Universitą degli studi di Pavia


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Curriculum Numerical analysis and mathematical modeling

Numerical approximation of partial differential equations

The expertise of our group in this field is widely recognized at international level. Important contributions are given by faculty members of the Ph. D. program as well as by other researchers of the institutions which promote this program. We refer, in particular, to the Department of Mathematics of the University of Pavia, to the IMATI Institute of the CNR (Pavia), and to the Department of Mathematics and its Applications of the University of Milano-Bicocca.
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 and provide links where interested students can find more detailed informations.

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.
Researchers involved: Marini, Pietra, Russo.

The IsoGeometric Method (IGM) stands for a 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. 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.
Researchers involved: Buffa, Sangalli.

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.
Researchers involved: Boffi.

CAGD, numerical modeling of curves and surfaces, and approximation of data

Our primary interests in CAGD (Computer-Aided Geometric Design) are related to the construction, the analysis and the applications of subdivision schemes, efficient iterative methods for the generation of graphs of functions, curves and surfaces via the specification of an initial set of discrete data and a set of local refinement rules. Over the past 20 years subdivision schemes have shown their usefulness in several application contexts ranging from Computer-Aided Geometric Design to Computer Graphics and Animation. Recently, subdivision schemes have become of interest also in biomedical imaging applications, especially for the representation of active surfaces for the segmentation of 3D biomedical images.
Researchers involved: Romani.


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