A unifying theory of a posteriori finite element error control

SummaryResidual-based a posteriori error estimates are derived within a unified setting for lowest-order conforming, nonconforming, and mixed finite element schemes. The various residuals are identified for all techniques and problems as the operator norm ||ℓ|| of a linear functional of the form in the variable υ of a Sobolev space V. The main assumption is that the first-order finite element space is included in the kernel Ker ℓ of ℓ. As a consequence, any residual estimator that is a computable bound of ||ℓ|| can be used within the proposed frame without further analysis for nonconforming or mixed FE schemes. Applications are given for the Laplace, Stokes, and Navier-Lamè equations.

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