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

Martin Schneider, Timo Koch

TL;DR

This work develops and analyzes three CVFE strategies—non-overlapping, overlapping, and hybrid CVFE-FEM—within the MINI element framework to solve the Stokes equations with strong local mass and momentum conservation. By combining control-volume flux balances with inf-sup stable FE spaces, the methods deliver stable, convergent solutions in 2D and 3D, including complex geometries, while maintaining uniformly bounded solver iterations under preconditioning. The results show expected convergence rates, with velocity attaining $O(h^2)$ in $L^2$ and $O(h)$ in $H^1$, and pressure exhibiting $L^2$ super-convergence on structured meshes; stability and conservation are demonstrated across classical benchmarks and challenging domains. The approaches are poised for coupling to other PDE systems (e.g., Navier–Stokes–Darcy) and for extension to higher-order spaces, though pressure robustness remains an inherent limitation of the MINI-based schemes.

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.

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

TL;DR

This work develops and analyzes three CVFE strategies—non-overlapping, overlapping, and hybrid CVFE-FEM—within the MINI element framework to solve the Stokes equations with strong local mass and momentum conservation. By combining control-volume flux balances with inf-sup stable FE spaces, the methods deliver stable, convergent solutions in 2D and 3D, including complex geometries, while maintaining uniformly bounded solver iterations under preconditioning. The results show expected convergence rates, with velocity attaining in and in , and pressure exhibiting super-convergence on structured meshes; stability and conservation are demonstrated across classical benchmarks and challenging domains. The approaches are poised for coupling to other PDE systems (e.g., Navier–Stokes–Darcy) and for extension to higher-order spaces, though pressure robustness remains an inherent limitation of the MINI-based schemes.

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.
Paper Structure (29 sections, 1 theorem, 27 equations, 6 figures, 9 tables)

This paper contains 29 sections, 1 theorem, 27 equations, 6 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weak form of the Stokes problem
  4. Spatial discretization
  5. Discrete solution representation and nodal bases
  6. Control-volume test function space
  7. Control-volume finite-element (CVFE) schemes
  8. Non-overlapping CVFE schemes
  9. Overlapping CVFE schemes
  10. Hybrid CVFE-FEM schemes
  11. Choice of stable discrete function spaces
  12. Velocity function space
  13. Pressure function space
  14. Construction of control volumes
  15. Non-overlapping CVFE method
  16. ...and 14 more sections

Key Result

Proposition 1

Given a space $V(\Omega) \subset L^2(\Omega)$, and $r \in V(\Omega)$, the following equivalence holds

Figures (6)

  • Figure 1: CVFE discretizations for the Stokes poblem. The control volume construction and the degrees of freedom for non-overlapping (top row), overlapping (middle row), and hybrid CVFE-FEM schemes (bottom row) when using the MINI element are shown. The brown vertex-centered ($\boldsymbol{v}$, $p$) sub-control volumes of a mesh form a disjoint partition of the domain $\Omega$. Turquoise element-centered control volumes are additional "overlapping" control volumes that only contain a strict subset of each element $E \in \mathcal{M}$ of the mesh. The corner vertices of turquoise control volumes coincide with the face centers of $E$. The figures on the right show the velocity control volumes around degrees of freedom. The control volumes for pressure are not shown but coincide with the brown control volumes when using the overlapping or hybrid approach. The construction trivially generalizes to $3$ dimensions for overlapping and hybrid CVFE-FEM schemes.
  • Figure 2: $\mathbb{P}_{1+b}$ basis functions on the two-dimensional reference element. The function $\varphi_E$ is a bubble function (vanishes on all faces) on the shown element, and $\varphi_k$, $k=1,2,3$, are modified nodal basis functions.
  • Figure 3: Grids
  • Figure 4: Boundary conditions and grid (coarse refinement level) used for the Poiseuille pipe flow test case; $R=0.1\m$, $L=1\m$, $\mu = 1mPa\s$, $p_\text{in} = 0.08Pa$, $p_\text{out} = 0Pa$, $\mathbf{n}$ denotes the outward-pointing unit normal on the cylinder surface and $\boldsymbol{t}$ the normal traction.
  • Figure 5: Stokes flow around a turtle-shaped obstacle. Grid used for the simulations (left), pressure (middle), and velocity solutions (right). The pressure approximation is physically consistent at the corner singularities. Moffatt eddies behind the extremities become obvious in the streamline pattern.
  • ...and 1 more figures

Theorems & Definitions (13)

  • Remark 1
  • Definition 1: Grid discretization
  • Definition 2: Nodal basis
  • Remark 2
  • Definition 3: Control-volume function space
  • Proposition 1
  • proof
  • Remark 3
  • Example 1: Vertex-centered $\mathbb{P}_1$- or $\mathbb{Q}_1$-CVFE
  • Remark 4
  • ...and 3 more