NEMESIS

The purpose of this project is to lay the ground for the submission of an ERC Synergy Grant (SyG), whose goal will be the development of the next generation of numerical simulators for problems governed by Partial Differential Equation (PDEs). The SyG project will involve four Principal Investigators (PIs) and their teams: Paola Antonietti (Politecnico di Milano, Italy), Lourenço Beirão da Veiga (Università di Milano Bicocca, Italy), Daniele Di Pietro (Université de Montpellier, France), and Jérôme Droniou (Monash University, Australia). The numerical simulation of physical models based on PDEs is an established tool in several domains. Classical (Finite Element or Finite Volume) discretization methods, however, are often unable to provide the flexibility required by modern applications. Their limitations include the type of computational meshes that can be handled, the possibility to modify the approximation order according to the (local) regularity of the solution, the lack of robustness with respect to the physics, etc. The envisaged SyG project aims at tackling these (and other) problems using novel tools and paradigms resulting from synergy of the PIs. This will be achieved through (i) methodological developments in numerical analysis; (ii) the design and analysis of novel discretization methods for challenging, so far untamed problems in engineering and sciences (fluid- and solid-mechanics, porous media flows, electromagnetism); (iii) the development of open source libraries. The four PIs have outstanding ten-years track records, and display an innovative and exceptional combination of knowledge and skills in numerical analysis, scientific computing, and applications. The PIs are pioneers in the development and analysis of next generation discretization methods for PDEs, and their combined knowledge essentially covers the full spectrum of polyhedral methods proposed up to now. Each PI also displays a specific and well-recognized expertise in various application fields, whose synergy will be essential to tackling complex (nonlinear, multi-scale, multi-physics) PDE problems. Finally, each PI has an extensive and specific set of skills in numerical analysis and scientific computing, whose combination will be essential for the synergy group to break new ground concerning fundamental aspects, methodological developments, and analysis tools. Cross-fertilization among the PIs' specific skills and expertise will be the main added value of the synergy, and occur at multiple levels. We expect the emergence of novel points of view in numerical analysis, in the design and analysis of discretization methods for complex problems that are out of reach for classical discretization methods, and the development of reference implementations for this new generation of numerical methods. The requested MRSEI funding will play a crucial role in the construction of the SyG project by (co-)funding three meetings of the PIs, as well as an international workshop in Montpellier, which will enable all the involved research teams to meet, exchange, and bond. The workshop will also be open to external experts in the domain, who will be consulted for advice on the development of the project. The main impact of the envisaged SyG project will be to develop, analyse and validate on challenging applications the next generation of discretization methods for the numerical approximations of problems governed by PDEs. The novel simulators developed within the SyG project will have ground breaking features that will make it possible to tackle problems so far inaccessible in terms of geometric features and complexity of the underlying physics. The SyG project will also have a major impact on the emerging community around next-generation discretization methods on polytopal meshes. Potential long-term applications include advanced manufacturing, biological processes, computational biomedicine, nuclear waste disposal, geological carbon dioxide storage, etc.