A posteriori error estimation based on numerical realization of the variational multiscale method

This paper presents a numerical realization of the variational multiscale method with the objective of providing a reliable and easy to implement local error estimation technique. The variational multiscale framework provides a systematic approach for solution scale decomposition into coarse scales captured by the mesh and fine or subgrid scales. In the proposed work, the coarse scale errors in the finite element solution are neglected in comparison to the fine scale errors. The fine scale variational equation is then localized using a general localization function over an element, or a patch of elements, to develop a local error estimation technique. Based on the proposed framework, a consistent formulation of a new subdomain error estimator for elliptic problems is derived, without the necessity of introducing an error locality assumption. The new subdomain error estimator is evaluated numerically within a mesh adaptivity algorithm and it is shown to produce very sharp error estimates that outperform the element residual method estimates.