NEw generation MEthods for numerical SImulationS

Introduction

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 of the NEMESIS project is to lay the groundwork for a novel generation of numerical simulators tackling all of the above difficulties at once.

Objectives

This will require the combination of skills and knowledge resulting from the synergy of the PIs, covering distinct and extremely technical fields of mathematics:

1. Numerical analysis
2. Analysis of nonlinear PDEs
3. Scientific computing

Research Program Structure

The research program is structured into four tightly interconnected clusters, whose goals are:

1. The development of Polytopal Exterior Calculus (PEC), a general theory of discrete Hilbert complexes on polytopal meshes
2. The design of innovative strategies to boost efficiency, embedded into a general abstract Multilevel Solvers Convergence Framework (MSCF)
3. The extension of the above tools to challenging nonlinear and hybrid-dimensional problems through Discrete Functional Analysis (DFA) tools
4. The demonstration through proof-of-concept applications in:
   • Magnetohydrodynamics (e.g., nuclear reactor models or aluminum smelting)
   • Geological flows (e.g., flows of gas/liquid mixtures in underground reservoirs with fractures, as occurring in CO2 storage)

Expected Outcomes

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.