Averaging Techniques for a Posteriori Error Control in Finite Element and Boundary Element Analysis

Averaging techniques for a posteriori error control are established for differential and integral equations within a unifying setting. The reliability and efficiency of the introduced estimator results from two grids T h and T H with different polynomial degrees for a smooth exact solution. The proofs are based on first order approximation operators and inverse estimates. For a finer and finer fine mesh T h, the estimator becomes asymptotically exact. The abstract framework is applicable to a finite element method for the Laplace equation, boundary element methods for Symm’s and the hypersingular integral equation or transmission problems.

[1]  W. Wendland.  Elliptic systems in the plane , 1979 .

[2]  Carsten Carstensen,et al.  Averaging Techniques for the Effective Numerical Solution of Symm's Integral Equation of the First Kind , 2005, SIAM J. Sci. Comput..

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

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

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

[6]  Christoph Schwab,et al.  On the extraction technique in boundary integral equations , 1999, Mathematics of Computation.

[7]  Carsten Carstensen,et al.  Averaging Techniques for the A Posteriori BEM Error Control for a Hypersingular Integral Equation in Two Dimensions , 2007, SIAM J. Sci. Comput..

[8]  Carsten Carstensen,et al.  RELIABLE AND EFFICIENT AVERAGING TECHNIQUES AS UNIVERSAL TOOL FOR A POSTERIORI FINITE ELEMENT ERROR CONTROL ON UNSTRUCTURED GRIDS , 2006 .

[9]  C. Schwab,et al.  An Extraction Technique for Boundary Element Methods , 1996 .

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

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

[12]  C. Schwab,et al.  On The Computation of Derivatives up to The Boundary and Recovery Techniques in BEM , 1999 .

[13]  Gabriel Wittum,et al.  Asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes. Part 1: A smooth problem and globally quasi-uniform meshes , 2001, Math. Comput..

[14]  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..

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

[16]  Claes Johnson,et al.  Computational Differential Equations , 1996 .

[17]  Carsten Carstensen,et al.  A posteriori error estimates for boundary element methods , 1995 .

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

[19]  E. Alarcon,et al.  p-adaptive boundary elements , 1986 .

[20]  Carsten Carstensen,et al.  Adaptive boundary-element methods for transmission problems , 1997, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[21]  Carsten Carstensen,et al.  Some remarks on the history and future of averaging techniques in a posteriori finite element error analysis , 2004 .

[22]  W. Hackbusch,et al.  Finite elements on degenerate meshes: inverse-type inequalities and applications , 2005 .

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

[24]  D. Braess.  Enhanced assumed strain elements and locking in membrane problems , 1998 .

[25]  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..

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

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

[28]  Martin Costabel,et al.  A direct boundary integral equation method for transmission problems , 1985 .

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

[30]  Carsten Carstensen,et al.  Averaging technique for FE – a posteriori error control in elasticity. Part I: Conforming FEM , 2001 .

[31]  S Sauter,et al.  Boundary elements methods. Analysis, numerics and implementation of fast algorithms. (Randelementmethoden. Analyse, Numerik und Implementierung schneller Algorithmen.) , 2004 .

[32]  Wolfgang Dahmen,et al.  Inverse inequalities on non-quasi-uniform meshes and application to the mortar element method , 2003, Math. Comput..

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

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

[36]  Carsten Carstensen,et al.  Adaptive coupling of boundary elements and finite elements , 1995 .

[37]  W. McLean.  Strongly Elliptic Systems and Boundary Integral Equations , 2000 .