Previous |  Up |  Next


a posteriori error estimation; error control in energy norm; two-sided error estimation; differential equation of elliptic type; mixed boundary conditions
The paper is devoted to verification of 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.
[1] M.  Ainsworth, J. T.  Oden: A Posteriori Error Estimation in Finite Element Analysis. John Wiley & Sons, , 2000. MR 1885308
[2] I.  Babuška, T.  Strouboulis: The Finite Element Method and Its Reliability. Oxford University Press, New York, 2001. MR 1857191
[3] W.  Bangerth, R.  Rannacher: Adaptive Finite Element Methods for Differential Equations. Lectures in Mathematics ETH Zürich. Birkhäuser-Verlag, Basel, 2003. MR 1960405
[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. MR 1430239
[5] J.  Brandts, M.  Křížek: Gradient superconvergence on uniform simplicial partitions of polytopes. IMA J.  Numer. Anal. 23 (2003), 489–505. DOI 10.1093/imanum/23.3.489 | MR 1987941
[6] 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. MR 1807259
[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. MR 0520174
[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. MR 1352472
[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. MR 2106499
[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. MR 2101057
[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. MR 2216003
[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. MR 2376087
[13] I.  Hlaváček, J.  Chleboun, and I.  Babuška: Uncertain Input Data Problems and the Worst Scenario Method. Elsevier, Amsterdam, 2004. MR 2285091
[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. MR 0895878
[15] S.  Korotov: A posteriori error estimation for linear elliptic problems with mixed boundary conditions. Preprint  A495, Helsinki University of Techology (March 2006). MR 2219926
[16] S.  Korotov: A posteriori error estimation of goal-oriented quantities for elliptic type BVPs. J.  Comput. Appl. Math. 191 (2006), 216–227. DOI 10.1016/ | MR 2219926 | Zbl 1089.65120
[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. DOI 10.1163/156939503322004882 | MR 1976438
[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. MR 1431889
[19] C.  Lovadina, R.  Stenberg: Energy norm a posteriori error estimates for mixed finite element methods. Math. Comput. 75 (2006), 1659–1674. DOI 10.1090/S0025-5718-06-01872-2 | MR 2240629
[20] S. G.  Mikhlin: Constants in Some Inequalities of Analysis. A Wiley-Interscience Publication. John Wiley & Sons, Chichester, 1986. MR 0853915
[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. MR 2095603
[22] J.  Nečas: Les Méthodes Directes en Théorie des Équations Elliptiques. Academia, Prague, 1967. MR 0227584
[23] J. T.  Oden, S.  Prudhomme: Goal-oriented error estimation and adaptivity for the finite element method. Comput. Math. Appl. 41 (2001), 735–756. DOI 10.1016/S0898-1221(00)00317-5 | MR 1822600
[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. MR 1629741 | 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. MR 2018375
[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). MR 1971889
[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. MR 2011610
[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. DOI 10.1016/S0377-0427(03)00491-6 | MR 2056902
[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. DOI 10.1007/s00466-006-0069-2 | MR 2298591
[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. DOI 10.1016/j.cma.2004.05.032 | MR 2186137
[31] T.  Vejchodský: Guaranteed and locally computable a posteriori error estimate. IMA J.  Numer. Anal. 26 (2006), 525–540. DOI 10.1093/imanum/dri043 | MR 2241313 | Zbl 1096.65112
[32] R.  Verfürth: A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, Stuttgart, 1996.
[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. DOI 10.1002/nme.1620240206 | MR 0875306
Partner of
EuDML logo