Simplified Subgrid multiscale stabilized finite element method in the transient framework for Stokes equations

Abstract

This article presents a novel stabilized finite element analysis for the transient Stokes model. The algebraic subgrid variational multiscale finite element scheme with dynamic subscales approach has been employed to arrive at the stabilized formulation. Both the coarse and the fine scale solutions are of time dependent nature and the unknown fine scale solution is completely eliminated in terms of the coarse scale solution during the derivation. This elimination results into the emergence of a new subgrid multiscale stabilized formulation in the transient framework. This formulation facilitates the theoretical derivations of the robustness properties of the scheme. The fully implicit backward Euler scheme has been applied for the time discretization. Here we have analysed the stability property of the approximate solution. As well as a detailed derivation of the a posteriori error estimate has been presented. The scheme is numerically validated for a benchmark problem and appropriate numerical experiments have been carried out to verify the theoretically established order of convergence results.