Research database

The Finite Element Method (FEM) is one of the most powerful tools for the numerical approximation of partial differential equations (PDEs). Real-life applications typically involve the closely coupled interaction of different physical phenomena occurring at multiple spatio-temporal scales. Multiphysics phenomena can be modelled via coupled systems of heterogeneous PDEs set in possibly complex geometries which may include embedded interfaces and free surfaces. The new paradigm of Polygonal Finite Element Methods (PolyFEMs) has been introduced in the last years. PolyFEMs are Galerkin-type projection methods where the finite-dimensional discretisation space is built by employing a computational grid of arbitrarily shaped polygonal/polyhedral (polytopic, for short) elements. This project focuses on two of such methods, namely, the Virtual Element Method (VEM) and the polyhedral Discontinuous Galerkin (PolyDG) Method, which stand at the forefront of the research in this area. As compared to other PolyFEMs, these two methods possess distinguishing features that can be exploited to tackle heterogeneous PDEs modelling multiphysics problems successfully. In particular, VEMs can support approximation spaces preserving the underlying physical models' fundamental properties (e.g., the divergence-free constraint for incompressible flow problems), while PolyDG schemes feature the capability to naturally guarantee a physically consistent coupling among the fields and suitable agglomeration/refinement procedures. This project aims to develop the theoretical foundations for Virtual Element and PolyDG methods for heterogeneous and hybrid-dimensional systems of PDEs modelling multiphysics problems and to enhance their accuracy and performance based on designing suitable Machine Learning-aided numerical algorithms. The capabilities of the methods will be assessed by tackling two challenging applications in the field of computational geophysics and advanced manufacturing processes. All algorithms will be implemented in open-source codes. The project will have a ground-breaking impact in the field of numerical approximation of PDEs for multiphysics problems and will lead to new Machine Learning-aided numerical algorithms, allowing to improve the accuracy and reduce the computational costs.