Volume 5, Issue 4
A Review of Unified A Posteriori Finite Element Error Control

Numer. Math. Theor. Meth. Appl., 5 (2012), pp. 509-558.

Published online: 2012-05

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• Abstract

This paper aims at a general guideline to obtain a posteriori error estimates for the finite element error control in computational partial differential equations. In the abstract setting of mixed formulations, a generalised formulation of the corresponding residuals is proposed which then allows for the unified estimation of the respective dual norms. Notably, this can be done with an approach which is applicable in the same way to conforming, nonconforming and mixed discretisations. Subsequently, the unified approach is applied to various model problems. In particular, we consider the Laplace, Stokes, Navier-Lam\'e, and the semi-discrete eddy current equations.

• Keywords

A~posteriori error analysis finite element method nonconforming finite element method mixed finite element method adaptive algorithm Poisson equation Lam\'e equations Stokes equations Maxwell equations unified a posteriori error analysis discontinuous Galerkin residual estimator

65N30 65N15 35J25

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@Article{NMTMA-5-509, author = {C. Carstensen, M. Eigel, R. H. W. Hoppe and C. Löbhard}, title = {A Review of Unified A Posteriori Finite Element Error Control}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2012}, volume = {5}, number = {4}, pages = {509--558}, abstract = {

This paper aims at a general guideline to obtain a posteriori error estimates for the finite element error control in computational partial differential equations. In the abstract setting of mixed formulations, a generalised formulation of the corresponding residuals is proposed which then allows for the unified estimation of the respective dual norms. Notably, this can be done with an approach which is applicable in the same way to conforming, nonconforming and mixed discretisations. Subsequently, the unified approach is applied to various model problems. In particular, we consider the Laplace, Stokes, Navier-Lam\'e, and the semi-discrete eddy current equations.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2011.m1032}, url = {http://global-sci.org/intro/article_detail/nmtma/5948.html} }
TY - JOUR T1 - A Review of Unified A Posteriori Finite Element Error Control AU - C. Carstensen, M. Eigel, R. H. W. Hoppe & C. Löbhard JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 509 EP - 558 PY - 2012 DA - 2012/05 SN - 5 DO - http://dor.org/10.4208/nmtma.2011.m1032 UR - https://global-sci.org/intro/article_detail/nmtma/5948.html KW - A~posteriori KW - error analysis KW - finite element method KW - nonconforming finite element method KW - mixed finite element method KW - adaptive algorithm KW - Poisson equation KW - Lam\'e equations KW - Stokes equations KW - Maxwell equations KW - unified a posteriori error analysis KW - discontinuous Galerkin KW - residual estimator AB -

This paper aims at a general guideline to obtain a posteriori error estimates for the finite element error control in computational partial differential equations. In the abstract setting of mixed formulations, a generalised formulation of the corresponding residuals is proposed which then allows for the unified estimation of the respective dual norms. Notably, this can be done with an approach which is applicable in the same way to conforming, nonconforming and mixed discretisations. Subsequently, the unified approach is applied to various model problems. In particular, we consider the Laplace, Stokes, Navier-Lam\'e, and the semi-discrete eddy current equations.

C. Carstensen, M. Eigel, R. H. W. Hoppe & C. Löbhard. (1970). A Review of Unified A Posteriori Finite Element Error Control. Numerical Mathematics: Theory, Methods and Applications. 5 (4). 509-558. doi:10.4208/nmtma.2011.m1032
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