Flux Recovery and A Posteriori Error Estimators: Conforming Elements for Scalar Elliptic Equations
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
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In this paper, we develop a postprocessing derivative recovery scheme for the finite element solution $u_h$ on general unstructured but shape regular triangulations. In the case of continuous piecewise polynomials of degree $p\geq 1$, by applying the global $L^2$ projection ($Q_h$) and a smoothing operator ($S_h$), the recovered $p$th derivatives ($S_h^m Q_h\partial^p u_h$) superconverge to the exact derivatives ($\partial^p u$). Based on this technique we are able to derive a local error indicator depending only on the geometry of corresponding element and the $(p+1)$st derivatives approximated by $\partial S_h^m Q_h\partial^p u_h$. We provide several numerical examples illustrating the effectiveness of our schemes. We also observe that higher order elements are likely to require more conservative refinement strategies to create meshes corresponding to optimal orders of convergence.