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://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 - <p style="text-align: justify;">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.</p>