We present an
overview of some recent
developments concerning the a posteriori error analysis of h–
and hp–version finite element approximations to compressible
fluid flows. After highlighting some of
the conceptual difficulties in error control for problems of
hyperbolic/nearly–hyperbolic type, such as the lack of
correlation between the local error and the local finite element
residual, we concentrate on a specific discretisation scheme: the
hp–version of the discontinuous Galerkin finite element
method. This method is capable of exploiting both local
(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.
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 lift and drag
coefficients for a body immersed into a fluid, the local mean value of
the field or its flux through the outflow boundary of the computational
domain, and the pointwise evaluation of a component of the solution.
By employing a duality argument we derive so–called weighted
or Type I a posteriori estimates which bound 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. 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.
For the hp–version of the
discontinuous Galerkin finite element method, the decision as to
whether to h–refine or p–refine an element is based
on estimating the local analyticity of the primal and dual solutions
via truncated Legendre series expansions. The performance of
the resulting hp–refinement algorithm is demonstrated through
a series of numerical experiments.
This research has been funded by the EU under the ADIGMA project.