Sharp upper global a posteriori error estimates for nonlinear elliptic variational problems. (English) Zbl 1212.65249

Summary: The paper is devoted to the problem of a verification of the accuracy of approximate solutions obtained in computer simulations. This problem is strongly related to a posteriori error estimates giving computable bounds for computational errors and detecting zones in the solution domain where such errors are too large and certain mesh refinements should be performed. A mathematical model embracing nonlinear elliptic variational problems is considered. Based on functional type estimates developed on an abstract level, we present a general technology for constructing computable sharp upper bounds for the global error for various particular classes of elliptic problems. Here, the global error is understood as a suitable energy type difference between the true and the computed solutions. The estimates obtained are completely independent of the numerical technique used to obtain approximate solutions, and are sharp in the sense that they can be, in principle, made as close to the true error as resources of the used computer allow. The latter can be achieved by suitably tuning the auxiliary parameter functions, involved in the proposed upper error bounds, in the course of the calculations.


65K10 Numerical optimization and variational techniques
49J20 Existence theories for optimal control problems involving partial differential equations
49M30 Other numerical methods in calculus of variations (MSC2010)
Full Text: DOI EuDML Link


[1] M. Ainsworth, J.T. Oden: A Posteriori Error Estimation in Finite Element Analysis. John Wiley & Sons, 2000. · Zbl 1008.65076
[2] O. Axelsson, A. Padiy: On a two-level Newton-type procedure applied for solving non-linear elasticity problems. Int. J. Numer. Methods Eng. 49 (2000), 1479–1493. · Zbl 0994.74066 · doi:10.1002/1097-0207(20001230)49:12<1479::AID-NME4>3.0.CO;2-4
[3] O. Axelsson, J. Maubach: On the updating and assembly of the Hessian matrix in finite elements. Comput. Methods Appl. Mech. Eng. 71 (1988), 41–67. · Zbl 0673.65068 · doi:10.1016/0045-7825(88)90095-3
[4] I. Babuska, T. Strouboulis: The Finite Element Method and Its Reliability. Oxford University Press, New York, 2001.
[5] W. Bangerth, R. Rannacher: Adaptive Finite Element Methods for Differential Equations. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel, 2003. · Zbl 1020.65058
[6] R. Becker, R. Rannacher: A feed-back approach to error control in finite element methods: Basic approach and examples. East-West J. Numer. Math. 4 (1996), 237–264. · Zbl 0868.65076
[7] R. Blaheta: Multilevel Newton methods for nonlinear problems with applications to elasticity. Copernicus 940820. Technical report.
[8] J. Brandts, M. Křížek: Gradient superconvergence on uniform simplicial partitions of polytopes. IMA J. Numer. Anal. 23 (2003), 489–505. · Zbl 1042.65081 · doi:10.1093/imanum/23.3.489
[9] C. Carstensen, S.A. Funken: Constants in Clément-interpolation error and residual based a posteriori error estimates in finite element methods. East-West J. Numer. Math. 8 (2000), 153–175. · Zbl 0973.65091
[10] Ph.G. Ciarlet: The Finite Element Method for Elliptic Problems. Studies in Mathematics and Its Applications Vol. 4. North-Holland Publishing, Amsterdam-New York-Oxford, 1978.
[11] P. Concus: Numerical solution of the nonlinear magnetostatic field equation in two dimensions. J. Comput. Phys. 1 (1967), 330–342. · Zbl 0154.41103 · doi:10.1016/0021-9991(67)90043-5
[12] K. Eriksson, D. Estep, P. Hansbo, C. Johnson: Introduction to adaptive methods for differential equations. Acta Numerica 1995. Cambridge University Press, Cambridge, 1995, pp. 105–158. · Zbl 0829.65122
[13] I. Faragó, J. Karátson: Numerical Solution of Nonlinear Elliptic Problems via Preconditioning Operators. Theory and Applications. Advances in Computation, Vol. 11. NOVA Science Publishers, New York, 2002.
[14] M.E. Frolov: On efficiency of the dual majorant method for the quality estimation of approximate solutions of fourth-order elliptic boundary value problems. Russ. J. Numer. Anal. Math. Model. 19 (2004), 407–418. · Zbl 1064.65129 · doi:10.1515/1569398042395970
[15] R. Glowinski, A. Marrocco: Analyse numérique du champ magnétique d’un alternateur par éléments finis et sur-relaxation ponctuelle non linéaire. Computer Methods Appl. Mech. Engin. 3 (1974), 55–85. · Zbl 0288.65068 · doi:10.1016/0045-7825(74)90042-5
[16] P. Grisvard: Elliptic Problems in Nonsmooth Domains. Pitman, Boston-London-Melbourne, 1985. · Zbl 0695.35060
[17] W. Han: A Posteriori Error Analysis via Duality Theory. With Applications in Modeling and Numerical Approximations. Advances in Mechanics and Mathematics Vol. 8. Springer, New York, 2005. · Zbl 1081.65065
[18] A. Hannukainen: Finite Element Methods for Maxwell’s Equations. Master Thesis. Institute of Mathematics, Helsinki University of Technology, Helsinki, 2007. · Zbl 1161.65081
[19] A. Hannukainen, S. Korotov: Techniques for a posteriori error estimation in terms of linear functionals for elliptic type boundary value problems. Far East J. Appl. Math. 21 (2005), 289–304. · Zbl 1092.65097
[20] I. Hlaváček, J. Chleboun, I. Babuška: Uncertain Input Data Problems and the Worst Scenario Method. Elsevier, 2004.
[21] I. Hlaváček, M. Křížek: On a superconvergent finite element scheme for elliptic systems. I. Dirichlet boundary condition; II. Boundary conditions of Newton’s or Neumann’s type; III. Optimal interior estimates. Apl. Mat. 32 (1987), 131–154; 200–213; 276–289.
[22] C.O. Horgan: Korn’s inequalities and their applications in continuum mechanics. SIAM Rev. 37 (1995), 491–511. · Zbl 0840.73010 · doi:10.1137/1037123
[23] L.M. Kachanov: Foundations of the Theory of Plasticity. North-Holland, Amstedram, 1971.
[24] J. Karátson: On the Lipschitz continuity of derivatives for some scalar nonlinearities. J. Math. Anal. Appl. 346 (2008), 170–176. · Zbl 1152.47047 · doi:10.1016/j.jmaa.2008.05.053
[25] J. Karátson, S. Korotov: Discrete maximum principles for FEM solutions of some nonlinear elliptic interface problems. Int. J. Numer. Anal. Model. 6 (2009), 1–16. · Zbl 1163.65076
[26] J. Karátson, S. Korotov: Sharp upper global a posteriori error estimates for nonlinear elliptic variational problems. Helsinki University of Technology, Institute of Mathematics, Research Report A527; July 2007. · Zbl 1212.65249
[27] S. Korotov: A posteriori error estimation of goal-oriented quantities for elliptic type BVPs. J. Comput. Appl. Math. 191 (2006), 216–227. · Zbl 1089.65120 · doi:10.1016/j.cam.2005.06.038
[28] S. Korotov: Two-sided a posteriori error estimates for linear elliptic problems with mixed boundary conditions. Appl. Math. 52 (2007), 235–249. · Zbl 1164.65485 · doi:10.1007/s10492-007-0012-7
[29] S. Korotov: Global a posteriori error estimates for convection-reaction-diffusion problems. Appl. Math. Modelling 32 (2008), 1579–1586. · Zbl 1176.65126 · doi:10.1016/j.apm.2007.04.013
[30] M. Křížek, P. Neittaanmäki: Mathematical and Numerical Modelling in Electrical Engineering: Theory and Applications. Kluwer Academic Publishers, Dordrecht, 1996.
[31] D. Kuzmin, A. Hannukainen, S. Korotov: A new a posteriori error estimate for convection-reaction-diffusion problems. J. Comput. Appl. Math. 218 (2008), 70–78. · Zbl 1143.65086 · doi:10.1016/j.cam.2007.04.033
[32] V.G. Maz’ja: Sobolev Spaces. Springer, Berlin, 1985.
[33] E. Miersemann: Zur Regularität verallgemeinerter Lösungen von quasilinearen elliptischen Differentialgleichungen zweiter Ordnung in Gebieten mit Ecken. Z. Anal. Anw. 1 (1982), 59–71. · Zbl 0518.35011
[34] S.G. Mikhlin: The Numerical Performance of Variational Methods. Walters Noordhoff Series of Monographs and Textbooks on Pure and Applied Mathematics. Walters Noordhoff Publishing, Groningen, 1971.
[35] S.G. Mikhlin: Constants in Some Inequalities of Analysis. A Wiley-Interscience Publication. John Wiley & Sons, Chichester, 1986.
[36] A.V. Muzalevsky, S.I. Repin: On two-sided error estimates for approximate solutions of problems in the linear theory of elasticity. Russ. J. Numer. Anal. Math. Model. 18 (2003), 65–85. · Zbl 1027.74070 · doi:10.1515/156939803322008209
[37] J. Necas: Les méthodes directes en théorie des équations elliptiques. Academia, Prague, 1967.
[38] J. Necas, I. Hlavácek: Mathematical Theory of Elastic and Elasto-Plastic Bodies: An Introduction. Studies in Applied Mechanics 3. Elsevier Scientific Publishing, Amsterdam-New York, 1980.
[39] P. Neittaanmäki, S. Repin: A posteriori error estimates for boundary-value problems related to the biharmonic operator. East-West J. Numer. Math. 9 (2001), 157–178. · Zbl 0986.65101
[40] P. Neittaanmäki, S. Repin: Reliable Methods for Computer Simulation. Error Control and A Posteriori Estimates. Studies in Mathematics and Its Applications 33. Elsevier Science B.V., Amsterdam, 2004. · Zbl 1076.65093
[41] A. Neumaier: Certified error bounds for uncertain elliptic equations. J. Comput. Appl. Math. 218 (2008), 125–136. · Zbl 1145.65088 · doi:10.1016/j.cam.2007.04.037
[42] J.T. Oden, S. Prudhomme: Goal-oriented error estimation and adaptivity for the finite element method. Comput. Math. Appl. 41 (2001), 735–756. · Zbl 0987.65110 · doi:10.1016/S0898-1221(00)00317-5
[43] S. Repin: A posteriori error estimation for nonlinear variational problems by duality theory. Zap. Nauchn. Semin. St. Petersburg, Otdel. Mat. Inst. Steklov (POMI) 243 (1997), 201–214. · Zbl 0904.65064
[44] S. Repin, S. Sauter, A. Smolianski: A posteriori error estimation for the Dirichlet problem with account of the error in the approximation of boundary conditions. Computing 70 (2003), 205–233. · Zbl 1128.35319
[45] S. Repin, S. Sauter, A. Smolianski: A posteriori error estimation for the Poisson equation with mixed Dirichlet/Neumann boundary conditions. J. Comput. Appl. Math. 164/165 (2004), 601–612. · Zbl 1038.65114 · doi:10.1016/S0377-0427(03)00491-6
[46] S. Repin, L. S. Xanthis: A posteriori error estimation for elastoplastic problems based on duality theory. Comput. Methods Appl. Mech. Eng. 138 (1996), 317–339. · Zbl 0886.73082 · doi:10.1016/S0045-7825(96)01136-X
[47] M. Rüter, S. Korotov, C. Steenbock: Goal-oriented error estimates based on different FE-spaces for the primal and the dual problem with applications to fracture mechanics. Comput. Mech. 39 (2007), 787–797. · Zbl 1178.74172 · doi:10.1007/s00466-006-0069-2
[48] T. Vejchodsky: Guaranteed and locally computable a posteriori error estimate. IMA J. Numer. Anal. 26 (2006), 525–540. · Zbl 1096.65112 · doi:10.1093/imanum/dri043
[49] R. Verfürth: A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. John Wiley & Sons-Teubner, Chichester-Stuttgart, 1996. · Zbl 0853.65108
[50] M. Vohralík: A posteriori error estimation in the conforming finite element method based on its local conservativity and using local minimization. C.R. Math., Acad. Sci. Paris 346 (2008), 687–690. · Zbl 1142.65086
[51] W.C. Waterhouse: The absolute-value estimate for symmetric multilinear forms. Linear Algebra Appl. 128 (1990), 97–105. · Zbl 0699.15012 · doi:10.1016/0024-3795(90)90284-J
[52] E. Zeidler: Nonlinear Functional Analysis and Its Applications. Springer, New York, 1986. · Zbl 0583.47050
[53] O.C. Zienkiewicz, J. Z. Zhu: A simple error estimator and adaptive procedure for practical engineering analysis. Int. J. Numer. Methods Eng. 24 (1987), 337–357. · Zbl 0602.73063 · doi:10.1002/nme.1620240206
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.