New error estimates of Lagrange-Galerkin methods for the advection equation

TL;DR

This paper introduces new error estimates and stabilized variants of the Lagrange-Galerkin method for the advection equation, enhancing stability and accuracy through local projections and discontinuity capturing techniques.

Contribution

It provides improved error bounds for the classical method and proposes novel stabilized and discontinuity-capturing Lagrange-Galerkin methods with analysis.

Findings

Enhanced error estimates for the classical Lagrange-Galerkin method

Development of a local projection stabilized Lagrange-Galerkin method

Introduction and analysis of a discontinuity-capturing Lagrange-Galerkin method

Abstract

We study in this paper new developments of the Lagrange-Galerkin method for the advection equation. In the first part of the article we present a new improved error estimate of the conventional Lagrange-Galerkin method. In the second part, we introduce a new local projection stabilized Lagrange-Galerkin method, whereas in the third part we introduce and analyze a discontinuity-capturing Lagrange-Galerkin method. Also, attention has been paid to the influence of the quadrature rules on the stability and accuracy of the methods via numerical experiments.