We study the hp version of three families of
Eulerian-Lagrangian mixed discontinuous finite element (MDFE)
methods for the numerical solution of advection-diffusion
problems. These methods are based on a space-time mixed
formulation of the advection-diffusion problems. In space, they
use discontinuous finite elements, and in time they approximately
follow the Lagrangian flow paths (i.e., the hyperbolic part of
the problems). Boundary conditions are incorporated in a natural
and mass conservative manner. In fact, these methods are locally
conservative. The analysis of this paper focuses on
advection-diffusion problems in one space dimension. Error
estimates are explicitly obtained in the grid size h, the polynomial degree p, and the solution regularity; arbitrary
space grids and polynomial degree are allowed. These estimates are
asymptotically optimal in both h and p for some of these
methods. Numerical results to show convergence rates in h and
p of the Eulerian-Lagrangian MDFE methods are presented. They
are in a good agreement with the theory.
