A mass-conservative Lagrange--Galerkin scheme of second order in time for convection-diffusion problems is presented, and convergence with optimal error estimates is proved in the framework of $L^2$-theory. The introduced scheme maintains the advantages of the Lagrange--Galerkin method, i.e., CFL-free robustness for convection-dominated problems and a symmetric and positive coefficient matrix resulting from the discretization. In addition, the scheme conserves the mass on the discrete level. Unconditional stability and error estimates of second order in time are proved by employing two new key lemmas on the truncation error of the material derivative in conservative form and on a discrete Gronwall inequality for multistep methods. The mass-conservation property is achieved by the Jacobian multiplication technique introduced by Rui and Tabata in 2010, and the accuracy of second order in time is obtained based on the idea of the multistep Galerkin method along characteristics originally introduced by Ewing and Russel in 1981. For the first time step, the mass-conservative scheme of first order in time by Rui and Tabata in 2010 is employed, which is efficient and does not cause any loss of convergence order in the $\ell^\infty(L^2)$- and $\ell^2(H^1_0)$-norms. For the time increment $\Delta t$, the mesh size $h$ and a conforming finite element space of polynomial degree $k$, the convergence order is of $O(\Delta t^2 + h^k)$ in the $\ell^\infty(L^2)\cap \ell^2(H^1_0)$-norm and of $O(\Delta t^2 + h^{k+1})$ in the $\ell^\infty(L^2)$-norm if the duality argument can be employed. Error estimates of $O(\Delta t^{3/2}+h^k)$ in discrete versions of the $L^\infty(H^1_0)$- and $H^1(L^2)$-norm are additionally proved. Numerical results confirm the theoretical convergence orders in one, two and three dimensions.
翻译:质量- 保守性 Lagrange- Galerkin 方法的优点, 即, 用于调和问题的无CFL 强度, 以及由离散产生的对数和正系数矩阵。 此外, 计划保存离散水平的质量。 对第二个顺序的不完全稳定性和误差的估计数, 以保守形式在材料衍生物的调离错误上使用两个新的基级的利玛 =2, 并使用离散的Gronwalkin 方法的优点。 由Rui 和 Tabata 于2010年推出的Cocobian 倍化技术所实现的质性, 并且根据离散 Galkin 方法的概念, 1981年由Ewon 和 Russelt 开始的特性 。 第一步, 在2010年1 美元 和 美元 的 Rix- holdal- dirstal 级中, 其质量- 和 Rixion 顺序由 Ral- dromaxyal- dromax 。