A Stable Mixed Finite Element Method for the Simulation of Stokes Flow Using Divergence Balanced H(Div)-L2 Pair of Approximation Spaces

Abstract

The Stokes equations are used to model the motion of fluid flows where inertial terms can be neglected. Traditional finite element approaches such as the Taylor–Hood element do not ensure local conservation pointwise of the mass. This can be achieved by employing a mixed formulation with the proper combination of $H\left(\text{div}\right)$ and $L^2$ spaces. In this context, this article presents a new hybrid-hybrid formulation to solve the Stokes equations. By applying Lagrange multipliers, the continuity of the tangential velocity is enforced, which is not intrinsically guaranteed by $H\left(\text{div}\right)$ spaces. In addition, a variation of the traditional $H\left(\text{div}\right)$ space, called Hdiv-C, is used to approximate the fields. The Hdiv-C space is created using concepts of the exact De Rham sequence and is shown to yield a smaller global system of equations than traditional finite element $H\left(\text{div}\right)$ spaces. A two-dimensional manufactured solution problem and the three-dimensional Annular-Couette flow problem are used to verify the hybrid-hybrid formulation's convergence rates, which are compared to Taylor–Hood's results. Application examples based on lab-on-chip mixers are analyzed to demonstrate the robustness of the proposed method. The examples consist of three different serpentine channel geometries: two in two dimensions (a sinusoidal and a “bumped” serpentine) and one in three dimensions (a C-shape serpentine). The results show that the hybrid-hybrid formulation combined with the Hdiv-C space is suitable for solving Stokes problems with optimal convergence rates, comparable to the Taylor–Hood element.