Local and parallel finite element methods based on two-grid discretizations for unsteady convection-diffusion problem

In this article, some local and parallel finite element methods are proposed and investigated for the time-dependent convection-diffusion problem. With backward Euler scheme for the temporal discretization, the basic idea of the present methods is that for a solution to the considered equations, low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a fine grid by some local and parallel procedure at each time step. The partition of unity is used to collect the local high frequency components to assemble a global continuous approximation. Theoretical results are obtained and numerical tests are reported to support the theoretical findings.