Mixed finite element projection methods for the unsteady Stokes equations

TL;DR

This paper introduces a new mixed finite element projection method for unsteady Stokes equations that ensures divergence-free velocity and achieves stability and first-order accuracy, suitable for unstructured grids.

Contribution

It develops an $H$(div)-conforming mixed finite element method with a projection approach for unsteady Stokes equations, featuring local stress elimination and divergence-free velocity computation.

Findings

Unconditionally stable scheme with first-order time accuracy.

Second-order accurate divergence-free velocity at each time step.

Effective implementation on unstructured triangular grids.

Abstract

We develop $H$(div)-conforming mixed finite element methods for the unsteady Stokes equations modeling single-phase incompressible fluid flow. A projection method in the framework of the incremental pressure correction methodology is applied, where a predictor and a corrector problems are sequentially solved, accounting for the viscous effects and incompressibility, respectively. The predictor problem is based on a stress-velocity mixed formulation, while the corrector projection problem uses a velocity-pressure mixed formulation. The scheme results in pointwise divergence-free velocity computed at the end of each time step. We establish unconditional stability and first order in time accuracy. In the implementation we focus on generally unstructured triangular grids. We employ a second order multipoint flux mixed finite element method based on the next-to-the-lowest order Raviart-Thomas space $RT_1$ and a suitable quadrature rule. In the predictor problem this approach allows for a local stress elimination, resulting in element-based systems for each velocity component with three degrees of freedom per element. Similarly, in the corrector problem, the velocity is locally eliminated and an element-based system for the pressure is solved. At the end of each time step we obtain a second order accurate $H$(div)-conforming piecewise linear velocity, which is pointwise divergence free. We present a series of numerical tests to illustrate the performance of the method.