The finite volume element method for non-stationary Stokes equations with an LC element pair

Jiehua Zhang; Jiehua Zhang

doi:10.3934/math.2025591

AIMS Mathematics

2025, Volume 10, Issue 6: 13166-13203. doi: 10.3934/math.2025591

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Research article

The finite volume element method for non-stationary Stokes equations with an LC element pair

Jiehua Zhang ^,

School of Science, Kaili University, Guizhou 556011, China

Received: 25 January 2025 Revised: 06 May 2025 Accepted: 09 May 2025 Published: 09 June 2025
MSC : 65N08, 65N12, 65N15, 76D05, 76D07

This paper proposes and analyzes a fully discrete spatial-temporal finite volume element method, which employs the LC element pair, for solving non-stationary Stokes equations on barycenter-refined triangular meshes. The proposed scheme utilizes an implicit first-order temporal discretization and is devoid of stabilization parameters. In terms of spatial discretization, the velocity is approximated using a quadratic conforming finite element space, while the pressure is approximated with a discontinuous piecewise linear function space. By utilizing a one-to-many mapping between the trial and test spaces for the velocity components in the finite volume element method, the equivalence between the bilinear forms resulting from the gradient and divergence operators is established, and thus under the mild restrictions of triangular meshes, the stability of the proposed scheme is demonstrated. By introducing a Stokes projection, error estimates for the proposed scheme are obtained. To validate the feasibility and efficiency of the proposed scheme, numerical experiments are presented for the non-stationary Stokes equations.
- finite volume element method,
- local mass conservation,
- non-stationary Stokes equations,
- LC element
Citation: Jiehua Zhang. The finite volume element method for non-stationary Stokes equations with an LC element pair[J]. AIMS Mathematics, 2025, 10(6): 13166-13203. doi: 10.3934/math.2025591

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Abstract

This paper proposes and analyzes a fully discrete spatial-temporal finite volume element method, which employs the LC element pair, for solving non-stationary Stokes equations on barycenter-refined triangular meshes. The proposed scheme utilizes an implicit first-order temporal discretization and is devoid of stabilization parameters. In terms of spatial discretization, the velocity is approximated using a quadratic conforming finite element space, while the pressure is approximated with a discontinuous piecewise linear function space. By utilizing a one-to-many mapping between the trial and test spaces for the velocity components in the finite volume element method, the equivalence between the bilinear forms resulting from the gradient and divergence operators is established, and thus under the mild restrictions of triangular meshes, the stability of the proposed scheme is demonstrated. By introducing a Stokes projection, error estimates for the proposed scheme are obtained. To validate the feasibility and efficiency of the proposed scheme, numerical experiments are presented for the non-stationary Stokes equations.

References

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