Flux Recovery and A Posteriori Error Estimators: Conforming Elements for Scalar Elliptic Equations

In this paper, we first study two flux recovery procedures for the conforming finite element approximation to general second-order elliptic partial differential equations. One is accurate in a weighted $L^2$ norm studied in [Z. Cai and S. Zhang, SIAM J. Numer. Anal., 47 (2009), pp. 2132-2156] for linear elements, and the other is accurate in a weighted $H(\mathrm{div})$ norm, up to the accuracy of the current finite element approximation. For the $L^2$ recovered flux, we introduce and analyze an a posteriori error estimator that is more accurate than the explicit residual-based estimator. Based on the $H(\mathrm{div})$ recovered flux, we introduce two a posteriori error estimators. One estimator may be regarded as an extension of the recovery-based estimator studied in [Z. Cai and S. Zhang, SIAM J. Numer. Anal., 47 (2009), pp. 2132-2156] to higher-order conforming elements. The global reliability and the local efficiency bounds for this estimator are established provided that the underlying problem is neither convection- nor reaction-dominant. The other is proved to be exact locally and globally on any given mesh with no regularity assumptions with respect to a norm depending on the underlying problem. Numerical results on test problems for these estimators are also presented.

[1]  Rüdiger Verfürth,et al.  Robust a posteriori error estimators for a singularly perturbed reaction-diffusion equation , 1998 .

[2]  P. Vassilevski.  Multilevel Block Factorization Preconditioners: Matrix-based Analysis and Algorithms for Solving Finite Element Equations , 2008 .

[3]  Ricardo H. Nochetto,et al.  Optimal multilevel methods for H(grad), H(curl), and H(div) systems on graded and unstructured grids , 2009 .

[4]  Shun Zhang,et al.  Recovery-Based Error Estimator for Interface Problems: Conforming Linear Elements , 2009, SIAM J. Numer. Anal..

[5]  Gerd Kunert,et al.  A Note on the Energy Norm for a Singularly Perturbed Model Problem , 2002, Computing.

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

[7]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[8]  Rüdiger Verfürth,et al.  A Note on Constant-Free A Posteriori Error Estimates , 2009, SIAM J. Numer. Anal..

[9]  Jeffrey S. Ovall,et al.  Fixing a ''Bug'' in Recovery-Type A Posteriori Error Estimators , 2006 .

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

[11]  Jean E. Roberts,et al.  Mixed and hybrid finite element methods , 1987 .

[12]  Jinchao Xu,et al.  Superconvergent Derivative Recovery for Lagrange Triangular Elements of Degree p on Unstructured Grids , 2007, SIAM J. Numer. Anal..

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

[14]  Dietrich Braess,et al.  Equilibrated residual error estimates are p-robust , 2009 .

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

[16]  Rolf Rannacher,et al.  An optimal control approach to a posteriori error estimation in finite element methods , 2001, Acta Numerica.

[17]  Rüdiger Verführt,et al.  A review of a posteriori error estimation and adaptive mesh-refinement techniques , 1996, Advances in numerical mathematics.

[18]  Joseph E. Pasciak,et al.  A least-squares approach based on a discrete minus one inner product for first order systems , 1997, Math. Comput..

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

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

[21]  Shun Zhang,et al.  Recovery-Based Error Estimators for Interface Problems: Mixed and Nonconforming Finite Elements , 2010, SIAM J. Numer. Anal..

[22]  Dietrich Braess,et al.  Equilibrated residual error estimator for edge elements , 2007, Math. Comput..

[23]  Douglas N. Arnold,et al.  Multigrid in H (div) and H (curl) , 2000, Numerische Mathematik.

[24]  Ricardo H. Nochetto,et al.  DESIGN AND CONVERGENCE OF AFEM IN H(DIV) , 2007 .

[25]  Tomáš Vejchodský,et al.  Local a posteriori error estimator based on the hypercircle method Combination of the equilibrated residual method and the method of hypercircle , 2004 .

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

[27]  Ricardo H. Nochetto,et al.  Convergence of Adaptive Finite Element Methods , 2002, SIAM Rev..

[28]  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..

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

[30]  T. Manteuffel,et al.  FIRST-ORDER SYSTEM LEAST SQUARES FOR SECOND-ORDER PARTIAL DIFFERENTIAL EQUATIONS : PART II , 1994 .

[31]  Endre Süli,et al.  Adaptive finite element methods for differential equations , 2003, Lectures in mathematics.

[32]  Andreas Veeser,et al.  A posteriori error estimators, gradient recovery by averaging, and superconvergence , 2006, Numerische Mathematik.

[33]  Zhimin Zhang,et al.  A New Finite Element Gradient Recovery Method: Superconvergence Property , 2005, SIAM J. Sci. Comput..

[34]  Randolph E. Bank,et al.  Hierarchical bases and the finite element method , 1996, Acta Numerica.

[35]  Jinchao Xu,et al.  Asymptotically Exact A Posteriori Error Estimators, Part I: Grids with Superconvergence , 2003, SIAM J. Numer. Anal..

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

[37]  Martin Petzoldt,et al.  A Posteriori Error Estimators for Elliptic Equations with Discontinuous Coefficients , 2002, Adv. Comput. Math..

[38]  J. Whiteman.  The Mathematics of Finite Elements and Applications. , 1983 .

[39]  P. Vassilevski,et al.  Multilevel iterative methods for mixed finite element discretizations of elliptic problems , 1992 .

[40]  R. Hiptmair,et al.  MULTIGRID METHOD FORH(DIV) IN THREE DIMENSIONS , 1997 .