A unifying theory of a posteriori error control for nonconforming finite element methods

Carsten Carstensen , Jun Hu

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 23 pages

MSC 2000

65N15 Error bounds

35J25 Boundary value problems for second-order, elliptic equations

Abstract
Residual-based a posteriori error estimates were derived within one unifying framework for lowest-order conforming, nonconforming, and mixed finite element schemes in [C. Carstensen, Numerische Mathematik 100 (2005) 617-637]. Therein, the key assumption is that the conforming first-order 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 Navier-Lame equations illustrate the presented unifying theory of a posteriori error control for nonconforming finite element methods.