On the asymptotic exactness of error estimators based on the equilibrated residual method for quadrilateral finite elements

We analyze the equilibrated residual method for a posteriori error estimation of finite element approximation on quadrilateral elements. We prove that the estimator obtained by solving the element residual problems over an infinite-dimensional space H1(K)/R is asymptotically exact in the energy norm for regular solutions, provided that the degree of approximation is of odd order and the elements are rectangles. Furthermore, when a finite-dimensional Lobatto approximate subspace is taken to solve the element residual problems, we derive a more favorable result, i.e., the error estimator is asymptotically exact for regular solutions, provided the mesh is parallel and the degree of approximation is of p-th order with p>1.