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 Carstensen [Numer Math 100:617–637, 2005]. Therein, the key assumption is that the conforming first-order finite element space $$V_h^c$$ annulates the linear and bounded residual ℓ written $$V_h^c \subseteq {\rm ker} \ell$$ . That excludes particular nonconforming finite element methods (NCFEMs) on parallelograms in that $$V_h^c \not\subset {\rm ker} \ell$$ . 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 $$\Pi: V_h^c \rightarrow V_h^{nc}$$ 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 NCFEM.

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