We introduce new control-volume finite-element discretization schemes suitable for solving the Stokes problem. Within a common framework, we present different approaches for constructing such schemes. The first and most established strategy employs a non-overlapping partitioning into control volumes. The second represents a new idea by splitting into two sets of control volumes, the first set yielding a partition of the domain and the second containing the remaining overlapping control volumes required for stability. The third represents a hybrid approach where finite volumes are combined with finite elements based on a hierarchical splitting of the ansatz space. All approaches are based on typical finite element function spaces but yield locally mass and momentum conservative discretization schemes that can be interpreted as finite volume schemes. We apply all strategies to the inf-sub stable MINI finite-element pair. Various test cases, including convergence tests and the numerical observation of the boundedness of the number of preconditioned Krylov solver iterations, as well as more complex scenarios of flow around obstacles or through a three-dimensional vessel bifurcation, demonstrate the stability and robustness of the schemes.

翻译：我们针对Stokes问题提出了新的控制体积有限元离散格式。在统一框架下，我们展示了构建此类格式的不同方法。第一种且最为成熟的策略采用非重叠的控制体积分区。第二种代表了一种新思路，通过将区域分为两组控制体积实现：第一组构成域的一个划分，第二组包含为保证稳定性所需的其余重叠控制体积。第三种则是一种混合方法，基于解空间的分层分解，将有限体积与有限元相结合。所有方法均基于典型的有限元函数空间，但能生成可解释为有限体积格式的局部质量和动量守恒的离散格式。我们将所有策略应用于inf-sub稳定的MINI有限元对。多种数值实验（包括收敛性测试、预处理Krylov求解器迭代次数有界性的数值观测，以及绕流障碍物或三维血管分叉等更复杂场景）验证了该格式的稳定性与鲁棒性。