Weighted Clement - Type Interpolation Anda

One of the main mathematical tools in the residual based a posteriori error analysis is a weak interpolation operator due to Cl ement. Based on a partition of unity, we introduce a modiied weak interpolation operator which enjoys a further orthogonality property. As a consequence , the volume contribution in standard residual based a posteriori error estimates can be replaced by a smaller one which is generically of higher order, and so neglectible. We show applications to model problems for conform, non-conform, and mixed nite element methods. 1. Introduction For a very special mesh, D. Yu proved that the edge-contributions dominate the residual based standard a posteriori error estimates for piecewise polynomials of odd degree Y1, Y2]. For unstructured grids, the author and R. Verf urth proved that the volume contributions can be replaced by a term which is generically of higher order CV]. In this paper, we continue , generalise, and reene this work as regards the Dirichlet boundary. The very general approximation results are formulated for Lipschitz continuous partitions of unity, and we investigate stability and approximation properties of the weighted weak Cl ement-type interpolation operator pro