A review of VMS a posteriori error estimation with emphasis in fluid mechanics

Abstract

This article outlines the research on the application of the variational multiscale theory (VMS) to a posteriori error estimation. VMS theory was initially developed by Professor Hughes to evince the origins of stabilized methods. In this paper it is shown that the stabilization parameters and the stabilization terms contain true error information that can be used to obtain explicit and implicit a posteriori error estimates. The technology consists of splitting the exact solution into resolved or coarse scales (finite element solution) and unresolved or fine scales (numerical error). By feeding this splitting into the variational formulation, an exact weak form can be derived for the fine scales as a function of the resolved scales. The way of solving or approximating this equation yields different algorithms and models for error estimation. Furthermore, using the so-called fine-scale Green’s function, an analytical representation of the fine scales is possible. Again, different approximations of this function give rise to various algorithms and models. This theory naturally suggests that the error can be computed by the combination of element interior and inter-element faces residuals with the corresponding error time-scales. From this standpoint, error estimators are developed for the transport equation and the Navier–Stokes equations. This technology can be further used for example to generate adapted meshes, to derive reduced order models and in verification and validation algorithms.