A Unifying Theory Of A Posteriori Finite Element Error Control

Abstract

Residual-based a posteriori error estimates are derived wihtin 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 \[ l(v) := \int_{\Omega} p_h : Dv dx + \int_{\Omega} g_{\Omega} \cdot v dx \] in the variable $v$ of a Soboloev space $V$. The main assumption is that the first-order finite element space $S^1_0 (\Omega) \subset \ker l \subset V$ is included in the kernel $\ker l$ of $l$. As a consequence, {\it any residual estimator} that is a computable bound of $\| l \|$ can be used within the proposed frame {\it without} further analysis for nonconforming or mixed FE schemes. Applications are given for the Laplace, Stokes, and Navier-Lam\'e equations.