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Two-sided a posteriori error estimates for linear elliptic problems with mixed boundary conditions. (English) Zbl 1164.65485
Summary: The paper is devoted to the 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 consisting of a linear elliptic (reaction-diffusion) equation with a mixed Dirichlet/Neumann/Robin boundary condition is considered in this work. On the base of this model, we present simple technologies for straightforward constructing computable upper and lower bounds for the error, which is understood as the difference between the exact solution of the model and its approximation measured in the corresponding energy norm. The estimates obtained are completely independent of the numerical technique used to obtain approximate solutions and are “flexible” in the sense that they can be, in principle, made as close to the true error as the resources of the used computer allow.

MSC:
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65G20 Algorithms with automatic result verification
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References:
[1] M. Ainsworth, J.T. Oden: A Posteriori Error Estimation in Finite Element Analysis. John Wiley & Sons, 2000. · Zbl 1008.65076
[2] I. Babuška, T. Strouboulis: The Finite Element Method and Its Reliability. Oxford University Press, New York, 2001.
[3] W. Bangerth, R. Rannacher: Adaptive Finite Element Methods for Differential Equations. Lectures in Mathematics ETH Zürich. Birkhäuser-Verlag, Basel, 2003. · Zbl 1020.65058
[4] R. Becker, R. Rannacher: A feed-back approach to error control in finite element methods: Basic analysis and examples. East-West J. Numer. Math. 4 (1996), 237–264. · Zbl 0868.65076
[5] 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
[6] C. Carstensen, S. A. Funken: Constants in Clément-interpolation error and residual based a posteriori error estimates in-nite element methods. East-West J. Numer. Math. 8 (2000), 153–175. · Zbl 0973.65091
[7] 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.
[8] K. Eriksson, D. Estep, P. Hansbo, C. Johnson: Introduction to adaptive methods for differential equations. Acta Numerica (A. Israel, ed.). Cambridge University Press, Cambridge, 1995, pp. 106–158. · Zbl 0829.65122
[9] I. Faragó, J. Karátson: Numerical Solution of Nonlinear Elliptic Problems via Preconditioning Operators: Theory and Applications. Advances in Computation. Theory and Practice, Vol. 11. Nova Science Publishers, Huntigton, 2002.
[10] W. Han: A Posteriori Error Analysis via Duality Theory. With Applications in Modeling and Numerical Approximations. Advances in Mechanics and Mathematics, Vol. 8. Springer-Verlag, New York, 2005. · Zbl 1081.65065
[11] 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
[12] A. Hannukainen, S. Korotov: Computational technologies for reliable control of global and local errors for linear elliptic type boundary value problems. Preprint A494. Helsinki University of Technology (February 2006); accepted by JNAIAM, J. Numer. Anal. Ind. Appl. Math. in 2007. · Zbl 1161.65081
[13] I. Hlaváček, J. Chleboun, and I. Babuška: Uncertain Input Data Problems and the Worst Scenario Method. Elsevier, Amsterdam, 2004.
[14] I. Hlaváček, M. Křížek: On a superconvergent finite element scheme for elliptic systems I, II, III. Apl. Mat. 32 (1987), 131–154, 200–213, 276–289.
[15] S. Korotov: A posteriori error estimation for linear elliptic problems with mixed boundary conditions. Preprint A495. Helsinki University of Techology (March 2006). · Zbl 1089.65120
[16] 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
[17] S. Korotov, P. Neittaanmäki, and S. Repin: A posteriori error estimation of goal-oriented quantities by superconvergence patch recovery. J. Numer. Math. 11 (2003), 33–59. · Zbl 1039.65075
[18] M. Křížek, P. Neittaanmäki: Mathematical and Numerical Modelling in Electrical Engineering. Theory and Practice. Mathematical Modelling: Theory and Applications, Vol. 1. Kluwer Academic Publishers, Dordrecht, 1996.
[19] C. Lovadina, R. Stenberg: Energy norm a posteriori error estimates for mixed finite element methods. Math. Comput. 75 (2006), 1659–1674. · Zbl 1119.65110 · doi:10.1090/S0025-5718-06-01872-2
[20] S. G. Mikhlin: Constants in Some Inequalities of Analysis. A Wiley-Interscience Publication. John Wiley & Sons, Chichester, 1986.
[21] P. Neittaanmäki, S. Repin: Reliable Methods for Computer Simulation. Error Control and A Posteriori Estimates. Studies in Mathematics and its Applications, Vol. 33. Elsevier, Amsterdam, 2004. · Zbl 1076.65093
[22] J. Nečas: Les Méthodes Directes en Théorie des Équations Elliptiques. Academia, Prague, 1967.
[23] 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
[24] S. Repin: A posteriori error estimation for nonlinear variational problems by duality theory. Zap. Nauchn. Semin. S.-Peterburg, Otdel. Mat. Inst. Steklov. (POMI) 243 (1997), 201–214. · Zbl 0904.65064
[25] S. Repin: Two-sided estimates of deviation from exact solutions of uniformly elliptic equations. Amer. Math. Soc. Transl. 209 (2003), 143–171. · Zbl 1039.65076
[26] S. Repin, M. Frolov: A posteriori estimates for the accuracy of approximate solutions of boundary value problems for equations of elliptic type. Zh. Vychisl. Mat. Mat. Fiz. 42 (2002), 1774–1787 (in Russian); translation in Comput. Math. Math. Phys. 42 (2002), 1704–1716. · Zbl 1116.65324
[27] 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
[28] 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
[29] M. Rüter, S. Korotov, and Ch. Steenbock: Goal-oriented error estimates based on different FE-solution 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
[30] M. Rüter, E. Stein: Goal-oriented a posteriori error estimates in linear elastic fracture mechanics. Comput. Methods Appl. Mech. Eng. 195 (2006), 251–278. · Zbl 1193.74128 · doi:10.1016/j.cma.2004.05.032
[31] T. Vejchodský: Guaranteed and locally computable a posteriori error estimate. IMA J. Numer. Anal. 26 (2006), 525–540. · Zbl 1096.65112 · doi:10.1093/imanum/dri043
[32] R. Verfürth: A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, Stuttgart, 1996. · Zbl 0853.65108
[33] 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
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