Uniform estimates for a family of Eulerian-Lagrangian methods for time-dependent convection-diffusion equations with degenerate diffusion

We prove a priori error estimates for a family of Eulerian-Lagrangian methods for time-dependent convection-diffusion equations with degenerate diffusion. The estimates depend only on certain Sobolev norms of the initial and right side data of the problem but not on the lower bound of the diffusion or any norms of the true solution. Thus these estimates hold uniformly with respect to the degenerate diffusion. On a general unstructured mesh, these estimates are suboptimal but sharp when the Courant number is less than unity and are optimal otherwise. We further prove an optimal-order error estimate and a superconvergence estimate for a special case of d-linear approximations on a d-dimensional rectangular domain with a uniform rectangular partition. We then use the interpolation of spaces and stability estimates to derive an estimate for problems with minimal or intermediate regularity, where the convergence rates are proportional to certain Besov norms of the initial and right side data. Numerical experiments are presented to confirm the theoretical results.