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$$hp$$-adaptive discontinuous Galerkin finite element methods for first-order hyperbolic problems. (English) Zbl 1029.65130
The a posteriori error analysis of $$hp$$-discontinuous Galerkin finite element approximations to first-order hyperbolic problems is considered. In particular, the question of error estimation for linear functionals, such as the outflow flux and the local average of the solution is discussed.
Based on that a posteriori error bound, the corresponding adaptive algorithm is designed and implemented to ensure reliable and efficient control of the error in the prescribed functional to within a given tolerance. This involves exploiting both local polynomial-degree variation and local mesh subdivision.
The theoretical results are illustrated by a series of numerical experiments.

##### MSC:
 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 65N15 Error bounds for boundary value problems involving PDEs 65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs 35L50 Initial-boundary value problems for first-order hyperbolic systems
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