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

C. Carstensen, M. Eigel, R. H. W. Hoppe & C. Löbhard

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

Published online: 2012-05

[An open-access article; the PDF is free to any online user.]

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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-Lamé, and the semi-discrete eddy current equations.

  • AMS Subject Headings

65N30, 65N15, 35J25

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-5-509, author = {}, 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-Lamé, 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 JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 509 EP - 558 PY - 2012 DA - 2012/05 SN - 5 DO - http://doi.org/10.4208/nmtma.2011.m1032 UR - https://global-sci.org/intro/article_detail/nmtma/5948.html KW - A posteriori, error analysis, finite element method, nonconforming finite element method, mixed finite element method, adaptive algorithm, Poisson equation, Lamé equations, Stokes equations, Maxwell equations, unified a posteriori error analysis, discontinuous Galerkin, 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-Lamé, and the semi-discrete eddy current equations.

C. Carstensen, M. Eigel, R. H. W. Hoppe & C. Löbhard. (2020). 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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