A posteriori error estimates for elliptic variational inequalities. (English) Zbl 0857.65071

A posteriori error estimates play a crucial role in the approximative solution of partial differential equations by adaptive finite element methods. In the paper, the author describes a hierachical error estimate which results from the following two steps: (i) Discretize the defect problem with respect to an enlarged space. (ii) Localize the discrete defect problem by domain decomposition.
This method reduces the evaluation amounts to the solution of corresponding scalar local subproblems. Some upper bounds for the effectivity rates and the numerical properties of the method are derived and illustrated by typical examples.


65K10 Numerical optimization and variational techniques
49M15 Newton-type methods
49J40 Variational inequalities
35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)


Full Text: DOI


[1] Zienkiewicz, O. C.; De, J. P.; Gago, S. R.; Kelly, D. W., The hierarchical concept in finite element analysis, Computers & Structures, 16, 53-65 (1983) · Zbl 0498.73072
[2] Deuflhard, P.; Leinen, P.; Yserentant, H., Concepts of an adaptive hierarchical finite element code, IMPACT Comput. Sci. Engrg., 1, 3-35 (1989) · Zbl 0706.65111
[3] Bornemann, F. A.; Erdmann, B.; Kornhuber, R., Adaptive multilevel methods in three space dimensions, J. Numer. Meth. Engrg., 36, 3187-3203 (1993) · Zbl 0780.73073
[5] Verfürth, R., A review of a posteriori error estimation and adaptive mesh-refinement techniques (1993), Manuscript · Zbl 1189.76394
[6] Verfürth, R., A posteriori error estimation and adaptive mesh-refinement techniques, J. Comp. Appl. Math., 50, 67-83 (1994) · Zbl 0811.65089
[7] Bank, R. E.; Smith, R. K., A posteriori error estimates based on hierarchical bases, SIAM J. Num. Anal., 30, 921-935 (1993) · Zbl 0787.65078
[8] Kornhuber, R.; Roitzsch, R., Self adaptive computation of the breakdown voltage of planar pn-junctions with multistep field plates, (Fichtner, W.; Aemmer, D., Simulation of Semiconductor Devices and Processes (1991), Hartung-Gorre: Hartung-Gorre Konstanz), 535-543
[9] Kornhuber, R.; Roitzsch, R., Self adaptive finite element simulation of bipolar, strongly reverse biased pn-junctions, Comm. Numer. Meth. Engrg., 9, 243-250 (1993) · Zbl 0781.65085
[10] Hoppe, R. H.W.; Kornhuber, R., Adaptive multilevel-methods for obstacle problems, SIAM J. Numer. Anal., 31, 2, 301-323 (1994) · Zbl 0806.65064
[11] Deuflhard, P., Cascadic conjugate gradient methods for elliptic partial differential equations. Algorithm and results, (Keyes, D. E.; Xu, J., Proceedings of the \(7^{th}\) International Conference on Domain Decomposition Methods 1993 (1994), AMS: AMS Providence, RI), 29-42 · Zbl 0817.65090
[12] Shaidurov, V. V., Some estimates of the rate of convergence for the cascadic conjugate-gradient method, (Preprint Nr. 4 (1994), Otto-von-Guericke-Universität: Otto-von-Guericke-Universität Magdeburg) · Zbl 0886.65107
[14] Crank, J., Free and Moving Boundary Problems (1988), Oxford University Press: Oxford University Press Oxford
[15] Duvaut, G.; Lions, J. L., Les inéquations en mécanique et en physique (1972), Dunaud: Dunaud Paris · Zbl 0298.73001
[16] Glowinski, R., Numerical Methods for Nonlinear Variational Problems (1984), Springer-Verlag: Springer-Verlag New York · Zbl 0575.65123
[17] Brezzi, F.; Hager, W. W.; Raviart, P. A., Error estimates for the finite element solution of variational inequalities I, Numer. Math., 28, 431-443 (1977) · Zbl 0369.65030
[18] Elliot, C. M., Error analysis of the enthalpy method for the Stefan problem, IMA J. Numer. Anal., 7, 61-71 (1987) · Zbl 0638.65088
[19] Kornhuber, R., Monotone multigrid methods for elliptic variational inequalities I, Numer. Math., 69, 167-184 (1994) · Zbl 0817.65051
[22] Babus̆ka, I.; Rheinboldt, W. C., Error estimates for adaptive finite element computations, SIAM J. Numer. Anal., 15, 736-754 (1978) · Zbl 0398.65069
[23] Bank, R. E., PLTMG—A Software Package for Solving Elliptic Partial Differential Equations, User’s Guide 6.0, (Frontiers in Applied Mathematics (1990), SIAM: SIAM Philadelphia) · Zbl 0717.68001
[24] Dörfler, W., Orthogonale Fehlermethoden (1993), Manuscript
[25] Beck, R.; Erdmann, B.; Roitzsch, R., An object-oriented adaptive finite element code, (Technical Report TR 95-4 (1995), Konrad-Zuse-Zentrum: Konrad-Zuse-Zentrum Berlin), KASKADE 3.0 · Zbl 0882.65095
[26] Rodrigues, J. F., Obstacle Problems in Mathematical Physics, (Mathematical Studies (1987), North-Holland: North-Holland Amsterdam), Number 134 · Zbl 1335.35311
[27] Glowinski, R.; Lions, J. L.; Trémolières, Numerical Analysis of Variational Inequalities (1981), North-Holland: North-Holland Amsterdam · Zbl 0508.65029
[28] Hoppe, R. H.W., Multigrid algorithms for variational inequalities, SIAM J. Numer. Anal., 24, 1046-1065 (1987) · Zbl 0628.65046
[29] Kuznetsov, Y.; Neittaanmäki, P.; Tarvainen, P., Overlapping block relaxation and Schwarz methods for the obstacle problem with a convection diffusion operator, (Keyes, D. E.; Xu, J., Proceedings of the \(7^{th}\) International Conference on Domain Decomposition Methods 1993 (1994), AMS: AMS Providence, RI), 251-257 · Zbl 0817.65052
[30] Jerome, J. W., Approximation of Nonlinear Evolution Equations (1983), Academic Press: Academic Press New York
[31] Nocchetto, R. H.; Paolini, M.; Verdi, C., An adaptive finite element method for two-phase Stefan problems in two space dimensions. Part I. Stability and error estimates, Math. Comp., 57, 195, 73-108 (1991) · Zbl 0733.65087
[32] Nocchetto, R. H.; Paolini, M.; Verdi, C., An adaptive finite element method for two-phase Stefan problems in two space dimensions. Part II. Implementation and numerical experiments, SIAM J. Sci. Stat. Comput., 12, 5, 1207-1244 (1991) · Zbl 0733.65088
[33] Ciavaldini, J. F., Analyse numérique d’un problème de Stefan a deux phases par une méthode d’éléments finis, SIAM J. Numer. Anal., 12, 464-487 (1975) · Zbl 0272.65101
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.