Optimal a priori error estimates for the ℎ𝑝-version of the local discontinuous Galerkin method for convection–diffusion problems

Castillo, Paul; Cockburn, Bernardo; Schötzau, Dominik; Schwab, Christoph

doi:10.1090/S0025-5718-01-01317-5

Optimal a priori error estimates for the $hp$-version of the local discontinuous Galerkin method for convection–diffusion problems
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by Paul Castillo, Bernardo Cockburn, Dominik Schötzau and Christoph Schwab HTML | PDF

Math. Comp. 71 (2002), 455-478 Request permission

Abstract:

We study the convergence properties of the $hp$-version of the local discontinuous Galerkin finite element method for convection-diffusion problems; we consider a model problem in a one-dimensional space domain. We allow arbitrary meshes and polynomial degree distributions and obtain upper bounds for the energy norm of the error which are explicit in the mesh-width $h$, in the polynomial degree $p$, and in the regularity of the exact solution. We identify a special numerical flux for which the estimates are optimal in both $h$ and $p$. The theoretical results are confirmed in a series of numerical examples.

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