We present a discontinuous Galerkin multilevel method with hp-adaptivity for acoustic problems. The main ad-
vantage of this multilevel method is that the size of the finite element space is independent on the presence of
complicated or tiny features in the domain. In other words, even on a very complicated domain, an approximation
of the solution can be computed with only a fistful of degrees of freedom. This is possible because two meshes are
used: a fine mesh is used to describe the geometry of the domain with all its features, but the problem is solved on
a coarse mesh that is, in general, too coarse to describe all the geometrical features of the domain. Unlikely other
multilevel methods, this method does not perturb the problem, in the sense that the problem solved on the coarse
mesh is always a discretization of the continuous problem, no matter how coarse the mesh is.
The hp-adaptivity algorithm that we present for this multilevel method is completely automatic and capable of ex-
ploiting 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 alone. The error estimator employs a duality argument for which we derive so-called weighted (or Type
I) a posteriori estimate, which bounds the error between the true value of the prescribed functional and the actual
computed value. In this error estimator the element residuals of the computed numerical solution are multiplied by
local weights involving the solution of a certain dual or adjoint problem. On the basis of the resulting a posteriori
error bound, 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 user-defined tolerance. Our adaptive finite element algo-
rithm decides automatically between either h-refine or p-refine. The performance of the resulting hp-refinement
algorithm is demonstrated through a series of numerical experiments.
This research has been funded by the EPSRC and by the EU.