The CSMNA (Calcolo Scientifico: Metodi Numerici e Applicazioni = Scientific Computing: Numerical Methods and Applications) deals with the development and analysis of innovative methods for the numerical solution of partial differential equations with applications in different areas of practical relevance such as solid mechanics, fluid mechanics, fluid-structure interaction, electromagnetism, and computational electrocardiogram.

### Research methods

#### Finite element method (FEM)

The leading method which distinguished the group in the past years is the Finite Element Method (FEM), one of the most widely used methods to numerically solve real-life problems, modeled by partial differential equations.

The group has produced internationally recognised contributions, both regarding the theoretical foundations of different types of finite element methods (conforming, non-conforming, mixed, discontinuous) and their application to numerous problems of practical interest related to: fluid-dynamics, diffusion-transport, linear elasticity, magnetostatic, plates and shells, semiconductor devices, fluid-structure interaction, etc.

The research also concerns other theoretical aspects of FEM and applications: The approximation of eigenvalue problems associated with partial differential equations, the simulation of fluid-structure interaction problems (immersed boundary method), the analysis and implementation of schemes to adaptive finite elements (a posteriori estimates and convergence of the adaptive scheme), approximation properties of finite element spaces on distorted meshes, application of finite elements to electromagnetism problems.

The numerical approach is based on the rigorous analysis of numerical schemes (well-posedness, stability, convergence, etc.) and on the numerical validation of the theoretical results. Parallel and scalable solvers are also studied for the resulting FEM systems, based on domain decomposition and multigrid methods.

#### Virtual Element Method (VEM)

The Method of Virtual Elements (VEM) introduced and analyzed recently, is enjoying success in the international scientific community. It is an evolution of the finite element method that allows the use of computational domain decompositions into polygons of arbitrary shape, thus avoiding the restrictions imposed on the mesh by the finite element method. It has already proved effective and robust method in a number of applications, including linear elasticity problems, thin plate bending problems, diffusion-transport-reaction problems, but several aspects yet to be examined.

#### IsoGeometric Method (IGM)

The IsoGeometric Method (IGM) includes a class of discretization techniques for partial differential equations (PDE), based on the interaction between Computer Aided Design (CAD) and numerical simulation of PDE. CAD software, used in the geometric modelling industry, typically describes the physical domain by means of Non-Uniform Rational B-Splines (NURBS) and the interface between CAD output and classical numerical schemes requires discretization techniques expensive and result in only an approximation of the physical domain. IGM methods are NURBS-based schemes for solving PDEs whose benefits go far beyond better integration with CAD. This research activity aims at developing the basic techniques to make IGM a highly accurate and stable methodology to be used in numerical simulations, especially for cases where accuracy is essential for the geometry and the representation of the solution.

More specifically, the following aspects are currently being studied:

• Theory of spline/NURBS spaces (high order splines, hierarchical splines, T-splines);
• Development of spline discretizations for complex geometries,obtained with Boolean operations of pasting, clipping, etc., and interfacing with solid modelling codes;
• Stability and well-posedness of the IGM for various classes of applications: Solid mechanics, contact problems, fluid dynamics, electromagnetism (compatible De Rham splines);

This activity takes place at the Department of Mathematics, University of Pavia, and IMATI-CNR (Institute of Applied Mathematics and Information Technologies E. Magenes, CNR) and involves a large group of PhD students and post-docs. The research activity concerns purely theoretical aspects, but also advanced applications (in collaboration with industrial partners: Total, Michelin, Hutchinson, and Alenia Aeronautica).

#### Domain Decomposition Methods (DDM)

Domain Decomposition methods are iterative methods for the parallel solution of linear or nonlinear systems generated by the discretisation of partial differential problems with finite elements, spectral, virtual, or isogeometric elements. The original problem is decomposed into local problems on subdomains, with or without overlap, and possibly into a sparse problem with one or more unknowns per subdomain. These subproblems are then parallelly solved by assigning them to different processors of distributed computing architectures, obtaining scalable preconditioned iterative methods.

### Seminars

The group co-organises, together with the group of Mathematical Analysis and Applications, the seminars of applied mathematics. The complete list of the seminars is available at the following link.