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

TL;DR

This paper establishes convergence theorems for a finite element method solving the incompressible Euler equations, providing theoretical error estimates and validating them through numerical experiments.

Contribution

It introduces convergence theorems for a finite element scheme using RT0/P0 elements and derives error estimates via the relative energy method.

Findings

Proved Lax-Wendroff-type and Lax equivalence theorems for the scheme.

Derived convergence rates for the finite element method.

Validated theoretical results with numerical experiments.

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.