Proceedings of the 24 th UK Conference of the Association for Computational Mechanics in Engineering 31 March - 01 April 2016, Cardiff University, Cardiff Comparison and improvement of implicit residual error estimates based on reduction to local Dirichlet Patch Problems through an unified framework Pedro Bonilla 1 , Abhishek Kundu 1 and *Pierre Kerfriden 1 1 Institute of Mechanics Materials and Advanced Manufacturing, Cardiff University, CF24 3AA - United Kingdom *KerfridenP@cardiff.ac.uk ABSTRACT A posteriori error estimation has become essential to assess the accuracy and robustness of Finite Element simulations. Furthermore, it is a key feature for any adaptivity procedure. The present work aims for an unified framework for different Reduction of the Residual to Local Patch approaches with Dirichlet boundary conditions, allowing its comparison and improvement through hybridization. The paper introduces the ideas of treating the residual as an element-wise pre-stress for the error problem rather than averaging patch contributions and decoupling the error fluctuation inside an element from the error fluctuation over its edges. Computational results in terms of accuracy, computational time and cost-efficiency for the different approaches and some new estimates based on the combination of the known and new concepts are included. Poisson and linear elasticity problems are used to compare the methods. Key Words: error estimation ; patch residual approach ; comparison 1. Introduction The Finite Element Method is a powerful tool to approximate boundary value problems for partial differential equations. The aim of the simulations is to compute a solution with minimal computational cost. However, we need reliable bounds for all the errors involved in them in order to extract useful conclusions from the simulations. A posteriori error estimation is recognized as a major tool not only to assess the accuracy of the computations, but to reduce the computational cost of the estimations through adaptive schemes. Techniques such as goal-oriented estimation or linearization can be applied to the estimators in order to measure the error in a norm relevant to the physical phenomena involved and to do it for non-linear problems. The energy norm of linear problems is enough for the comparison of different methods, and consequently the aforementioned techniques are not considered. There are two main families of a posteriori error estimator approaches. The first one is the Recovery- Based family, consisting in the approximation of a better recovered gradient post-processing the data computed for the coarse solution (for instance making it continuous) then use it to minimize the error. The other family of estimators is based on the Residual of the true error. Neumann or Dirichlet approaches can be used for approximate the global residual problem to a set of local ones. M. Ainsworth and J.T. Oden [1] published a monograph about this. The reduction of the Residual to local Dirichlet Problems retains most of the good properties of the Residual family, specifically the ones concerning the robustness for local non-linear phenomena. Fur- thermore its ease of implementation is on the scale of the Gradient Recovery methods. On the other hand, it lacks the property of a guaranteed upper bound provided by the Equilibrated Residual approaches. The method uses the assumption that the contribution to the error of neighbour elements is of higher order than the distant ones, enhancing prediction of local phenomena.