×

Verification of functional a posteriori error estimates for obstacle problem in 2D. (English) Zbl 1312.49035

Summary: We verify functional a posteriori error estimates proposed by S. Repin for a class of obstacle problems in two space dimensions. New benchmarks with known analytical solution are constructed based on the one-dimensional benchmark introduced by P. Harasim and J. Valdman. A numerical approximation of the solution of the obstacle problem is obtained by the finite element method using bilinear elements on a rectangular mesh. The error of the approximation is measured by a functional majorant. The majorant value contains three unknown fields: a gradient field discretized by Raviart-Thomas elements, a Lagrange multiplier field discretized by piecewise constant functions and a scalar parameter \(\beta\). The minimization of the majorant value is realized by an alternate minimization algorithm, whose convergence is discussed. Numerical results validate two estimates, the energy estimate bounding the error of the approximation in the energy norm by the difference of the energies of the discrete and exact solutions and the majorant estimate bounding the difference of the energies of the discrete and the exact solutions by the value of the functional majorant.

MSC:

49M25 Discrete approximations in optimal control
49J40 Variational inequalities
65K15 Numerical methods for variational inequalities and related problems
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
74M15 Contact in solid mechanics
74S05 Finite element methods applied to problems in solid mechanics
74K05 Strings
PDF BibTeX XML Cite
Full Text: arXiv Link

References:

[1] Ainsworth, M., Oden, J. T.: A Posteriori Error Estimation in Finite Element Analysis. Wiley and Sons, New York 2000. · Zbl 1008.65076
[2] Babuška, I., Strouboulis, T.: The finite Element Method and its Reliability. Oxford University Press, New York 2001. · Zbl 0997.74069
[3] Bangerth, W., Rannacher, R.: Adaptive Finite Element Methods for Differential Equations. Birkhäuser, Berlin 2003. · Zbl 1020.65058
[4] Braess, D., Hoppe, R. H. W., Schöberl, J.: A posteriori estimators for obstacle problems by the hypercircle method. Comput. Vis. Sci. 11 (2008), 351-362.
[5] Brezi, F., Hager, W. W., Raviart, P. A.: Error estimates for the finite element solution of variational inequalities I. Numer. Math. 28 (1977), 431-443. · Zbl 0369.65030
[6] Buss, H., Repin, S.: A posteriori error estimates for boundary value problems with obstacles. Proc. 3nd European Conference on Numerical Mathematics and Advanced Applications, Jÿvaskylä 1999, World Scientific 2000, pp. 162-170. · Zbl 0968.65041
[7] Carstensen, C., Merdon, C.: A posteriori error estimator competition for conforming obstacle problems. Numer. Methods Partial Differential Equations 29 (2013), 667-692. · Zbl 1364.65243
[8] Dostál, Z.: Optimal Quadratic Programming Algorithms. Springer 2009. · Zbl 1401.90013
[9] Falk, R. S.: Error estimates for the approximation of a class of variational inequalities. Math. Comput. 28 (1974), 963-971. · Zbl 0297.65061
[10] Fuchs, M., Repin, S.: A posteriori error estimates for the approximations of the stresses in the Hencky plasticity problem. Numer. Funct. Anal. Optim. 32 (2011), 610-640. · Zbl 1419.74076
[11] Glowinski, R., Lions, J. L., Trémolieres, R.: Numerical Analysis of Variational Inequalities. North-Holland 1981. · Zbl 0463.65046
[12] Gustafsson, B.: A simple proof of the regularity theorem for the variational inequality of the obstacle problem. Nonlinear Anal. 10 (1986), 12, 1487-1490. · Zbl 0612.49005
[13] Valdman, P. Harasim AD J.: Verification of functional a posteriori error estimates for obstacle problem in 1D. Kybernetika 49 (2013), 5, 738-754. · Zbl 1278.49035
[14] Hlaváček, I., Haslinger, J., Nečas, J., Lovíšek, J.: Solution of variational inequalities in mechanics. Applied Mathematical Sciences 66, Springer-Verlag, New York 1988. · Zbl 0654.73019
[15] Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York 1980. · Zbl 0988.49003
[16] Lions, J. L., Stampacchia, G.: Variational inequalities. Comm. Pure Appl. Math. 20 (1967), 493-519. · Zbl 0152.34601
[17] Neittaanmäki, P., Repin, S.: Reliable Methods for Computer Simulation (Error Control and A Posteriori Estimates). Elsevier, 2004. · Zbl 1076.65093
[18] Nochetto, R. H., Seibert, K. G., Veeser, A.: Pointwise a posteriori error control for elliptic obstacle problems. Numer. Math. 95 (2003), 631-658. · Zbl 1027.65089
[19] Rahman, T., Valdman, J.: Fast MATLAB assembly of FEM matrices in 2D and 3D: nodal elements. Appl. Math. Comput. 219 (2013), 7151-7158. · Zbl 1288.65169
[20] Repin, S.: A posteriori error estimation for variational problems with uniformly convex functionals. Math. Comput. 69 (230) (2000), 481-500. · Zbl 0949.65070
[21] Repin, S.: A posteriori error estimation for nonlinear variational problems by duality theory. Zapiski Nauchn. Semin. POMI 243 (1997), 201-214. · Zbl 0904.65064
[22] Repin, S.: Estimates of deviations from exact solutions of elliptic variational inequalities. Zapiski Nauchn. Semin, POMI 271 (2000), 188-203. · Zbl 1118.35320
[23] Repin, S.: A Posteriori Estimates for Partial Differential Equations. Walter de Gruyter, Berlin 2008. · Zbl 1162.65001
[24] Repin, S., Valdman, J.: Functional a posteriori error estimates for problems with nonlinear boundary conditions. J. Numer. Math. 16 (2008), 1, 51-81. · Zbl 1146.65054
[25] Repin, S., Valdman, J.: Functional a posteriori error estimates for incremental models in elasto-plasticity. Centr. Eur. J. Math. 7 (2009), 3, 506-519. · Zbl 1269.74202
[26] Ulbrich, M.: Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces. SIAM, 2011. · Zbl 1235.49001
[27] Valdman, J.: Minimization of functional majorant in a posteriori error analysis based on \(H(div)\) multigrid-preconditioned CG method. Adv. Numer. Anal. (2009). · Zbl 1200.65095
[28] Zou, Q., Veeser, A., Kornhuber, R., Gräser, C.: Hierarchical error estimates for the energy functional in obstacle problems. Numer. Math. 117 (2012), 4, 653-677. · Zbl 1218.65067
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.