Implicit a posteriori error estimation using patch recovery techniques

Implicit a posteriori error estimators are developed for elliptic boundary value problems.Local problems are formulated for the error and the corresponding Neumann type boundary conditions are approximated using a new family of gradient averaging procedure.The convergence properties of the implicit error estimator are discussed independently from that of the residual type error estimators, which provides a freedom in the choice of the boundary conditions. General assumptions are elaborated for the gradient averaging which define a family of implicit a posteriori error estimators. The performance and the favor of the method is demonstrated trough some numerical experiments.