High order Lagrange-Galerkin methods for the conservative formulation of the advection-diffusion equation

We introduce in this paper the numerical analysis of high order both in time and space Lagrange-Galerkin methods for the conservative formulation of the advection-diffusion equation. As time discretization scheme we consider the Backward Differentiation Formulas up to order $q=5$. The development and analysis of the methods are performed in the framework of time evolving finite elements presented in C. M. Elliot and T. Ranner, IMA Journal of Numerical Analysis $\mathbf{\text{41}}$, 1696-1845 (2021). The error estimates show through their dependence on the parameters of the equation the existence of different regimes in the behavior of the numerical solution; namely, in the diffusive regime, that is, when the diffusion parameter $\mu$ is large, the error is $O(h^{k+1}+\mathrm{\Delta }{t}^q)$, whereas in the advective regime, $\mu \ll 1$, the convergence is $O(\min (h^k,\frac{h^{k+1}}{\mathrm{\Delta }t})+\mathrm{\Delta }{t}^q)$. It is worth remarking that the error constant does not have exponential ${\mu }^{-1}$ dependence.