An Optimal Adaptive Finite Element Method for an Obstacle Problem

Abstract This paper provides a refined a posteriori error control for the obstacle problem with an affine obstacle which allows for a proof of optimal complexity of an adaptive algorithm. This is the first adaptive mesh-refining finite element method known to be of optimal complexity for some variational inequality. The result holds for first-order conforming finite element methods in any spacial dimension based on shape-regular triangulation into simplices for an affine obstacle. The key contribution is the discrete reliability of the a posteriori error estimator from [Numer. Math. 107 (2007), 455–471] in an edge-oriented modification which circumvents the difficulties caused by the non-existence of a positive second-order approximation [Math. Comp. 71 (2002), 1405–1419].

[1]  Andreas Veeser,et al.  Efficient and Reliable A Posteriori Error Estimators for Elliptic Obstacle Problems , 2001, SIAM J. Numer. Anal..

[2]  Christian Kreuzer,et al.  Quasi-Optimal Convergence Rate for an Adaptive Finite Element Method , 2008, SIAM J. Numer. Anal..

[3]  Andreas Veeser,et al.  Hierarchical error estimates for the energy functional in obstacle problems , 2011, Numerische Mathematik.

[4]  R. Glowinski,et al.  Numerical Analysis of Variational Inequalities , 1981 .

[5]  R. Hoppe,et al.  Adaptive multilevel methods for obstacle problems , 1994 .

[6]  Rob P. Stevenson,et al.  Optimality of a Standard Adaptive Finite Element Method , 2007, Found. Comput. Math..

[7]  Ricardo H. Nochetto,et al.  Pointwise a posteriori error control for elliptic obstacle problems , 2003, Numerische Mathematik.

[8]  L. R. Scott,et al.  Finite element interpolation of nonsmooth functions satisfying boundary conditions , 1990 .

[9]  Carsten Carstensen,et al.  Convergence analysis of a conforming adaptive finite element method for an obstacle problem , 2007, Numerische Mathematik.

[10]  Andreas Veeser,et al.  A Posteriori Error Estimators for Regularized Total Variation of Characteristic Functions , 2003, SIAM J. Numer. Anal..

[11]  Wolfgang Dahmen,et al.  Adaptive Finite Element Methods with convergence rates , 2004, Numerische Mathematik.

[12]  Rüdiger Verfürth,et al.  A posteriori error estimation and adaptive mesh-refinement techniques , 1994 .

[13]  I. Babuska,et al.  The finite element method and its reliability , 2001 .

[14]  J. Tinsley Oden,et al.  Local a posteriori error estimators for variational inequalities , 1993 .

[15]  Dietrich Braess,et al.  A posteriori error estimators for obstacle problems – another look , 2005, Numerische Mathematik.

[16]  ROB STEVENSON,et al.  The completion of locally refined simplicial partitions created by bisection , 2008, Math. Comput..

[17]  D. Praetorius,et al.  Convergence of adaptive FEM for some elliptic obstacle problem , 2010 .

[18]  W. Dörfler.  A convergent adaptive algorithm for Poisson's equation , 1996 .

[19]  Ricardo H. Nochetto,et al.  Residual type a posteriori error estimates for elliptic obstacle problems , 2000, Numerische Mathematik.

[20]  Ricardo H. Nochetto,et al.  Data Oscillation and Convergence of Adaptive FEM , 2000, SIAM J. Numer. Anal..

[21]  Carsten Carstensen,et al.  Averaging techniques yield reliable a posteriori finite element error control for obstacle problems , 2004, Numerische Mathematik.

[22]  Ricardo H. Nochetto,et al.  Fully Localized A posteriori Error Estimators and Barrier Sets for Contact Problems , 2004, SIAM J. Numer. Anal..