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

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 ⊆ ker l. That excludes particular nonconforming finite element methods (NCFEMs) on parallelograms in that V c h 6⊂ 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 Π : V c h → 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-Lamé equations illustrate the presented unifying theory of a posteriori error control for nonconforming finite element methods. 1. Unified Mixed Approach to Error Control Suppose that the primal variable u ∈ V (e.g., the displacement field) is accompanied by a dual variable p ∈ L (e.g., the flux or stress field). Typically L is some Lebesgue and V is some Sobolev space; suppose throughout this paper that L and V are Hilbert spaces and X := L× V . Given bounded bilinear forms (1.1) a : L× L → R and b : L× V → R and well established conditions on a and b [16, 20], the linear and bounded operator A : X → X, defined by (1.2) (A(p, u))(q, v) := a(p, q) + b(p, v) + b(q, u), is bijective. Then, given right-hand sides f ∈ L and g ∈ V , there exists some unique (p, u) ∈ X with a(p, q) + b(q, u) = f(q) for all q ∈ L, (1.3) b(p, v) = g(v) for all v ∈ V . (1.4) Date: October 31, 2006. 2000 Mathematics Subject Classification. 65N10, 65N15, 35J25.