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The weak Galerkin finite element method for Stokes interface problems with curved interface

Lin Yang, Qilong Zhai, Ran Zhang

TL;DR

This work develops a weak Galerkin finite element method for 2D Stokes interface problems with curved interfaces, formulating a scheme that incorporates interface conditions through vector-valued traces and uses non-affine transformations to handle curved cells. The authors establish stability via an inf-sup condition, derive error equations, and prove optimal convergence in the energy norm $|||\cdot|||$ and in the $L^2$ norm for velocity and pressure under standard regularity assumptions. They also provide a duality-based $L^2$ error estimate, showing $\|Q_0\mathbf{u}-\mathbf{u}_0\| \le C h^{k+1}$, and confirm the theory with numerical results on curved meshes that exhibit the expected rates. Overall, the paper demonstrates that the WG scheme with curved-edge handling preserves optimal convergence and avoids geometric error, making it effective for Stokes interface problems with curved interfaces.

Abstract

In this paper, we develop a new weak Galerkin finite element scheme for the Stokes interface problem with curved interfaces. We take a unique vector-valued function at the interface and reflect the interface condition in the variational problem. Theoretical analysis and numerical experiments show that the errors can reach the optimal convergence order under the energy norm and $L^2$ norm.

The weak Galerkin finite element method for Stokes interface problems with curved interface

TL;DR

This work develops a weak Galerkin finite element method for 2D Stokes interface problems with curved interfaces, formulating a scheme that incorporates interface conditions through vector-valued traces and uses non-affine transformations to handle curved cells. The authors establish stability via an inf-sup condition, derive error equations, and prove optimal convergence in the energy norm and in the norm for velocity and pressure under standard regularity assumptions. They also provide a duality-based error estimate, showing , and confirm the theory with numerical results on curved meshes that exhibit the expected rates. Overall, the paper demonstrates that the WG scheme with curved-edge handling preserves optimal convergence and avoids geometric error, making it effective for Stokes interface problems with curved interfaces.

Abstract

In this paper, we develop a new weak Galerkin finite element scheme for the Stokes interface problem with curved interfaces. We take a unique vector-valued function at the interface and reflect the interface condition in the variational problem. Theoretical analysis and numerical experiments show that the errors can reach the optimal convergence order under the energy norm and norm.
Paper Structure (11 sections, 15 theorems, 119 equations, 8 figures, 1 algorithm)

This paper contains 11 sections, 15 theorems, 119 equations, 8 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Weak Galerkin Numerical Scheme
  3. The notations
  4. Weak Galerkin finite element space
  5. The numerical scheme
  6. Stability
  7. Error Equations
  8. Error Estimate in the Energy Norm
  9. Error Estimate in the $L^2$ Norm
  10. Numerical Results
  11. Conclusion

Key Result

Lemma 2.1

${|\space|\space|} \cdot {|\space|\space|}$ provides a norm in $V_h^0$.

Figures (8)

  • Figure 1: The model of Stokes-Stokes coupling flow
  • Figure 2: The numerical results for Example \ref{['example3']} on the curved triangular meshes (left) and straight triangular meshes (right) with $k=1$.
  • Figure 3: The numerical results for Example \ref{['example3']} on the curved triangular meshes (left) and straight triangular meshes (right) with $k=2$.
  • Figure 4: The numerical results for Example \ref{['example3']} on the curved triangular meshes (left) and straight triangular meshes (right) with $k=3$.
  • Figure 5: The numerical solutions by $P_2$ WG element on curved triangular meshes level 3 in Example \ref{['example3']}. Left: The first component of ${\bf u}_h$. Middle: The second component of ${\bf u}_h$. Right: Pressure function $p_h$.
  • ...and 3 more figures

Theorems & Definitions (32)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 22 more