Polynomial Preserving Gradient Recovery and a Posteriori Estimate for Bilinear Element on Irregular Quadrilaterals

Abstract

A polynomial preserving gradient recovery method is proposed and analyzed for bilinear element under quadrilateral meshes. It has been proven that the recovered gradient converges at a rate $O(h^{1+\rho})$ for $\rho = min(\alpha, 1)$, when the mesh is distorted $O(h^{1+\alpha})$ ($\alpha > 0$) from a regular one. Consequently, the a posteriori error estimator based on the recovered gradient is asymptotically exact.