zbMATH — the first resource for mathematics

On the numerical performance of a sharp a posteriori error estimator for some nonlinear elliptic problems. (English) Zbl 1340.65248
Summary: J. Karátson and S. Korotov developed a sharp upper global a posteriori error estimator for a large class of nonlinear problems of elliptic type [Appl. Math., Praha 54, No. 4, 297–336 (2009; Zbl 1212.65249)]. The goal of this paper is to check its numerical performance, and to demonstrate the efficiency and accuracy of this estimator on the base of quasilinear elliptic equations of the second order. The focus will be on the technical and numerical aspects and on the components of the error estimation, especially on the adequate solution of the involved auxiliary problem.
65N15 Error bounds for boundary value problems involving PDEs
35J62 Quasilinear elliptic equations
65N06 Finite difference methods for boundary value problems involving PDEs
Maple; Matlab
Full Text: DOI
[1] M. Ainsworth, J. T. Oden: A Posteriori Error Estimation in Finite Element Analysis. Pure and Applied Mathematics. A Wiley-Interscience Series of Texts, Monographs, and Tracts, Chichester, Wiley, 2000. · Zbl 1008.65076
[2] Axelsson, O.; Maubach, J., On the updating and assembly of the Hessian matrix in finite element methods, Comput. Methods Appl. Mech. Eng., 71, 41-67, (1988) · Zbl 0673.65068
[3] Becker, R.; Rannacher, R., A feed-back approach to error control in finite element methods: basic analysis and examples, East-West J. Numer. Math., 4, 237-264, (1996) · Zbl 0868.65076
[4] Brezinski, C., A classification of quasi-Newton methods. international conference on numerical algorithms, vol. I (marrakesh, 2001), Numer. Algorithms, 33, 123-135, (2003) · Zbl 1030.65053
[5] Faragó, I.; Karátson, J., Numerical solution of nonlinear elliptic problems via preconditioning operators, No. 11, (2002), Huntington · Zbl 1030.65117
[6] Faragó, I.; Karátson, J., The gradient-finite element method for elliptic problems. numerical methods and computational mechanics (miskolc, 1998), Comput. Math. Appl., 42, 1043-1053, (2001) · Zbl 0987.65121
[7] W. Han: A Posteriori Error Analysis via Duality Theory. With Applications in Modeling and Numerical Approximations. Advances in Mechanics and Mathematics 8, Springer, New York, 2005. · Zbl 1081.65065
[8] Hannukainen, A.; Korotov, S., Techniques for a posteriori error estimation in terms of linear functionals for elliptic type boundary value problems, Far East J. Appl. Math., 21, 289-304, (2005) · Zbl 1092.65097
[9] Hlaváček, I.; Křížek, M., On a superconvergent finite element scheme for elliptic systems, I. Dirichlet boundary condition, Apl. Mat., 32, 131-154, (1987) · Zbl 0622.65097
[10] Karátson, J., On the Lipschitz continuity of derivatives for some scalar nonlinearities, J. Math. Anal. Appl., 346, 170-176, (2008) · Zbl 1152.47047
[11] Karátson, J.; Faragó, I., Variable preconditioning via quasi-Newton methods for nonlinear problems in Hilbert space, SIAM J. Numer. Anal. (electronic), 41, 1242-1262, (2003) · Zbl 1130.65309
[12] Karátson, J.; Korotov, S., Sharp upper global a posteriori error estimates for nonlinear elliptic variational problems, Appl. Math., Praha, 54, 297-336, (2009) · Zbl 1212.65249
[13] Karátson, J.; Kovács, B., Variable preconditioning in complex Hilbert space and its application to the nonlinear Schrödinger equation, Comput. Math. Appl., 65, 449-459, (2013) · Zbl 1319.65047
[14] Korotov, S., Global a posteriori error estimates for convection-reaction-diffusion problems, Appl. Math. Modelling, 32, 1579-1586, (2008) · Zbl 1176.65126
[15] Kovács, B., A comparison of some efficient numerical methods for a nonlinear elliptic problem, Cent. Eur. J. Math., 10, 217-230, (2012) · Zbl 1247.65148
[16] Mikhlin, S. G., Constants in some inequalities of analysis. transl. from the German, (1986), Chichester · Zbl 0593.41001
[17] P. Neittaanmäki, S. Repin: Reliable Methods for Computer Simulation. Error Control and a Posteriori Estimates. Studies in Mathematics and its Applications 33, Elsevier, Amsterdam, 2004. · Zbl 1076.65093
[18] Repin, S. I., A posteriori error estimation for nonlinear variational problems by duality theory, J. Math. Sci., New York, 99, 927-935, (2000) · Zbl 0904.65064
[19] R. Verfürth: A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner Series Advances in Numerical Mathematics, John Wiley & Sons, Stuttgart, Chichester, 1996. · Zbl 0853.65108
[20] V. S. Vladimirov: Equations of Mathematical Physics. Transl. from the Russian. Mir, Moskva, 1984.
[21] E. Zeidler: Nonlinear Functional Analysis and its Applications. III: Variational Methods and Optimization. Springer, New York, 1985. · Zbl 0583.47051
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.