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The $$hp$$ version of Eulerian-Lagrangian mixed discontinuous finite element methods for advection-diffusion problems. (English) Zbl 1028.65105
Summary: 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.

##### MSC:
 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 35K65 Degenerate parabolic equations 35A35 Theoretical approximation in context of PDEs
##### Keywords:
space-time mixed formulation; one space dimension
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