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

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(div)$$ H\left(\operatorname{div}\right) $$ and L2$$ {L}<^>2 $$ 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(div)$$ H\left(\operatorname{div}\right) $$ spaces. In addition, a variation of the traditional H(div)$$ H\left(\operatorname{div}\right) $$ 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(div)$$ H\left(\operatorname{div}\right) $$ 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.