Refined Fully Explicit a Posteriori Residual-Based Error Control

The explicit residual-based a posteriori error estimator for elliptic partial differential equations is reliable up to the multiplication of some generic constant which needs to be involved for full error control. The present mathematical literature takes this constant from the stability and approximation properties of Clement-type quasi-interpolation operators and so results in an overestimation of the error which is bigger than for implicit and more expensive a posterori error estimators. This paper propagates a paradigm shift to start with an equilibration error estimator technique followed by its efficiency analysis. The outcome is a refined residual-based a posteriori error estimate with explicit constants which leads to slightly sharper error control than the work of Veeser and Verfurth in 2009. A first application to guaranteed explicit error estimation for two-dimensional nonconforming and a generalization to higher-order finite element methods conclude the paper.

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