A Gradient Recovery Operator Based on an Oblique Projection

We present a construction of a gradient recovery operator based on an oblique projection, where the basis functions of two involved spaces satisfy a condition of biorthogonality. The biorthogonality condition guarantees that the recovery operator is local. 1. Introduction. One reason for the success of the finite element method for solving partial differential equations is that a reliable a posteriori error estimator can be applied to measure the approximation of the finite element solution in any local region [1, 2]. The a posteriori error estimator uses the finite element solution itself to assess the accuracy of the numerical solution. Based on this assessment, the finite element mesh can be locally refined resulting in an adaptive process of controling the global error. The adaptive refinement process is much more efficient than the uniform refinement process in finite element computation. One of the most popular a posteriori estimators is based on recovery of the gradient of the numerical solution. If the recovered gradient approximates the exact gradient better than the gradient computed directly by using the finite element solution, the comparison gives an a posteriori error estimator. The asymptotic exactness of the estimator is based on some superconvergence results [1, 5, 9, 15, 18, 19]. One can compute the orthogonal projection of the computed gradient of the finite element solution onto the actual finite element space to reconstruct the gradient [3, 7, 11]. As the mass matrix is not diagonal, the recovery process is not local. Although one can use a mass lumping procedure to diagonalize the computed mass matrix, the projection property is not valid and the superconvergence property is, in general, lost after doing the mass lumping procedure. Therefore, in this paper we focus on an oblique projection of the directly computed gradient of the numerical solution. The oblique projection is obtained by using two different finite element spaces, where these two spaces satisfy a biorthogonality property. The trial and test spaces for projecting the finite element gradient are chosen such that arising Gram matrix is diagonal. The biorthogonality property allows the local computation of the recovery operator. We show that the error estimator obtained by using the oblique projection is equivalent to the one obtained by using the orthogonal projection. We introduce our oblique projection in the next section and prove some properties of the recovered gradient.