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Lax convergence theorems and error estimates of a finite element method for the incompressible Euler system

Mária Lukáčová-Medviďová, Bangwei She

Abstract

In this paper, we present convergence theorems for numerical solutions of the incompressible Euler equations. The first result is the Lax-Wendroff-type theorem, while the second can be formulated in the framework of the Lax equivalence theorem. To illustrate their application, we study a finite element method that uses a pair of $RT_0/P_0$ elements to approximate the velocity and pressure, respectively. Applying the concept of the relative energy, we derive the convergence rates of our numerical method using two different approaches. Finally, we validate the theoretical convergence results through numerical experiments.

Lax convergence theorems and error estimates of a finite element method for the incompressible Euler system

Abstract

In this paper, we present convergence theorems for numerical solutions of the incompressible Euler equations. The first result is the Lax-Wendroff-type theorem, while the second can be formulated in the framework of the Lax equivalence theorem. To illustrate their application, we study a finite element method that uses a pair of elements to approximate the velocity and pressure, respectively. Applying the concept of the relative energy, we derive the convergence rates of our numerical method using two different approaches. Finally, we validate the theoretical convergence results through numerical experiments.

Paper Structure

This paper contains 19 sections, 12 theorems, 77 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. The incompressible Euler system
  3. Solution concepts
  4. Consistent approximations
  5. Convergence
  6. A finite element method
  7. Notations.
  8. Function spaces.
  9. Finite element method.
  10. Error estimates
  11. Convergence rate -- I
  12. Convergence rate -- II
  13. Numerical experiment
  14. Experiment 1
  15. Experiment 2
  16. ...and 4 more sections

Key Result

Lemma 2.3

Let $\Omega \subset R^d, d=2,3$. Suppose that $\bm{u} \in L^\infty(0,T;L^2(\Omega;\mathbb{R}^d))$ is a dissipative weak (DW) solution of the incompressible Euler system in the sense of Definition defi_DWs. Let $\widetilde{\bm{u}}$ be a strong solution of the same system and the same initial data in $\blacktriangleleft$$\blacktriangleleft$

Figures (4)

  • Figure 1: Experiment 1 (Taylor-Green vortex). Time evolution of streamline and pressure contour, from left to right are $t=0, 1$.
  • Figure 2: Experiment 2. Streamline and pressure contours computed by $RT_k/P_k$ elements with $\mu_h =0$ and $h=0.1851$. From top to bottom are $t=2,4,6,8$, from left to right are $k=0,1,2$.
  • Figure 3: Experiment 2. Streamline and pressure contours computed by $RT_k/P_k$ elements with $\mu_h =h=0.1851$. From top to bottom are $t=2,4,6,8$, from left to right are $k=0,1,2$.
  • Figure 4: Experiment 2. Vorticity contour at time $t=6$, computed by $RT_k/P_k$ elements with $\mu_h =0$. From top to bottom are mesh sizes $h=0.7405, 0.3702, 0.1851,0.0926$. From left to right are $k=1,2.$

Theorems & Definitions (33)

  • Definition 2.1: Dissipative weak solution
  • Remark 1
  • Definition 2.2: Strong solution
  • Remark 2
  • Lemma 2.3: Dissipative weak-strong uniqueness
  • Definition 2.4: Consistent approximations
  • Lemma 2.5: Unconditional limit of consistent approximations
  • proof
  • Theorem 2.6: Lax-Wendroff theorem
  • proof
  • ...and 23 more