We present an overview of some recent developments concerning the a posteriori error analysis of h- and hp-
version finite element approximations to acoustic problems for a specific discretisation scheme: the hp-version
of the discontinuous Galerkin (DG) finite element method. This 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. Moreover, by exploiting the flexibility of the method it is possible to handle easily complex geometries with
different materials. Also, we apply the DG methods to two classes of problems: the class of source problems used
to compute the response of a structure under the action of an external force and the class of eigenvalue problems
used to compute the eigenmodes of a structure.
We shall be particularly concerned with the derivation of a posteriori bounds on the error in certain output functionals of the solution of practical interest; relevant examples include the response at a certain point, the mean value
of the response, the accurate localization of the eigenvalues. We have derived two types of a posteriori estimates,
the first one is explicit, which makes the error estimator very cheap to compute. Secondly, by employing a duality
argument we derived the so-called weighted a posteriori estimates which bounds the error between the true value
of the prescribed functional and the actual computed value. In these error estimates, the element residuals of the
computed numerical solution are multiplied by local weights involving the solution of a certain dual or adjoint
problem. The great advantage of the latter error estimator is the possibility to compute reliable bounds on the
error for the quantity of interest, which can be safely used for finite element analysis and model validation. On the
basis of the resulting a posteriori error estimates, we design and implement an adaptive finite element algorithm to
ensure reliable and efficient control of the error in the computed functional with respect to a userdefined tolerance.
Our adaptive finite element algorithm decides automatically between isotropic and anisotropic refinement either in
h−refine or in p−refine.
We present a comprehensive series of numerical experiments for both classes of problems, both in 2D and in 3D.
The examples cover a wide range of frequency from low to high and both error estimators are used on the whole
spectrum of frequencies.
This research has been funded by the Marie Curie Foundation and by the EU under the MIDEA project.