonlinear problems of parabolic type with degenerate or singular character. Cited in 3 ReviewsCited in 90 Documents MSC: 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 34G20 Nonlinear differential equations in abstract spaces 65L70 Error bounds for numerical methods for ordinary differential equations 35K90 Abstract parabolic equations 65J15 Numerical solutions to equations with nonlinear operators 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 35K55 Nonlinear parabolic equations 35K65 Degenerate parabolic equations Keywords:degenerate type; Cauchy problem; a posteriori error estimates; backward Euler method; variable time steps; abstract evolution equations; Hilbert spaces; strongly nonlinear problems PDFBibTeX XMLCite \textit{R. H. Nochetto} et al., Commun. Pure Appl. Math. 53, No. 5, 525--589 (2000; Zbl 1021.65047) Full Text: DOI References: [1] Amann, Aequationes Math 1 pp 242– (1968) · Zbl 0165.13701 [2] A monotone convergence theorem for sequences of nonlinear mappings. Nonlinear functional analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 1, Chicago, Ill., 1968), 1-9. Amer. Math. Soc., Providence, R.I., 1970. [3] Baillon, Israel J Math 26 pp 137– (1977) · Zbl 0352.47023 [4] Discretization of evolution variational inequalities. Partial differential equations and the calculus of variations, Vol. I, 59-92. Progr Nonlinear Differential Equations Appl, 1. Birkhäuser Boston, Boston, 1989. [5] Baiocchi, Calcolo 20 pp 143– (1983) · Zbl 0538.65077 [6] Variational and quasivariational inequalities. Applications to free boundary problems. Translated from the Italian by Lakshmi Jayakar. Wiley, New York, 1984. [7] Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations. Contributions to nonlinear functional analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971), 101-156. Academic Press, New York, 1971. [8] Brézis, J Math Pures Appl (9) 51 pp 1– (1972) [9] Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. (French) North-Holland Mathematics Studies, No. 5. Notas de Matemática (50). North-Holland, Amsterdam-London; American Elsevier, New York, 1973. [10] Brézis, Ad-vances in Math 18 pp 115– (1975) · Zbl 0318.45011 [11] Brezis, Bull Soc Math France 96 pp 153– (1968) · Zbl 0165.45601 [12] Brézis, J Math Soc Japan 25 pp 565– (1973) · Zbl 0278.35041 [13] Caffarelli, Comm Pure Appl Math 50 pp 563– (1997) · Zbl 0901.46034 [14] ; Convex analysis and measurable multifunctions. Lecture Notes in Mathematics, Vol. 580. Springer, Berlin-New York, 1977. · Zbl 0346.46038 [15] Chen, Proc Roy Soc London Ser A 444 pp 429– (1994) · Zbl 0814.35044 [16] ; Degenerate evolution systems modeling cardiac electric field at micro and macroscopic level. I.A.N. C.N.R., Pavia, no. 1007, 1996. [17] ; Time discretization of Stefan problems with singular heat flux. Free bound-ary problems, theory and applications (Zakopane, 1995), 16-28. Pitman Res Notes Math Ser, 363. Longman, Harlow, 1996. [18] Nonlinear semigroups and evolution governed by accretive operators. Non-linear functional analysis and its applications, Part 1 (Berkeley, Calif., 1983), 305-337. Proc Sympos Pure Math, 45, Part 1. Amer. Math. Soc., Providence, R.I., 1986. [19] Crandall, Israel J Math 21 pp 261– (1975) · Zbl 0351.34037 [20] Crandall, Amer J Math 93 pp 265– (1971) · Zbl 0226.47038 [21] ; Analyse numérique des équations différentielles. (French) [Numerical analysis of differential equations] Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master’s Degree] Masson, Paris, 1984. [22] Error analysis for a class of methods for stiff non-linear initial value problems. Numerical analysis (Proc. 6th Biennial Dundee Conf., Univ. Dundee, Dun-Dee, 1975), 60-72. Lecture Notes in Mathematics, Vol. 506, Springer, Berlin, 1976. [23] Damlamian, Nonlinear Anal 23 pp 115– (1994) · Zbl 0820.35143 [24] de Mottoni, Trans Amer Math Soc 3475 pp 1533– (1995) [25] Degenerate parabolic equations. Universitext. Springer, New York, 1993. [26] ; Analyse convexe et problèmes variationnels. (French) Collection Études Mathématiques. Dunod Gauthier-Villars, Paris-Brussels-Montreal, 1974. [27] ; ; ; Computational differential equations. Cam-bridge University Press, Cambridge, 1996. [28] Eriksson, SIAM J Numer Anal 28 pp 43– (1991) · Zbl 0732.65093 [29] Eriksson, SIAM J Numer Anal 32 pp 1729– (1995) · Zbl 0835.65116 [30] Eriksson, SIAM J Numer Anal 35 pp 1315– (1998) · Zbl 0909.65063 [31] Numerical initial value problems in ordinary differential equations. Prentice-Hall, Englewood Cliffs, N.J., 1971. [32] Minimal surfaces and functions of bounded variation. Monographs in Mathematics, 80. Birkhäuser, Basel-Boston, 1984. · Zbl 0545.49018 [33] ; ; Numerical analysis of variational inequalities. Translated from the French. Studies in Mathematics and Its Applications, 8. North-Holland, Amsterdam-New York, 1981. [34] Elliptic problems in nonsmooth domains. Monographs and Studies in Mathematics, 24. Pitman (Advanced Publishing Program), Boston-London, 1985. · Zbl 0695.35060 [35] ; Solving ordinary differential equations. II. Stiff and differential-algebraic problems. Second edition. Springer Series in Computational Mathematics, 14. Springer, Berlin, 1996. · Zbl 0859.65067 [36] Discrete variable methods in ordinary differential equations. Wiley, New York-London, 1962. · Zbl 0112.34901 [37] Johnson, SIAM J Numer Anal 25 pp 908– (1988) · Zbl 0661.65076 [38] Perturbation theory for linear operators. Second edition. Grundlehren der Mathematischen Wissenschaften, Band 132. Springer, Berlin-New York, 1976. [39] Systems of nonlinear PDEs arising from dynamical phase transitions. Phase transitions and hysteresis (Montecatini Terme, 1993), 39-86. Lecture Notes in Mathematics, 1584. Springer, Berlin, 1994. [40] ; An introduction to variational inequalities and their applications. Pure and Applied Mathematics, 88. Academic Press [Harcourt Brace Jovanovich], New York-London, 1980. [41] Lichnewsky, J Differential Equations 30 pp 340– (1978) · Zbl 0368.49016 [42] Quelques méthodes de résolution des problèmes aux limites non linéaires. (French) Dunod; Gauthier-Villars, Paris, 1969. [43] Lions, Comm Pure Appl Math 20 pp 493– (1967) · Zbl 0152.34601 [44] ; Introduction to the theory of (nonsymmetric) Dirichlet forms. Universitext. Springer, Berlin, 1992. [45] Stefan problems in a concentrated capacity. Advances in mathematical computation and applications, 82-90. N.C.C., Novosibirsk, 1995. [46] Multiple integrals in the calculus of variations. Die Grundlehren der mathematischen Wissenschaften, Band 130. Springer New York, New York, 1966. [47] Nochetto, Ann Scuola Norm Sup Pisa Cl Sci (4) 21 pp 193– (1994) [48] ; ; Adaptive solution of phase change problems over unstructured tetrahedral meshes. Grid generation and adaptive algorithms, 163-181. The IMA Volumes in Mathematics and Its Applications, 113. Springer, New York, 1999. [49] Nochetto, Math Comp 69 pp 1– (2000) · Zbl 0942.65111 [50] Rockafellar, Pacific J Math 17 pp 497– (1966) · Zbl 0145.15901 [51] Convex analysis. Princeton Mathematical Series, No. 28. Princeton University Press, Princeton, N.J., 1970. · Zbl 0932.90001 [52] Rulla, SIAM J Numer Anal 33 pp 68– (1996) · Zbl 0855.65102 [53] Savaré, Rend Accad Naz Sci XL Mem Mat (5) 17 pp 83– (1993) [54] Savaré, Adv Math Sci Appl 6 pp 377– (1996) [55] Savaré, J Funct Anal 152 pp 176– (1998) · Zbl 0889.35018 [56] Savaré, Atti Accad Naz Lincei Cl Sci Fis Mat Natur Rend Lincei (9) Mat Appl 8 pp 49– (1997) [57] Temam, Arch Rational Mech Anal 44 pp 121– (1971) · Zbl 0252.49002 [58] Applications de l’analyse convexe au calcul des variations. (French) Nonlinear operators and the calculus of variations (Summer School, Univ. Libre Bruxelles, Brussels, 1975), 208-237. Lecture Notes in Mathematics, Vol. 543, Springer, Berlin, 1976. [59] Galerkin finite element methods for parabolic problems. Springer Series in Computational Mathematics, 25. Springer, Berlin, 1997. · Zbl 0884.65097 [60] Tomarelli, Numer Math 45 pp 23– (1984) · Zbl 0527.65036 [61] Visintin, IMA J Appl Math 34 pp 225– (1985) · Zbl 0585.35053 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.