Consistency and convergence of flux-corrected finite element methods for nonlinear hyperbolic problems

TL;DR

This paper analyzes the consistency and convergence of flux-corrected finite element methods for nonlinear hyperbolic conservation laws, establishing theoretical results and convergence properties including weak and strong convergence under certain conditions.

Contribution

It introduces a monolithic convex limiting approach and proves a Lax--Wendroff-type theorem, advancing the theoretical understanding of flux-corrected finite element schemes for hyperbolic problems.

Findings

Proves weak convergence of flux-corrected schemes for Euler equations.

Establishes strong convergence to strong solutions when they exist.

Provides a weak BV estimate from entropy stability.

Abstract

We investigate the consistency and convergence of flux-corrected finite element approximations in the context of nonlinear hyperbolic conservation laws. In particular, we focus on a monolithic convex limiting approach and prove a Lax--Wendroff-type theorem for the corresponding semi-discrete problem. A key component of our analysis is the use of a weak estimate on bounded variation, which follows from the semi-discrete entropy stability property of the method under investigation. For the Euler equations of gas dynamics, we prove the weak convergence of the flux-corrected finite element scheme to a dissipative weak solution. If a strong solution exists, the sequence of numerical approximations converges strongly to the strong solution.