Prof Emmanuil Georgoulis, Heriot Watt and NTU Athens

Affiliation: Heriot Watt and National Technical University Athens

Academic Webpage

Title: Discontinuous Galerkin methods on arbitrarily-shaped elements: stability, analysis, and adaptivity

Abstract: I will present a recent generalisation of the popular interior-penalty discontinuous Galerkin (dG) method discretizing general classes of linear and nonlinear advection-diffusion-reaction problems to meshes comprising extremely general, essentially arbitrarily-shaped element shapes. In particular, our analysis allows for curved element shapes, without the use of non-linear elemental maps. The feasibility of the method relies on the definition of a suitable choice of the discontinuity-penalisation, which turns out to be explicitly dependent on the particular element shape, but essentially independent on small shape variations.  A priori error bounds for the resulting method will be given, under very mild structural assumptions restricting the magnitude of the local curvature of element boundaries. In the second part of the talk, I plan to present new, rigorous, a posteriori  error estimates in various norms for the above method for elliptic as well as a class of semilinear parabolic problems. The key challenge of robustness with respect to local element shape variations. I will present some ideas on how this is addressed theoretically, by generalising existing concepts of proof of a posteriori error estimates for non-conforming Galerkin methods. Numerical experiments will be also presented throughout the talk aiming to motivate and showcase the practicality and the potential advantages of the proposed numerical framework.