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Convergence analysis of dynamically regularized Lagrange multiplier pressure correction method for the incompressible Navier-Stokes equations

Yi Shen, Rihui Lan, Hua Wang

Abstract

We propose first-order pressure-correction scheme for the incompressible Navier-Stokes equations, incorporating the recently developed the Dynamically Regularized Lagrange Multiplier (DRLM) methods. The resulting algorithms are fully decoupled and require solving only Poisson-type equations at each time step. Moreover, it exhibits unconditional energy stability. This paper provides a rigorous error analysis for the first-order scheme, establishing optimal error estimates for both velocity and pressure. Specifically, we employ mathematical induction to derive sharp velocity error bounds, while leveraging the inf-sup condition to prove optimal convergence rate for the pressure. To validate our theoretical findings, we present two numerical experiments demonstrating the accuracy and robustness of the method.

Convergence analysis of dynamically regularized Lagrange multiplier pressure correction method for the incompressible Navier-Stokes equations

Abstract

We propose first-order pressure-correction scheme for the incompressible Navier-Stokes equations, incorporating the recently developed the Dynamically Regularized Lagrange Multiplier (DRLM) methods. The resulting algorithms are fully decoupled and require solving only Poisson-type equations at each time step. Moreover, it exhibits unconditional energy stability. This paper provides a rigorous error analysis for the first-order scheme, establishing optimal error estimates for both velocity and pressure. Specifically, we employ mathematical induction to derive sharp velocity error bounds, while leveraging the inf-sup condition to prove optimal convergence rate for the pressure. To validate our theoretical findings, we present two numerical experiments demonstrating the accuracy and robustness of the method.
Paper Structure (13 sections, 8 theorems, 127 equations, 2 figures, 1 table)

This paper contains 13 sections, 8 theorems, 127 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Pressure-correction DRLM scheme and its properties
  3. Pressure-correction DRLM scheme
  4. Energy stability
  5. Decoupling Strategy and well-posedness of solutions
  6. Temporal error estimate
  7. Preliminaries
  8. Optimal error estimates for the velocity and Lagrange multiplier
  9. Optimal error estimates for the pressure
  10. Numerical experiments
  11. Convergence test
  12. Lid-driven cavity flow
  13. Conclusion

Key Result

Lemma 2.2

Assume $\bm u_0\in \boldsymbol{V}$ and $\bm f\in \boldsymbol{L}^\infty(0,T;\boldsymbol{L}^2(\Omega))$. There exists a positive time $T^*$, with $T^* = T$ if $d = 2$ and $T^* = T^*(\bm u_0) \leq T$ if $d = 3$, such that NS:orig admits a unique strong solution $(\bm u, p)$ satisfying Moreover, if $d = 2$ or, in the case $d = 3$, if $\Vert\bm u_0\Vert_1$ and $\Vert\bm f\Vert_{\boldsymbol{L}^\infty(0

Figures (2)

  • Figure 1: Contour plots of the velocity magnitude at times $t=2, 4, 8, 10, 20$, and $80$ by P-DRLM1 scheme.
  • Figure 2: The velocity at the center line with $x$-component velocity at $x = 0.5$ (left) and $y$-component velocity at $y = 0.5$ (right) for $\text{Re} = 5000$ at $t=80$.

Theorems & Definitions (19)

  • Remark 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Remark 2.7
  • Lemma 2.8
  • ...and 9 more