High-order polyhedral Discontinuous Galerkin methods for multiphysics problems

The numerical approximation of multiphysics problems modelled by (possibly heterogeneous) partial differential equations (PDEs) is relevant for many Engineering and Applied Science fields. Two relevant examples are environmental problems (e.g., establishing the safety of soil exploitation activities) and biological systems (e.g., modelling functional changes in the brain). Typically, the PDEs governing the applications mentioned above are posed on complex domains, possibly involve the mutual interaction of mechanical and/or thermodynamical effects, and are characterised by problem coefficients (and/or solutions) that vary across multiple space and/or time length scales. This complexity demands for the development of “innovative” numerical schemes that are: i) flexible, to represent complex geometries with a reasonable effort; ii) accurate in the approximation of the coupled physical fields; iii) efficient, to handle large-scale simulations. In this talk we discuss recent developments in the construction and analysis of high-order polyhedral Discontinuous Galerkin (PolyDG) methods for PDEs modelling multiphysics problems. We also discuss how to enhance PolyDG accuracy and algorithmic efficiency based on employing novel Machine Learning-aided techniques. We test the proposed approach on two diverse applications in the fields of geothermal energy production and brain modelling.