Discontinuous Galerkin methods on arbitrarily shaped elements.

We extend the applicability of the popular interior-penalty discontinuous Galerkin (dG) method discretizing 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 (iso-)parametric elemental maps. The feasibility of the method relies on the definition of a suitable choice of the discontinuity-penal-ization parameter, which turns out to be essentially independent on the particular element shape. A priori error bounds for the resulting method are given under very mild structural assumptions restricting the magnitude of the local curvature of element boundaries. Numerical experiments are also presented, indicating the practicality of the proposed approach. Moreover, we shall discuss a number of perspectives on the possible applications of the proposed framework in parabolic problems on moving domains as well as on multiscale problems. The above is an overview of results from joint works with A. Cangiani (Nottingham, UK), Z. Dong (FORTH, Greece / Cardiff UK) and T. Kappas (Leicester, UK).