A posteriori error estimation for discontinuous Galerkin discretizations of H(curl)-elliptic partial differential equations

We develop the a posteriori error estimation of interior penalty discontinuous Galerkin discretizations for H(curl)-elliptic problems that arise in eddy current models. Computable upper and lower bounds on the error measured in terms of a natural (mesh-dependent) energy norm are derived. The proposed a posteriori error estimator is validated by numerical experiments, illustrating its reliability and efficiency for a range of test problems.

Symmetric Interior Penalty DG Methods for the Compressible Navier-Stokes Equations II: Goal--Oriented A Posteriori Error Estimation

In this article we consider the application of the generalization of the symmetric version of the interior penalty discontinuous Galerkin finite element method to the numerical approximation of the compressible Navier--Stokes equations. In particular, we consider the a posteriori error analysis and adaptive mesh design for the underlying discretization method. Indeed, by employing a duality argument (weighted) Type I a posteriori bounds are derived for the estimation of the error measured in terms of general target functionals of the solution; these error estimates involve the product of the finite element residuals with local weighting terms involving the solution of a certain dual problem that must be numerically approximated. This general approach leads to the design of economical finite element meshes specifically tailored to the computation of the target functional of interest, as well as providing efficient error estimation. Numerical experiments demonstrating the performance of the proposed approach will be presented.

Error estimation and adaptive mesh refinement for aerodynamic flows

This lecture course covers the theory of so-called duality-based a posteriori error estimation of DG finite element methods. In particular, we formulate consistent and adjoint consistent DG methods for the numerical approximation of both the compressible Euler and Navier-Stokes equations; in the latter case, the viscous terms are discretized based on employing an interior penalty method. By exploiting a duality argument, adjoint-based a posteriori error indicators will be established. Moreover, application of these computable bounds within automatic adaptive finite element algorithms will be developed. Here, a variety of isotropic and anisotropic adaptive strategies, as well as $hp$-mesh refinement will be investigated.