A posteriori error estimation for variational problems with uniformly convex functionals. (English) Zbl 0949.65070

Repin, Sergey I.

Math. Comput. 69, No. 230, 481-500 (2000).

Variational problems of the form \[ \inf_{v\in V} (F(v)+ G(\Lambda v)), \] where \(F:V\to \mathbb{R}\) is a convex lower semicontinuous functional, \(G: Y\to\mathbb{R}\) is a uniformly convex functional, \(V\) and \(Y\) are reflexive Banach spaces and \(\Lambda: V\to Y\) is a bounded linear operator is considered. A general scheme for deriving a posteriori error estimates by using the duality theory from the calculus of variations is introduced. It is shown that the main classes of such a posteriori estimates can be justified via the duality theory.

Reviewer: J.Vaníček (Praha)

MSC:

65K10	Numerical optimization and variational techniques
49J27	Existence theories for problems in abstract spaces

Keywords:

variational problems; a posteriori error estimation; uniformly convex functional; Banach spaces; bounded linear operator; duality theory

References:

[1]	Mark Ainsworth and J. Tinsley Oden, A unified approach to a posteriori error estimation using element residual methods, Numer. Math. 65 (1993), no. 1, 23 – 50. · Zbl 0797.65080
[2]	Jean Pierre Aubin and Hermann G. Burchard, Some aspects of the method of the hypercircle applied to elliptic variational problems, Numerical Solution of Partial Differential Equations, II (SYNSPADE 1970) (Proc. Sympos., Univ. of Maryland, College Park, Md., 1970) Academic Press, New York, 1971, pp. 1 – 67. · Zbl 0264.65069
[3]	I. Babu\(\check{\text{s}}\)ka and W. C. Rheinboldt, A-posteriori error estimates for the finite element method, Internat. J. Numer. Meth. Engrg., 12 (1978), 1597-1615. · Zbl 0396.65068
[4]	I. Babuška and W. C. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal. 15 (1978), no. 4, 736 – 754. · Zbl 0398.65069
[5]	Ivo Babuška and Werner C. Rheinboldt, A posteriori error analysis of finite element solutions for one-dimensional problems, SIAM J. Numer. Anal. 18 (1981), no. 3, 565 – 589. · Zbl 0487.65060
[6]	I. Babu\(\check{\text{s}}\)ka and R. Rodriguez, The problem of the selection of an a posteriori error indicator based on smoothing techniques, Internat. J. Numer. Meth. Engrg., 36 (1993), 539-567.
[7]	R. E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp. 44 (1985), no. 170, 283 – 301. · Zbl 0569.65079
[8]	Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. · Zbl 0383.65058
[9]	Ph. Clément, Approximations by finite element functions using local regularization, RIARO Anal. Numer., 2 (1975), 77-84. · Zbl 0368.65008
[10]	M. Crouzeix and V. Thomée, The stability in \?_{\?} and \?\textonesuperior _{\?} of the \?\(_{2}\)-projection onto finite element function spaces, Math. Comp. 48 (1987), no. 178, 521 – 532. · Zbl 0637.41034
[11]	Jim Douglas Jr., Todd Dupont, and Lars Wahlbin, The stability in \?^{\?} of the \?²-projection into finite element function spaces, Numer. Math. 23 (1974/75), 193 – 197. · Zbl 0297.41022
[12]	Ivar Ekeland and Roger Temam, Analyse convexe et problèmes variationnels, Dunod; Gauthier-Villars, Paris-Brussels-Montreal, Que., 1974 (French). Collection Études Mathématiques. Ivar Ekeland and Roger Temam, Convex analysis and variational problems, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1976. Translated from the French; Studies in Mathematics and its Applications, Vol. 1.
[13]	Kenneth Eriksson and Claes Johnson, An adaptive finite element method for linear elliptic problems, Math. Comp. 50 (1988), no. 182, 361 – 383. · Zbl 0644.65080
[14]	Roland Glowinski, Numerical methods for nonlinear variational problems, Springer Series in Computational Physics, Springer-Verlag, New York, 1984. · Zbl 0536.65054
[15]	Claes Johnson and Peter Hansbo, Adaptive finite element methods in computational mechanics, Comput. Methods Appl. Mech. Engrg. 101 (1992), no. 1-3, 143 – 181. Reliability in computational mechanics (Kraków, 1991). · Zbl 0778.73071
[16]	Claes Johnson, Yi Yong Nie, and Vidar Thomée, An a posteriori error estimate and adaptive timestep control for a backward Euler discretization of a parabolic problem, SIAM J. Numer. Anal. 27 (1990), no. 2, 277 – 291. · Zbl 0701.65063
[17]	D. W. Kelly, The self-equilibration of residuals and complementary a posteriori error estimates in the finite element method, Internat. J. Numer. Methods Engrg. 20 (1984), no. 8, 1491 – 1506. · Zbl 0575.65100
[18]	P. Ladevèze and D. Leguillon, Error estimate procedure in the finite element method and applications, SIAM J. Numer. Anal. 20 (1983), no. 3, 485 – 509. · Zbl 0582.65078
[19]	P. Ladevèze, J.-P. Pelle, and Ph. Rougeot, Error estimation and mesh optimization for classical finite elements, Engrg. Comput. 8 (1991), no. 1, 69 – 80.
[20]	S. G. Mikhlin, Variational methods in mathematical physics, Translated by T. Boddington; editorial introduction by L. I. G. Chambers. A Pergamon Press Book, The Macmillan Co., New York, 1964. · Zbl 0119.19002
[21]	Solomon G. Mikhlin, Error analysis in numerical processes, Pure and Applied Mathematics (New York), John Wiley & Sons, Ltd., Chichester, 1991. Translated and revised from the German; Translated by Reinhard Lehmann; A Wiley-Interscience Publication. · Zbl 0786.65038
[22]	Механика жесткопластических сред, ”Наука”, Мосцощ, 1981 (Руссиан). · Zbl 0551.73002
[23]	J. T. Oden, L. Demkowicz, W. Rachowicz, and T. A. Westermann, A posteriori error analysis in finite elements: the element residual method for symmetrizable problems with applications to compressible Euler and Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg. 82 (1990), no. 1-3, 183 – 203. Reliability in computational mechanics (Austin, TX, 1989). · Zbl 0727.73072
[24]	S. I. Repin, A priori estimates for the convergence of variational-difference methods in problems of ideal plasticity, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 221 (1995), no. Kraev. Zadachi Mat. Fiz. i Smezh. Voprosy Teor. Funktsiĭ. 26, 226 – 234, 260 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (New York) 87 (1997), no. 2, 3421 – 3427. · Zbl 0927.74080
[25]	Sergey I. Repin, Errors of finite element method for perfectly elasto-plastic problems, Math. Models Methods Appl. Sci. 6 (1996), no. 5, 587 – 604. · Zbl 0856.73071
[26]	S. I. Repin, A posteriori error estimation for nonlinear variational problems by duality theory, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 243 (1997), no. Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funktsiĭ. 28, 201 – 214, 342 (English, with English and Russian summaries); English transl., J. Math. Sci. (New York) 99 (2000), no. 1, 927 – 935. · Zbl 0904.65064
[27]	S. I. Repin, A posteriori error estimates for approximate solutions of variational problems with power growth functionals, Zapiski Nauchnych Seminarov POMI, 249 (1997), 244-255. · Zbl 0961.49016
[28]	S. I. Repin, A posteriori error estimates for approximate solutions of variational problems, In the Proceedings of the Second European Conference on Numerical Mathematics and Advanced Applications, Heidelberg 1997. VSP, Utrecht (1999). · Zbl 0961.49016
[29]	S. I. Repin, A posteriori error estimates for approximate solutions of variational problems with strongly convex functionals, Problems of Mathematical Analysis, 17 (1997), 199-226. (in Russian, English translation in the Journal of Mathematical Sciences 1998). · Zbl 0941.65059
[30]	S. I. Repin, A posteriori estimates of the accuracy of variational methods in problems with nonconvex functionals, Algebra i Analiz, 11 (1999) (in Russian). · Zbl 0964.49009
[31]	S. I. Repin, A unified approach to a posteriori error estimation by duality error majorants, Mathematics and Computers in Simulation (1999).
[32]	S. Repin and G. Serëgin, Error estimates for stresses in the finite element analysis of the two-dimensional elasto-plastic problems, Internat. J. Engrg. Sci. 33 (1995), no. 2, 255 – 268. · Zbl 0899.73532
[33]	Sergey I. Repin and Leonidas S. Xanthis, A posteriori error estimation for elastoplastic problems based on duality theory, Comput. Methods Appl. Mech. Engrg. 138 (1996), no. 1-4, 317 – 339. · Zbl 0886.73082
[34]	Sergey I. Repin and Leonidas S. Xanthis, A posteriori error estimation for nonlinear variational problems, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), no. 10, 1169 – 1174 (English, with English and French summaries). · Zbl 0877.65045
[35]	S. L. Sobolev, Some applications of functional analysis in mathematical physics, Translations of Mathematical Monographs, vol. 90, American Mathematical Society, Providence, RI, 1991. Translated from the third Russian edition by Harold H. McFaden; With comments by V. P. Palamodov. · Zbl 0732.46001
[36]	R. Verfürth, A review of a posteriori error estimation and adaptive mesh-refinement techniques. Wiley, Teubner, 1996. · Zbl 0853.65108
[37]	O. C. Zienkiewicz and J. Z. Zhu, A simple error estimator and adaptive procedure for practical engineering analysis, Internat. J. Numer. Methods Engrg. 24 (1987), no. 2, 337 – 357. · Zbl 0602.73063
[38]	O. C. Zienkiewicz and J. Z. Zhu, The superconvergent patch recovery (SPR) and adaptive finite element refinement, Comput. Methods Appl. Mech. Engrg. 101 (1992), no. 1-3, 207 – 224. Reliability in computational mechanics (Kraków, 1991). · Zbl 0779.73078