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

Stability analysis and error estimates of local discontinuous Galerkin  method for convection-diffusion equations on overlapping mesh with  non-periodic boundary conditions