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

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.