Hierarchical error estimates for the energy functional in obstacle problems. (English) Zbl 1218.65067

Authors’ abstract: We present a hierarchical a posteriori error analysis for the minimum value of the energy functional in symmetric obstacle problems. The main result is that the error in the energy minimum is, up to oscillation terms, equivalent to an appropriate hierarchical estimator. The proof does not invoke any saturation assumption. We even show that small oscillation implies a related saturation assumption. In addition, we prove efficiency and reliability of an a posteriori estimate of the discretization error and thereby cast some light on the theoretical understanding of previous hierarchical estimators. Finally, we illustrate our theoretical results by numerical computations.


65K15 Numerical methods for variational inequalities and related problems
49J40 Variational inequalities
49M25 Discrete approximations in optimal control
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[1] Ainsworth M., Oden J.T.: A posteriori error estimation in finite element analysis. Wiley, New York (2000) · Zbl 1008.65076
[2] Bank R.E., Smith R.K.: A posteriori error estimates based on hierarchical bases. SIAM J. Numer. Anal. 30, 921–935 (1993) · Zbl 0787.65078
[3] Bartels S., Carstensen C.: Averaging techniques yield reliable a posteriori finite element error control for obstacle problems. Numer. Math. 99(2), 225–249 (2004) · Zbl 1063.65050
[4] Bornemann F.A., Erdmann B., Kornhuber R.: A posteriori error estimates for elliptic problems in two and three space dimensions. SIAM J. Numer. Anal. 33, 1188–1204 (1996) · Zbl 0863.65069
[5] Braess D.: A posteriori error estimators for obstacle problems–another look. Numer. Math. 101, 415–421 (2005) · Zbl 1118.65068
[6] Braess D., Hoppe R.H.W., Schöberl J.: A posteriori estimator for obstacle problems by the hypercircle method. Comp. Vis. Sci. 11, 351–362 (2008)
[7] Brezzi F., Caffarelli L.A.: Convergence of the discrete free boundaries for finite element approximations. RAIRO Numer. Anal. 17, 385–395 (1983)
[8] 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
[9] Deuflhard P., Leinen P., Yserentant H.: Concepts of an adaptive hierarchical finite element code. IMPACT Comput. Sci. Eng. 1, 3–35 (1989) · Zbl 0706.65111
[10] Dörfler W., Nochetto R.H.: Small data oscillation implies the saturation assumption. Numer. Math. 91, 1–12 (2002) · Zbl 0995.65109
[11] Fierro, F., Veeser, A.: A posteriori error estimators for regularized total variation of characteristic functions. SIAM J. Numer. Anal. 41(6), 2032–2055 (2003, electronic) · Zbl 1058.65066
[12] Hoppe R.H.W., Kornhuber R.: Adaptive multilevel-methods for obstacle problems. SIAM J. Numer. Anal. 31(2), 301–323 (1994) · Zbl 0806.65064
[13] Kornhuber R.: A posteriori error estimates for elliptic variational inequalities. Comp. Math. Appl. 31, 49–60 (1996) · Zbl 0857.65071
[14] Kornhuber R.: Adaptive monotone multigrid methods for nonlinear variational problems. Teubner, Stuttgart (1997) · Zbl 0879.65041
[15] Kornhuber R., Zou Q.: Efficient and reliable hierarchical error estimates for the discretization error of elliptic obstacle problems. Preprint 519, DFG Research Center Matheon. Math. Comp. 80, 69–88 (2011) · Zbl 1211.65079
[16] Nochetto R.H., Siebert K.G., Veeser A.: Pointwise a posteriori error control for elliptic obstacle problems. Numer. Math. 95, 631–658 (2003) · Zbl 1027.65089
[17] Sander, O.: Multidimensional coupling in a human knee model. PhD thesis, FU Berlin (2008)
[18] Siebert K.G., Veeser A.: A unilaterally constrained quadratic minimization with adaptive finite elements. SIAM J. Optim. 18, 260–289 (2007) · Zbl 1154.90008
[19] Veeser A.: Efficient and reliable a posteriori error estimators for elliptic obstacle problems. SIAM J. Numer. Anal. 39(1), 146–167 (2001) · Zbl 0992.65073
[20] Veeser A.: Convergent adaptive finite elements for the nonlinear Laplacian. Numer. Math. 92(4), 743–770 (2002) · Zbl 1016.65083
[21] Veeser, A., Verfürth, R.: Poincaré constants of finite element stars. Report Dp. di Mathematica, U Milano and Fak. Mathematik, U Bochum (2010) · Zbl 1235.65127
[22] Verfürth R.: A review of a posteriori error estimation and adaptive mesh–refinement techniques. Wiley-Teubner, New York (1996) · Zbl 0853.65108
[23] Zienkiewicz O.C., De S.R. Gago J.P., Kelly D.W.: The hierarchical concept in finite element analysis. Comp. Str. 16, 53–65 (1983) · Zbl 0498.73072
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