Estimator competition for Poisson problems

We compare 13 different a posteriori error estimators for the Poisson problem with lowest-order finite element discretization. Residual-based error estimators compete with a wide range of averaging estimators and estimators based on local problems. Among our five benchmark problems we also look on two examples with discontinuous isotropic diffusion and their impact on the performance of the estimators.

[1]  Carsten Carstensen,et al.  Merging the Bramble-Pasciak-Steinbach and the Crouzeix-Thomée criterion for H1-stability of the L2-projection onto finite element spaces , 2002, Math. Comput..

[2]  P. Raviart,et al.  A mixed finite element method for 2-nd order elliptic problems , 1977 .

[3]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis , 2000 .

[4]  Barbara I. Wohlmuth,et al.  A Local A Posteriori Error Estimator Based on Equilibrated Fluxes , 2004, SIAM J. Numer. Anal..

[5]  Rodolfo Rodríguez.  A Posteriori Error Analysis in the Finite Element Method , 1994 .

[6]  Ricardo H. Nochetto,et al.  Local problems on stars: A posteriori error estimators, convergence, and performance , 2003, Math. Comput..

[7]  Carsten Carstensen,et al.  Averaging technique for FE – a posteriori error control in elasticity. Part II: λ-independent estimates , 2001 .

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

[9]  Stefan A. Sauter,et al.  A Posteriori Error Estimation for the Dirichlet Problem with Account of the Error in the Approximation of Boundary Conditions , 2003, Computing.

[10]  Claes Johnson,et al.  Introduction to Adaptive Methods for Differential Equations , 1995, Acta Numerica.

[11]  I. Babuska,et al.  A feedback element method with a posteriori error estimation: Part I. The finite element method and some basic properties of the a posteriori error estimator , 1987 .

[12]  J. Tinsley Oden,et al.  A posteriori error estimators for second order elliptic systems: Part 1. Theoretical foundations and a posteriori error analysis , 1993 .

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

[14]  J. Oden,et al.  A unified approach to a posteriori error estimation using element residual methods , 1993 .

[15]  Ronald H. W. Hoppe,et al.  Element-oriented and edge-oriented local error estimators for nonconforming finite element methods , 1996 .

[16]  Carsten Carstensen,et al.  An experimental survey of a posteriori Courant finite element error control for the Poisson equation , 2001, Adv. Comput. Math..

[17]  Carsten Carstensen,et al.  Fully Reliable Localized Error Control in the FEM , 1999, SIAM J. Sci. Comput..

[18]  Rolf Rannacher,et al.  A Feed-Back Approach to Error Control in Finite Element Methods: Basic Analysis and Examples , 1996 .

[19]  Sergey Repin.  Two-sided estimates of deviation from exact solutions of uniformly elliptic equations , 2003 .

[20]  Pierre Ladevèze,et al.  Error Estimate Procedure in the Finite Element Method and Applications , 1983 .

[21]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[22]  Rüdiger Verfürth,et al.  Adaptive finite element methods for elliptic equations with non-smooth coefficients , 2000, Numerische Mathematik.

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

[24]  Ricardo H. Nochetto,et al.  Small data oscillation implies the saturation assumption , 2002, Numerische Mathematik.

[25]  C. Carstensen.  QUASI-INTERPOLATION AND A POSTERIORI ERROR ANALYSIS IN FINITE ELEMENT METHODS , 1999 .

[26]  M. Petzoldt.  Regularity and error estimators for elliptic problems with discontinuous coefficients , 2001 .

[27]  S. Repin.  A Posteriori Estimates for Partial Differential Equations , 2008 .

[28]  Carsten Carstensen,et al.  All first-order averaging techniques for a posteriori finite element error control on unstructured grids are efficient and reliable , 2003, Math. Comput..

[29]  Carsten Carstensen,et al.  Inhomogeneous Dirichlet conditions in a priori and a posteriori finite element error analysis , 2004, Numerische Mathematik.

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

[31]  R. Rodríguez.  Some remarks on Zienkiewicz‐Zhu estimator , 1994 .

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

[33]  D. Braess,et al.  EQUILIBRATED RESIDUAL ERROR ESTIMATOR FOR MAXWELL ’ S EQUATIONS , 2006 .

[34]  Ricardo H. Nochetto,et al.  Removing the saturation assumption in a posteriori error analysis , 1993 .

[35]  Mark Ainsworth,et al.  A posteriori error estimators for second order elliptic systems part 2. An optimal order process for calculating self-equilibrating fluxes , 1993 .

[36]  Carsten Carstensen,et al.  Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part I: Low order conforming, nonconforming, and mixed FEM , 2002, Math. Comput..

[37]  Carsten Carstensen,et al.  Averaging techniques for reliable a posteriori FE-error control in elastoplasticity with hardening , 2003 .

[38]  O. C. Zienkiewicz,et al.  A simple error estimator and adaptive procedure for practical engineerng analysis , 1987 .

[39]  C. Bahriawati,et al.  Three Matlab Implementations of the Lowest-order Raviart-Thomas Mfem with a Posteriori Error Control , 2005 .

[40]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[41]  R. Verfürth,et al.  Edge Residuals Dominate A Posteriori Error Estimates for Low Order Finite Element Methods , 1999 .

[42]  Carsten Carstensen,et al.  A posteriori error control in low-order finite element discretisations of incompressible stationary flow problems , 2001, Math. Comput..

[43]  Carsten Carstensen,et al.  Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part II: Higher order FEM , 2002, Math. Comput..