Stable and locally mass- and momentum-conservative control-volume finite-element schemes for the Stokes problem

TL;DR

This paper introduces new control-volume finite-element schemes for the Stokes problem that ensure local mass and momentum conservation, combining finite element and finite volume techniques for improved stability and robustness.

Contribution

It presents novel control-volume discretization schemes, including a hybrid approach, that are stable, locally conservative, and applicable to complex flow scenarios.

Findings

Schemes are stable and locally conservative.

Numerical tests confirm convergence and robustness.

Effective in complex flow simulations like obstacle flow and vessel bifurcation.

Abstract

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.