a posteriori error analysis
finite element method
nonconforming finite element method
error estimates reliability
A unifying theory of a posteriori error control for nonconforming finite element methods
Carsten Carstensen
Carstensen
Carsten
Jun Hu
Hu
Jun
Institut für Mathematik, HumboldtUniversität zu Berlin (ISSN 08630976), 23 pages
A unifying theory of a posteriori error control for nonconforming finite element methods
Carsten Carstensen
,
Jun Hu
Preprint series:
Institut für Mathematik, HumboldtUniversität zu Berlin (ISSN 08630976), 23 pages
MSC 2000
 65N15 Error bounds

35J25 Boundary value problems for secondorder, elliptic equations
Abstract
Residualbased a posteriori error estimates were derived within one unifying framework for lowestorder conforming, nonconforming, and mixed finite element schemes in [C. Carstensen, Numerische Mathematik 100 (2005) 617637].
Therein, the key assumption is that the conforming firstorder finite element space $V^c_h$ annulates the linear and bounded residual $l$ written $V^c_h \subseteq \ker l$.
That excludes particular nonconforming finite element methods (NCFEMs) on parallelograms in that $V^c_h \not\subset \ker l$. The present paper generalises the aforementioned theory to more
general situations to deduce new a posteriori error estimates, also for mortar and discontinuous Galerkin methods. The key assumption is the existence of some bounded linear operator $\Pi : V^c_h \to V^nc_h$ with some elementary properties.
It is conjectured that the more general hypothesis (H1)(H3) can be established for all known NCFEMs. Applications on various nonstandard finite element schemes for the Laplace, Stokes, and NavierLame equations illustrate the presented unifying theory of a posteriori error control for nonconforming finite element methods.
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