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International Journal of Mathematics and Mathematical Sciences
Volume 2003, Issue 53, Pages 3385-3411

The hp version of Eulerian-Lagrangian mixed discontinuous finite element methods for advection-diffusion problems

1Department of Mathematics, University of Wyoming, Laramie, WY 82071, USA
2Department of Mathematics, P.O. Box 750156, Southern Methodist University, Dallas, TX 75275-0156, USA

Received 26 December 2001

Copyright © 2003 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.