Variational Multi-Scale Algebraic Subgrid Stabilized Finite Element Method for Nonlinear PDEs modeling flow systems

Abstract: The study of the transportation of solute in a fluid flowing through various domains has been an interesting area of research for more than six decades due to its wide range of applications in the fields of biomedical engineering, environmental sciences, fluid mechanics, chemical engineering, etc. This physical phenomenon is mathematically expressed in terms of a coupled system of fluid flow and transport equations. Several difficulties faced in finding the analytical solutions have given rise to the development of various numerical schemes to approximate the solution of the complex coupled systems efficiently. The standard Galerkin finite element method (FEM), though being a good numerical scheme, suffers from numerical instabilities for convection-dominated problems. Besides the pair of the velocity-pressure finite-dimensional spaces is required to satisfy an inf-sup compatibility condition, and hence many computationally convenient pairs of spaces are debarred to be used. Several stabilization techniques have been developed in order to stabilize the standard FEM. In this talk, the Sub-Grid Scales (SGS) stabilized FEM, the most general approach among the stabilized FE schemes, has been introduced first for steady variable coefficients Advection-Diffusion-Reaction (VADR) equation, and later the idea has been extended to time-dependent problems. The stabilization parameter corresponding to the VADR equation has been derived. Various robustness properties of the method for both the steady and transient problems have been analyzed theoretically and numerically. Furthermore, the technique has been employed to approximate the solution of a strongly coupled system of transient Navier-Stokes VADR equations. The strong coupling indicates the dynamic viscosity of the fluid to be dependent upon the solute mass concentration. Appropriate theoretical and numerical investigations verify the accuracy, efficiency, and performance of the method in solving such complex coupled systems of equations.