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$$hp$$-discontinuous Galerkin finite element methods for hyperbolic problems: Error analysis and adaptivity. (English) Zbl 1021.76027
Summary: We develop a posteriori error analysis of the $$hp$$-version of discontinuous Galerkin finite element method for linear and nonlinear hyperbolic problems. By employing a duality argument, sharp a posteriori error bounds are derived for certain output functionals of practical interest. These bounds exhibit an exponential rate of convergence under $$hp$$-refinement if either the primal or dual solution is an analytic function over the computational domain. Based on our a posteriori error bounds, we design and implement the corresponding $$hp$$-adaptive finite element algorithm to ensure the reliable and efficient control of the error in a prescribed functional to within a user-defined tolerance.

##### MSC:
 76M10 Finite element methods applied to problems in fluid mechanics 76N15 Gas dynamics (general theory)
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##### References:
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