Progetti di ricerca

Relevant partial differential equations (PDEs) problems of the 21st century, including those encountered in magnetohydrodynamics and geological flows, involve severe difficulties linked to: the presence of incomplete differential operators related to Hilbert complexes; nonlinear and hybrid-dimensional physical behaviors; embedded/moving interfaces. The goal… Leggi tutto of the NEMESIS project is to lay the groundwork for a novel generation of numerical simulators tackling all of the above difficulties at once. This will require the combination of skills and knowledge resulting from the synergy of the PIs, covering distinct and extremely technical fields of mathematics: numerical analysis, analysis of nonlinear PDEs, and scientific computing. The research program is structured into four tightly interconnected clusters, whose goals are: the development of Polytopal Exterior Calculus (PEC), a general theory of discrete Hilbert complexes on polytopal meshes; the design of innovative strategies to boost efficiency, embedded into a general abstract Multilevel Solvers Convergence Framework (MSCF); the extension of the above tools to challenging nonlinear and hybriddimensional problems through Discrete Functional Analysis (DFA) tools; the demonstration through proof-of-concept applications in magnetohydrodynamics (e.g., nuclear reactor models or aluminum smelting) and geological flows (e.g., flows of gas/liquid mixtures in underground reservoirs with fractures, as occurring in CO2 storage). This project will bring key advances in numerical analysis through the introduction of entirely novel paradigms such as the PEC and DFA, and in scientific computing through MSCF. The novel mathematical tools developed in the project will break long-standing barriers in engineering and applied sciences, and will be implemented in a practitioner-oriented open-source library that will boost design and prediction capabilities in these fields.

The Virtual Element Method (VEM) is a novel technology for the discretization of partial differential equations (PDEs), that shares the same variational background as the Finite Element Method. First but not only, the VEM responds to the strongly increasing interest… Leggi tutto in using general polyhedral and polygonal meshes in the approximation of PDEs without the limit of using tetrahedral or hexahedral grids. By avoiding the explicit integration of the shape functions that span the discrete space and introducing an innovative construction of the stiffness matrixes, the VEM acquires very interesting properties and advantages with respect to more standard Galerkin methods, yet still keeping the same coding complexity. For instance, the VEM easily allows for polygonal/polyhedral meshes (even non-conforming) with non-convex elements and possibly with curved faces; it allows for discrete spaces of arbitrary C^k regularity on unstructured meshes. The main scope of the project is to address the recent theoretical challenges posed by VEM and to assess whether this promising technology can achieve a breakthrough in applications. First, the theoretical and computational foundations of VEM will be made stronger. A deeper theoretical insight, supported by a wider numerical experience on benchmark problems, will be developed to gain a better understanding of the method's potentials and set the foundations for more applicative purposes. Second, we will focus our attention on two tough and up-to-date problems of practical interest: large deformation elasticity (where VEM can yield a dramatically more efficient handling of material inclusions, meshing of the domain and grid adaptivity, plus a much stronger robustness with respect to large grid distortions) and the cardiac bidomain model (where VEM can lead to a more accurate domain approximation through MRI data, a flexible refinement/derefinement procedure along the propagation front, to an exact satisfaction of conservation laws).