Averaging techniques for reliable and efficient a posteriori finite element error control : analysis and applications.

Local averaging techniques, which are used to postprocess discrete flux or stress approximations of low-order finite element schemes for elliptic boundary value problems, are applied for error control and adaptive mesh refinement. We put particular emphasis on the explicit calculation of all constants, arising in the proofs of reliability and efficiency, in terms of the known data and quantify the equivalence of local averaging techniques. We highlight and discuss a wide selection of applications for which averaging-based estimators provide highly accurate error control.

[1]  Carsten Carstensen,et al.  Numerical analysis of the primal problem of elastoplasticity with hardening , 1999, Numerische Mathematik.

[2]  John W. Barrett,et al.  A priori and a posteriori error bounds for a nonconforming linear finite element approximation of a non-newtonian flow , 1998 .

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

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

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

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

[7]  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..

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

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

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

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

[12]  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..

[13]  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..

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

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

[16]  Carsten Carstensen,et al.  Adaptive Finite Element Methods for Microstructures? Numerical Experiments for a 2-Well Benchmark , 2003, Computing.

[17]  Carsten Carstensen,et al.  Numerical Analysis of Time-Depending PrimalElastoplasticity with Hardening , 2000, SIAM J. Numer. Anal..