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.
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