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 \textbf{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 $O(h^{k+1}+\Delta t^{q})$, whereas in the advective regime, $\mu \ll 1$, the convergence is $O(\min (h^{k},\frac{h^{k+1} }{\Delta t})+\Delta t^{q})$. It is worth remarking that the error constant does not have exponential $\mu ^{-1}$ dependence.

翻译：本文针对对流扩散方程的守恒格式，提出时间与空间均达到高阶精度的拉格朗日-伽辽金方法的数值分析。时间离散格式采用阶数最高为$q=5$的向后差分公式。方法的构建与分析均在C. M. Elliot与T. Ranner（IMA Journal of Numerical Analysis \textbf{41}, 1696-1845 (2021)）提出的时间演化有限元框架下完成。误差估计通过其对方程参数的依赖关系揭示了数值解行为的不同形态：在扩散主导形态下（即扩散参数$\mu$较大时），误差为$O(h^{k+1}+\Delta t^{q})$；而在对流主导形态下（$\mu \ll 1$），收敛阶为$O(\min (h^{k},\frac{h^{k+1} }{\Delta t})+\Delta t^{q})$。值得指出的是，误差常数不存在指数形式的$\mu ^{-1}$依赖性。