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 problem and a corrector problem 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 RT1 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.

中文翻译：

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非定常 Stokes 方程的混合有限元投影方法


我们开发了符合 H（div） 的混合有限元方法，用于模拟单相不可压缩流体流动的非定常斯托克斯方程。在增量压力校正方法的框架中应用了投影方法，其中预测器问题和校正器问题依次求解，分别考虑了粘性效应和不可压缩性。预测变量问题基于应力-速度混合公式，而校正变量投影问题使用速度-压力混合公式。该方案导致在每个时间步结束时计算无点发散速度。我们建立了无条件稳定性和一阶时间精度。在实现中，我们关注通常是非结构化的三角形网格。我们采用基于次低阶 Raviart-Thomas 空间 RT1 和合适的正交规则的二阶多点磁通混合有限元方法。在预测器问题中，这种方法允许局部应力消除，从而为每个速度分量生成基于单元的系统，每个单元具有三个自由度。同样，在校正器问题中，速度被局部消除，并求解基于单元的压力系统。在每个时间步结束时，我们获得符合H（div）的二阶精确分段线速度，该速度是无点发散的。我们提出了一系列数值测试来说明该方法的性能。