We present an overview of recent developments concerning the a posteriori error analysis of hp-version finite element approximations to hyperbolic problems. After highlighting some of the conceptual difficulties in error control for hyperbolic problems, such as the lack of correlation between the local error and the local finite element residual, we concentrate on a specific discretisation: the hp-version of the discontinuous Galerkin finite element method. The method is capable of exploiting both local polynomial-degree-variation (p-refinement) and local mesh subdivision (h-refinement), thereby offering greater flexibility and efficiency than numerical techniques which only incorporate h-refinement or p-refinement in isolation. The decision as to whether to h-refine or p-refine is based on a new algorithm for Sobolev-index estimation via truncated Legendre series expansions.

We shall be particularly concerned with the derivation of a posteriori bounds on the error in output functionals of the solution; relevant examples include the lift and drag coefficients for a body immersed into an inviscid fluid, the local mean value of the field, or its flux through the outflow boundary of the computational domain.

The theoretical results will be illustrated by numerical experiments.

References:

Technical reports are available from:     http://web.comlab.ox.ac.uk/oucl/work/endre.suli/biblio.html

[1] R Hartmann, P Houston, and E Süli. Adaptive discontinuous Galerkin finite element methods for nonlinear hyperbolic problems.
NA-01/06, Oxford University Computing Laboratory, 2001.

[2] P Houston and E Süli. Adaptive Lagrange-Galerkin methods for  unsteady convection-dominated diffusion problems. Mathematics of Computation. Volume 70, No 233, pp.77--106, 2001.

[3] P Houston and E Süli. hp-Adaptive discontinuous Galerkin finite element methods for first-order hyperbolic problems. SIAM Journal on Scientific Computing, 23(4):1225-1251, 2001.

[4] P Houston and E Süli. Adaptive Finite Element Approximation of Hyperbolic Problems. NA-02/01, Oxford University Computing Laboratory, 2002.

[5] P Houston, B Senior and E Süli. Sobolev Regularity Estimation for hp-Adaptive Finite Element Methods. NA-02/02, Oxford University Computing Laboratory, 2002.