A new moving mesh scheme based on the Lagrange-Galerkin method for the approximation of the one-dimensional convection-diffusion equation is studied. The mesh movement, which is prescribed by a discretized dynamical system for the nodal points, follows the direction of convection. It is shown that under a restriction of the time increment the mesh movement cannot lead to an overlap of the elements and therefore an invalid mesh. For the linear element, optimal error estimates in the $\ell^\infty(L^2) \cap \ell^2(H_0^1)$ norm are proved in case of both, a first-order backward Euler method and a second-order two-step method in time. These results are based on new estimates of the time dependent interpolation operator derived in this work. Preservation of the total mass is verified for both choices of the time discretization. Numerical experiments are presented that confirm the error estimates and demonstrate that the proposed moving mesh scheme can circumvent limitations that the Lagrange-Galerkin method on a fixed mesh exhibits.

翻译：研究了一种基于拉格朗日-伽辽金方法逼近一维对流扩散方程的新型移动网格格式。网格移动由节点离散动力系统定义，并沿对流方向进行。研究表明，在时间增量限制下，网格移动不会导致单元重叠及无效网格。对于线性单元，在时间离散采用一阶向后欧拉法和二阶两步法两种情况下，均证明了$\ell^\infty(L^2) \cap \ell^2(H_0^1)$范数下的最优误差估计。这些结果基于本文推导的时变插值算子的新估计。两种时间离散选择均验证了总质量的守恒性。数值实验证实了误差估计，并表明所提出的移动网格格式能够克服固定网格上拉格朗日-伽辽金方法存在的局限性。