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Numerical analysis of 2D Navier--Stokes equations with nonsmooth initial value in the critical space

Buyang Li, Qiqi Rao, Hui Zhang, Zhi Zhou

Abstract

This paper addresses the numerical solution of the two-dimensional Navier--Stokes (NS) equations with nonsmooth initial data in the $L^2$ space, which is the critical space for the two-dimensional NS equations to be well-posed. In this case, the solutions of the NS equations exhibit certain singularities at $t=0$, e.g., the $H^s$ norm of the solution blows up as $t\rightarrow 0$ when $s>0$. To date, the best convergence result proved in the literature are first-order accuracy in both time and space for the semi-implicit Euler time-stepping scheme and divergence-free finite elements (even high-order finite elements are used), while numerical results demonstrate that second-order convergence in time and space may be achieved. Therefore, there is still a gap between numerical analysis and numerical computation for the NS equations with $L^2$ initial data. The primary challenge to realizing high-order convergence is the insufficient regularity in the solutions due to the rough initial condition and the nonlinearity of the equations. In this work, we propose a fully discrete numerical scheme that utilizes the Taylor--Hood or Stokes-MINI finite element method for spatial discretization and an implicit-explicit Runge--Kutta time-stepping method in conjunction with graded stepsizes. By employing discrete semigroup techniques, sharp regularity estimates, negative norm estimates and the $L^2$ projection onto the divergence-free Raviart--Thomas element space, we prove that the proposed scheme attains second-order convergence in both space and time. Numerical examples are presented to support the theoretical analysis. In particular, the convergence in space is at most second order even higher-order finite elements are used. This shows the sharpness of the convergence order proved in this article.

Numerical analysis of 2D Navier--Stokes equations with nonsmooth initial value in the critical space

Abstract

This paper addresses the numerical solution of the two-dimensional Navier--Stokes (NS) equations with nonsmooth initial data in the space, which is the critical space for the two-dimensional NS equations to be well-posed. In this case, the solutions of the NS equations exhibit certain singularities at , e.g., the norm of the solution blows up as when . To date, the best convergence result proved in the literature are first-order accuracy in both time and space for the semi-implicit Euler time-stepping scheme and divergence-free finite elements (even high-order finite elements are used), while numerical results demonstrate that second-order convergence in time and space may be achieved. Therefore, there is still a gap between numerical analysis and numerical computation for the NS equations with initial data. The primary challenge to realizing high-order convergence is the insufficient regularity in the solutions due to the rough initial condition and the nonlinearity of the equations. In this work, we propose a fully discrete numerical scheme that utilizes the Taylor--Hood or Stokes-MINI finite element method for spatial discretization and an implicit-explicit Runge--Kutta time-stepping method in conjunction with graded stepsizes. By employing discrete semigroup techniques, sharp regularity estimates, negative norm estimates and the projection onto the divergence-free Raviart--Thomas element space, we prove that the proposed scheme attains second-order convergence in both space and time. Numerical examples are presented to support the theoretical analysis. In particular, the convergence in space is at most second order even higher-order finite elements are used. This shows the sharpness of the convergence order proved in this article.

Paper Structure

This paper contains 10 sections, 12 theorems, 145 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Spatial semi-discretization by finite element method
  3. Weak solution
  4. Spatial semi-discretization
  5. Fully discretization
  6. Runge-Kutta method and error equations
  7. Regularities of numerical solutions and estimates for operators
  8. Error analysis
  9. Numerical examples
  10. Conclusion

Key Result

Lemma 2.1

For any given initial value $u_0\in\dot L^{2}(\varOmega)$, the exact solution $u$ of problem system-1 satisfy the following regularity result.

Figures (4)

  • Figure 4.1: $L^{2}$ errors of $u$
  • Figure 4.2: Isocontours of the velocity $u$
  • Figure 4.3: $L^{2}$ errors of $u$
  • Figure 4.4: Isocontours of the velocity $u$

Theorems & Definitions (24)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • Corollary 2.7
  • ...and 14 more