A conforming-nonconforming mixed immersed finite element method for unsteady Stokes equations with moving interfaces

Derrick Jones; Xu Zhang; Derrick Jones; Xu Zhang

doi:10.3934/era.2021032

Electronic Research Archive

2021, Volume 29, Issue 5: 3171-3191. doi: 10.3934/era.2021032

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A conforming-nonconforming mixed immersed finite element method for unsteady Stokes equations with moving interfaces

Derrick Jones ^{1
,},
Xu Zhang ^{2
,
,}

1.
Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS 39762, USA
2.
Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, USA

Received: 01 November 2020 Revised: 01 February 2021 Published: 15 April 2021
Primary: 65N15, 65N30; Secondary: 35R05

In this article, we develop a new mixed immersed finite element discretization for two-dimensional unsteady Stokes interface problems with unfitted meshes. The proposed IFE spaces use conforming linear elements for one velocity component and non-conforming linear elements for the other velocity component. The pressure is approximated by piecewise constant. Unisolvency, among other fundamental properties of the new vector-valued IFE functions, is analyzed. Based on the new IFE spaces, semi-discrete and full-discrete schemes are developed for solving the unsteady Stokes equations with a stationary or a moving interface. Re-meshing is not required in our numerical scheme for solving the moving-interface problem. Numerical experiments are carried out to demonstrate the performance of this new IFE method.
- Conforming-nonconforming immersed finite element,
- Stokes interface problem,
- moving interface
Citation: Derrick Jones, Xu Zhang. A conforming-nonconforming mixed immersed finite element method for unsteady Stokes equations with moving interfaces[J]. Electronic Research Archive, 2021, 29(5): 3171-3191. doi: 10.3934/era.2021032

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Abstract

In this article, we develop a new mixed immersed finite element discretization for two-dimensional unsteady Stokes interface problems with unfitted meshes. The proposed IFE spaces use conforming linear elements for one velocity component and non-conforming linear elements for the other velocity component. The pressure is approximated by piecewise constant. Unisolvency, among other fundamental properties of the new vector-valued IFE functions, is analyzed. Based on the new IFE spaces, semi-discrete and full-discrete schemes are developed for solving the unsteady Stokes equations with a stationary or a moving interface. Re-meshing is not required in our numerical scheme for solving the moving-interface problem. Numerical experiments are carried out to demonstrate the performance of this new IFE method.

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