Dipartimento di Matematica ''F. Casorati''

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Scientific Computing: Numerical Methods and Applications

The CSMNA (Calcolo Scientifico: Metodi Numerici ed Applicazioni) group at the Mathematics Department of the University of Pavia is active in the design, analysis and development of innovative methods for the numerical solution of partial differential equations, with applications to many areas of practical interest such as solid mechanics, fluid mechanics, fluid-structure interaction, electromagnetism, and in modeling and computational electrocardiology.
Our primary interests are related to the Finite Element Method (FEM), one of most widely used methods to numerically solve problems in the applications that can be written in terms of partial differential equations, and to its variants. We have developed great expertise, internationally recognized, in the foundations, the theoretical analysis, and the application of various kinds of FEM (conforming, non-conforming, mixed, discontinuous) to several interesting real-life problems: fluid-dynamics, advection-diffusion, linear elasticity, magnetostatic, plates and shells, fluid-structure interaction, etc.
The following paragraphs give an overview of the most active research areas.

The Finite Element Method
Other theoretical aspects of FEM and its applications are active object of research. Among those, we recall in particular the approximation of eigenvalue problems arising from partial differential equations, the simulation of fluid-structure interaction problems (Immersed boundary method), the analysis and the implementation of adaptive finite element schemes (a posteriori estimates and convergence of the adaptive scheme), approximation properties of finite element spaces on distorted meshes, the application of FEM to electromagnetism.
For all these problems, the numerical approach is based on a rigorous analysis of the approximation scheme (well-posedness, stability, convergence, etc.) and on the numerical validation of the theoretical results. 
Parallel and scalable solvers, based on domain decomposition and multigrid methods, for the resulting linear systems are also studied.

The Virtual Element Method (VEM)
The Virtual Element Method (VEM) is a new method that we developed in the last years, and which is gaining great attention by the scientific community at the international level. The VEM, that could be seen as an evolution of FEM, gives the possibility of using almost arbitrary decompositions of the computational domain, thus avoiding restrictions on the mesh required by FEM. Virtual Elements have already proved their efficiency in a number of applications: advection-diffusion-reaction, linear elasticity, plate bending problems and so on, but there is a lot to do yet.

The IsoGeometric Method
The IsoGeometric Method (IGM) stands for class of discretisation techniques for partial differential equations (PDEs) that addresses the interoperability between Computer Aided Design (CAD) and numerical simulation of PDEs. CAD software, used in industry for geometric modeling, typically describes physical domains by means of Non-Uniform Rational B-Splines (NURBS) and the interface between CAD output and classical numerical schemes calls for expensive re-meshing methods that result in approximate representation of domains. IGMs are NURBS-based schemes for solving PDEs whose benefits go beyond the improved interoperability with CAD. Indeed, they provide a substantial increase of the accuracy-to-computational-effort ratio and, thanks to the use of high-degree smooth NURBS within the numerical scheme, they outperform classical numerical schemes in most academic benchmarks. However, the mathematical understanding of the IGM is still incomplete and likely we are far from exploiting its full potential. The use of higher-degree IGM for real-world applications asks for new tools allowing for the efficient construction and solution of the linear system, time integration, flexible local mesh refinement, and so on. This research activity is aimed at providing the crucial knowledge to further develop the IGM into a highly accurate and stable methodology, having an impact in the field of numerical simulation, particularly when accuracy is essential both in geometry and fields representation.

In particular, the following topic are currently under investigation:

- theory of spline/NURBS spaces (high-degree splines, hierarchical splines, T-splines)
- design of spline spaces for complex geometries (multipatch, trimming, software interface to solid modelers)
- stability and well-posedness of IGMs for various class of applications: solid mechanics, contact problems, fluid dynamics, electromagnetism (De Rahm compatible splines).
- IGM parallel and scalable solvers, based on domain decomposition methods
- adaptive method for IGM based on hierarchical splines and T-splines

This activity is carried on in Pavia at the Mathematics Department and at IMATI-CNR (Istituto di Matematica Applicata e Tecnologie Informatiche E. Magenes, CNR), involving a large group of PhDs and Post-docs, and spans from very theoretical research to implementation and testing for complex applications (in collaboration with industrial partners: Total, Michelin, Hutchinson and Alenia Aeronautica)

Dipartimento di Matematica ''F. Casorati''

Università degli Studi di Pavia - Via Ferrata, 5 - 27100 Pavia
Tel +39.0382.985600 - Fax +39.0382.985602