A variational multiscale a posteriori error estimation method for mixed form of nearly incompressible elasticity

This paper presents an error estimation framework for a mixed displacement-pressure finite element method for nearly incompressible elasticity. The proposed method is based on Variational Multiscale (VMS) concepts, wherein the displacement field is decomposed into coarse scales that can be resolved by a given finite element mesh and fine scales that are beyond the resolution capacity of the mesh. Variational projection of fine scales onto the coarse-scale space via variational embedding of the fine-scale solution into the coarse-scale formulation leads to the stabilized method with two major attributes: first, it is free of volumetric locking and, second, it accommodates arbitrary combinations of interpolation functions for the displacement and pressure fields. This VMS-based stabilized method is equipped with naturally derived error estimators and offers various options for numerical computation of the error. Specifically, two error estimators are explored. The first method employs an element-based strategy and a representation of error via a fine-scale error equation defined over element interiors which is evaluated by a direct post-solution evaluation. This quantity when combined with the global pollution error results in a simple explicit error estimator. The second method involves solving the fine-scale error equation through localization to overlapping patches spread across the domain, thereby leading to an implicit calculation of the local error. This implicit calculation when combined with the global pollution error results in an implicit error estimator. The performance of the stabilized method and the error estimators is investigated through numerical convergence tests conducted for two model problems on uniform and distorted meshes. The sharpness and robustness of the estimators is shown to be consistent across the test cases employed.