Error estimates for shock capturing finite element approximations of the one dimensional Burgers' equation

TL;DR

This paper presents an error analysis for a shock capturing finite element method applied to the one-dimensional Burgers' equation, focusing on weak norms and robustness in the inviscid limit.

Contribution

It introduces a novel error estimation approach in weak norms for shock capturing finite element methods, applicable in the inviscid limit.

Findings

Error estimates relate to filtered quantities.

Estimates are robust as viscosity approaches zero.

Uses total variation bounds for error analysis.

Abstract

We propose an error analysis in weak norms of a shock capturing finite element method for the Burgers' equation. The estimates can be related to estimates of certain filtered quantities and are robust in the inviscid limit. Using a total variation apriori bound on the discrete solution and an interpolation inequality error estimates in $L^p$-norms can are obtained using interpolation.