Superconvergent patch recovery of finite‐element solution and a posteriori L₂ norm error estimate

Abstract

Abstract In the paper we present a superconvergent patch recovery technique for obtaining higher‐order‐accurate finite‐element solutions and thus a postprocessed type of L 2 norm error estimate. Two modifications make our procedure different from the one proposed by Zienkiewicz and Zhu (1992), in which higher‐order‐accurate derivatives of the finite‐element solution at nodes are determined. Firstly, the recovery process is made for element, not for nodes. An ‘element patch’, which represents the union of an element under consideration and the surrounding elements, is introduced. Secondly, the local error estimate is calculated directly from the improved solution for this element. Numerical tests on both 1D and 2D model problems show that this method can provide an asymptotically exact a posteriori L 2 norm error estimate if the used element possesses superconvergent points for the solutions.