A new local discontinuous Galerkin (LDG) method for convection-diffusion equations on overlapping meshes with periodic boundary conditions was introduced in \cite{Overlap1}. With the new method, the primary variable $u$ and the auxiliary variable $p=u_x$ are solved on different meshes. In this paper, we will extend the idea to convection-diffusion equations with non-periodic boundary conditions, i.e. Neumann and Dirichlet boundary conditions. The main difference is to adjust the boundary cells. Moreover, we study the stability and suboptimal error estimates. Finally, numerical experiments are given to verify the theoretical findings.

翻译：在“cite{Overlap1}”中引入了一种新的局部不连续的Galerkin(LDG)方法,用于对具有定期边界条件的重叠金属的对流-扩散方程式进行对流-扩散方程式。在采用新方法后,主要变量$u$和辅助变量$u_x$($p=u_x$)在不同的介质上得到解决。在本文中,我们将扩大这一想法,使之适用于具有非周期边界条件的对流-扩散方程式,即Neumann和Drichlet边界条件。主要区别在于调整边界单元格。此外,我们研究了稳定性和次优化误差估计。最后,我们进行了数字实验,以核实理论结论。