Uniform Estimates for Eulerian–Lagrangian Methods for Singularly Perturbed Time-Dependent Problems

We prove a priori optimal-order error estimates in a weighted energy norm for several Eulerian-Lagrangian methods for singularly perturbed, time-dependent convection-diffusion equations with full regularity. The estimates depend only on certain Sobolev norms of the initial and right-hand side data, but not on $\varepsilon$ or any norm of the true solution, and so hold uniformly with respect to $\varepsilon$. We use the interpolation of spaces and stability estimates to derive an $\varepsilon$-uniform estimate for problems with minimal or intermediate regularity, where the convergence rates are proportional to certain Besov norms of the initial and right-hand side data.