Local error estimates for a finite element method combining linear and nonlinear stabilization for the linear hyperbolic transport equation

TL;DR

This paper rigorously analyzes a finite element method combining linear and nonlinear stabilization techniques for the linear hyperbolic transport equation, demonstrating localized error control and optimal convergence rates.

Contribution

It provides a new theoretical proof of localized error estimates for a stabilized finite element method applied to the transport equation.

Findings

Error in smooth regions scales as h^{k+1/2} in L^2 norm.

Errors in rough regions remain localized and do not affect smooth parts.

The analysis confirms the effectiveness of combined stabilization in preserving solution quality.

Abstract

In this paper, we investigate the combination of a linear continuous interior penalty type and a non-linear artificial diffusion stabilisation applied to the transport problem, based on continuous Galerkin finite elements in space. This method was recently introduced and analysed for globally smooth solutions in [Burman 2023, SIAM J. Sci. Comput., 45, 1, A96-A122]. We provide a rigorous proof of a localisation principle in terms of weighted stability and a priori error bound results, which follow the widely known $\mathcal{O}(h^{k+1/2})$ scaling in the $L^2(\Omega; t=T)$ norm, where $k$ denotes the polynomial order of the finite element space and $h$ the mesh size. The analysis is semi-discrete in space and assumes sufficient local regularity of the continuous solution on the smooth part of the domain, where the continuous interior penalty stabilisation is active, whilst artificial diffusion operates on the remaining rough parts of the domain. Thereby, the analysis demonstrates that typical numerical errors in the rough part stay localised relative to the convection velocity and do not negatively affect the smooth parts of the solution, if the stabilisation combination is set up accordingly.