A posteriori error estimation for variational problems with uniformly convex functionals

Repin, Sergey

doi:10.1090/S0025-5718-99-01190-4

A posteriori error estimation for variational problems with uniformly convex functionals
HTML articles powered by AMS MathViewer

by Sergey I. Repin PDF

Math. Comp. 69 (2000), 481-500 Request permission

Abstract:

The objective of this paper is to introduce a general scheme for deriving a posteriori error estimates by using duality theory of the calculus of variations. We consider variational problems of the form \[ \inf \limits _{v\in V} \{ F(v)+G(\Lambda v) \}, \] where $F:V\rightarrow \mathbb {R}$ is a convex lower semicontinuous functional, $G: Y\rightarrow \mathbb {R}$ is a uniformly convex functional, $V$ and $Y$ are reflexive Banach spaces, and $\Lambda :V\rightarrow Y$ is a bounded linear operator. We show that the main classes of a posteriori error estimates known in the literature follow from the duality error estimate obtained and, thus, can be justified via the duality theory.

References

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. MR 1217437, DOI 10.1007/BF01385738
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. MR 0285136
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.
I. Babuška and W. C. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal. 15 (1978), no. 4, 736–754. MR 483395, DOI 10.1137/0715049
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. MR 615532, DOI 10.1137/0718036
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.
R. E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp. 44 (1985), no. 170, 283–301. MR 777265, DOI 10.1090/S0025-5718-1985-0777265-X
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, Approximations by finite element functions using local regularization, RIARO Anal. Numer., 2 (1975), 77–84.
M. Crouzeix and V. Thomée, The stability in $L_p$ and $W^1_p$ of the $L_2$-projection onto finite element function spaces, Math. Comp. 48 (1987), no. 178, 521–532. MR 878688, DOI 10.1090/S0025-5718-1987-0878688-2
Jim Douglas Jr., Todd Dupont, and Lars Wahlbin, The stability in $L^{q}$ of the $L^{2}$-projection into finite element function spaces, Numer. Math. 23 (1974/75), 193–197. MR 383789, DOI 10.1007/BF01400302
Ivar Ekeland and Roger Temam, Analyse convexe et problèmes variationnels, Collection Études Mathématiques, Dunod, Paris; Gauthier-Villars, Paris-Brussels-Montreal, Que., 1974 (French). MR 0463993
Kenneth Eriksson and Claes Johnson, An adaptive finite element method for linear elliptic problems, Math. Comp. 50 (1988), no. 182, 361–383. MR 929542, DOI 10.1090/S0025-5718-1988-0929542-X
Roland Glowinski, Numerical methods for nonlinear variational problems, Springer Series in Computational Physics, Springer-Verlag, New York, 1984. MR 737005, DOI 10.1007/978-3-662-12613-4
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). MR 1195583, DOI 10.1016/0045-7825(92)90020-K
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. MR 1043607, DOI 10.1137/0727019
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. MR 759999, DOI 10.1002/nme.1620200811
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. MR 701093, DOI 10.1137/0720033
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. MR 1182888, DOI 10.1108/eb023827
S. G. Mikhlin, Variational methods in mathematical physics, A Pergamon Press Book, The Macmillan Company, New York, 1964. Translated by T. Boddington; editorial introduction by L. I. G. Chambers. MR 0172493
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. MR 1129889
P. P. Mosolov and V. P. Myasnikov, Mekhanika zhestkoplasticheskikh sred, “Nauka”, Moscow, 1981 (Russian). MR 641694
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). MR 1077656, DOI 10.1016/0045-7825(90)90164-H
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. MR 1359758, DOI 10.1007/BF02355592
Sergey I. Repin, Errors of finite element method for perfectly elasto-plastic problems, Math. Models Methods Appl. Sci. 6 (1996), no. 5, 587–604. MR 1403722, DOI 10.1142/S0218202596000237
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. MR 1629741, DOI 10.1007/BF02673600
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.
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).
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).
S. I. Repin, A posteriori estimates of the accuracy of variational methods in problems with nonconvex functionals, Algebra i Analiz, 11 (1999) (in Russian).
S. I. Repin, A unified approach to a posteriori error estimation by duality error majorants, Mathematics and Computers in Simulation (1999).
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. MR 1311562, DOI 10.1016/0020-7225(94)00057-Q
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. MR 1422336, DOI 10.1016/S0045-7825(96)01136-X
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). MR 1451942, DOI 10.1016/S0764-4442(97)87906-2
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. MR 1125990, DOI 10.1090/mmono/090
R. Verfürth, A review of a posteriori error estimation and adaptive mesh–refinement techniques. Wiley, Teubner, 1996.
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. MR 875306, DOI 10.1002/nme.1620240206
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). MR 1195585, DOI 10.1016/0045-7825(92)90023-D