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A Mini Immersed Finite Element Method for Two-Phase Stokes Problems on Cartesian Meshes

Haifeng Ji, Dong Liang, Qian Zhang

TL;DR

This work tackles the two-phase Stokes interface problem with piecewise constant viscosities by introducing a mini immersed finite element method on Cartesian unfitted meshes. The method modifies only interface elements to enforce discrete interface jump conditions and provides explicit IFE basis functions and correction functions to accommodate surface forces, all while maintaining the same degrees of freedom as the standard mini element. The authors establish norm equivalence, inf-sup stability, and optimal a priori error estimates, with a stiffness matrix condition number bound that matches conventional mini-element behavior and remains independent of the interface location. A level-set based discretization of the interface and a fictitious box around interface elements enable stable handling of non-homogeneous jumps, making the approach suitable for moving interfaces. Numerical experiments in 2D and 3D corroborate the theory, showing optimal convergence and robustness against interface geometry and jump magnitudes, including cases with nonzero surface forces $\mathbf{g}$.

Abstract

This paper presents a mini immersed finite element (IFE) method for solving two- and three-dimensional two-phase Stokes problems on Cartesian meshes. The IFE space is constructed from the conventional mini element, with shape functions modified on interface elements according to interface jump conditions, while keeping the degrees of freedom unchanged. Both discontinuous viscosity coefficients and surface forces are taken into account in the construction. The interface is approximated using discrete level set functions, and explicit formulas for IFE basis functions and correction functions are derived, facilitating ease of implementation.The inf-sup stability and the optimal a priori error estimate of the IFE method, along with the optimal approximation capabilities of the IFE space, are derived rigorously, with constants that are independent of the mesh size and the manner in which the interface intersects the mesh, but may depend on the discontinuous viscosity coefficients. Additionally, it is proved that the condition number has the usual bound independent of the interface. Numerical experiments are provided to confirm the theoretical results.

A Mini Immersed Finite Element Method for Two-Phase Stokes Problems on Cartesian Meshes

TL;DR

This work tackles the two-phase Stokes interface problem with piecewise constant viscosities by introducing a mini immersed finite element method on Cartesian unfitted meshes. The method modifies only interface elements to enforce discrete interface jump conditions and provides explicit IFE basis functions and correction functions to accommodate surface forces, all while maintaining the same degrees of freedom as the standard mini element. The authors establish norm equivalence, inf-sup stability, and optimal a priori error estimates, with a stiffness matrix condition number bound that matches conventional mini-element behavior and remains independent of the interface location. A level-set based discretization of the interface and a fictitious box around interface elements enable stable handling of non-homogeneous jumps, making the approach suitable for moving interfaces. Numerical experiments in 2D and 3D corroborate the theory, showing optimal convergence and robustness against interface geometry and jump magnitudes, including cases with nonzero surface forces .

Abstract

This paper presents a mini immersed finite element (IFE) method for solving two- and three-dimensional two-phase Stokes problems on Cartesian meshes. The IFE space is constructed from the conventional mini element, with shape functions modified on interface elements according to interface jump conditions, while keeping the degrees of freedom unchanged. Both discontinuous viscosity coefficients and surface forces are taken into account in the construction. The interface is approximated using discrete level set functions, and explicit formulas for IFE basis functions and correction functions are derived, facilitating ease of implementation.The inf-sup stability and the optimal a priori error estimate of the IFE method, along with the optimal approximation capabilities of the IFE space, are derived rigorously, with constants that are independent of the mesh size and the manner in which the interface intersects the mesh, but may depend on the discontinuous viscosity coefficients. Additionally, it is proved that the condition number has the usual bound independent of the interface. Numerical experiments are provided to confirm the theoretical results.
Paper Structure (27 sections, 17 theorems, 170 equations, 4 figures, 3 tables)

This paper contains 27 sections, 17 theorems, 170 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Finite element discretization
  3. Unfitted meshes
  4. Discretization of the interface
  5. The IFE space for the case $\mathbf{g}=\mathbf{0}$
  6. The IFE method for the case $\mathbf{g}=\mathbf{0}$
  7. Extension to the case $\mathbf{g}\not=\mathbf{0}$
  8. Explicit formulas for IFE basis functions
  9. Explicit formulas for correction functions
  10. Approximation capabilities of IFE spaces
  11. Interpolation error decomposition
  12. Interpolation error estimates
  13. Analysis of the IFE method
  14. Norm-equivalence for IFE functions
  15. Boundedness and coercivity
  16. ...and 12 more sections

Key Result

Lemma 2.1

For all $\delta\in (0,\delta_0]$, there is a constant $C$ depending only on $\Gamma$ such that Furthermore, if $v|_{\Gamma}\not=0$, there holds $\|v\|_{L^2(U(\Gamma,\delta))}\leq C\delta^{1/2} \|\nabla v\|_{H^1(U(\Gamma,\delta_0))}.$

Figures (4)

  • Figure 1: Left: an example of $\Gamma_h$ in 2D; Middle: an interface element in 2D; Right: an interface element in 3D.
  • Figure 2: 2D illustration for the construction of $R_T$ and $\Gamma_{R_T}$.
  • Figure 3: Plots of $\log_{10}(error)$ versus $\log_{10} (h)$ for Example 1. The linear regression analysis is used to find an approximate order of convergence.
  • Figure 4: Plots of approximate pressures obtained by the conventional mini element method (left) and the mini IFE method (right) for Example 3, using the uniform Cartesian mesh with $M=32$.

Theorems & Definitions (35)

  • Lemma 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Lemma 2.7
  • proof
  • Lemma 2.8
  • Remark 2.9
  • ...and 25 more