Interests

Numerical methods for the solution of partial differential equations.

Keywords

• • PE1_17 - Numerical analysis

  • PE1_18 - Scientific computing and data processing

  • PE1_19 - Control theory and optimization

  • PE1_20 - Application of mathematics in sciences

  • PE1_21 - Application of mathematics in industry and society life

Major Research Topics

The research activity of the Numerical Analysis Group is oriented towards the approximation of Partial Differential Equations (PDE) and contributes by developing new computational methodologies based, in different ways, on the interaction of various differential models and/or of several numerical discretization methods.

1. Fluid-structure interaction

   1. The nonlinear coupling of the equations governing fluid-structure interaction systems requires appropriate numerical approaches in order to deal with the motion of the domains occupied by solid and fluid. One of the main issues is the construction of stable numerical schemes. Two different approaches have been considered:

      1. 1. Immersed boundary method

         2. Arbitrary Lagrangian-Eulerian formulation

2. Approximation of PDEs by finite element methods

• 1. The finite element method is one of the most popular methods available for the numerical resolution of PDEs of different types. In view of practical applications, the finite elements methods need to be robust, efficient, and accurate. In the case of finite elements for problems in mixed form, this requires that some compatibility conditions are satisfied.

     1. 1. Finite element methods for the approximation of eigenproblem in mixed form

        2. Finite element approximation of evolution problem in mixed form

        3. Edge finite elements for Maxwell and photonic crystal equations

        4. Finite elements for the Stokes problem

3. Domain Decomposition Methods (DDM) for Heterogeneous Problems

• 1. Subdomain splitting is an interesting path towards multiphysics (or heterogeneous problems) in which different kinds of PDE (modeling different physical phenomena) are set up in different subdomains. Examples are the coupling of Stokes equations with the Darcy equations to simulate the filtration of fluids in porous media; the coupling between advection-diffusion with dominated advection and pure advection phenomena; the coupling between the Navier-Stokes equation and the system of linear or nonlinear elasticity for fluid-structure interactions.

     1. 1. Interface Control Domain Decomposition (ICDD) methods are overlapping DDM that are well suited to face heterogeneous problems.

        2. INTERNODES: a general-purpose method to deal with non-conforming discretizations of partial differential equations on regions partitioned into two or several disjoint subdomains. It exploits two intergrid interpolation operators, one for transferring the Dirichlet trace across the interfaces, the others for the Neumann trace.

4. High-order methods for the approximation of PDE's

• 1. Spectral Methods are high-order methods for solving PDE's which offer the best performance (in terms of computational efficiency and in handling complex geometries) when they are coupled with low-order methods (such as finite elements) inside the preconditioning step, and domain decomposition techniques.

  2. Algebraic Fractional-Step Schemes are very efficient and accurate techniques to approximate time-dependent PDE's as, e.g., the incompressible Navier-Stokes equations.

     1. 1. Finite-element preconditioning of spectral methods

        2. Algebraic fractional step schemes for the incompressible Navier-Stokes equations

Major Research Project

1. MIUR/PRIN2017

   1. Project name: Modeling the heart across the scales: from cardiac cells to the whole organ.

   2. P.I.: Alfio Quarteroni, Politecnico di Milano

   3. Start period: August 19, 2019

   4. Period (months): 36

2. MIUR/PRIN2017

• 1. Project name: Numerical Analysis for Full and Reduced Order Methods for the efficient and accurate solution of complex systems governed by Partial Differential Equations (NA-FROM-PDEs)

  2. P.I.: Gianluigi Rozza, SISSA Trieste

  3. Start period: August 19, 2019

  4. Period (months): 36

3. MIUR/PRIN2012

• 1. Project name: Metodologie innovative nella modellistica differenziale numerica.

  2. P.I.: Claudio Canuto, Politecnico di Torino

  3. Start period: 8 Marzo 2014

  4. Period (months): 36

4. MIUR/PRIN2008

• 1. Project name: Analisi e sviluppo di metodi numerici avanzati per EDP.

  2. P.I.: Franco Brezzi, Università di Pavia

  3. Start period: March 22, 2010

  4. Period (months): 24

5. GNCS/Progetti di ricerca 2010

• 1. Project name: Approssimazione numerica di problemi di interazione fluido-struttura

  2. Start period: December 01, 2009

  3. Period (months): 12

6. MIUR/PRIN2006

• 1. Project name: Equazioni cinetiche e idrodinamiche di sistemi collisionali complessi.

  2. Start period: February 02, 2007

  3. Period (months): 24

7. MIUR/PRIN2004

• 1. Project name: Metodi numerici avanzati per equazioni alle derivate parziali di interesse applicativo.

  2. P.I.: Franco Brezzi, Università di Pavia

  3. Start period: November 30, 2004

  4. Period (months): 24

8. MIUR/PRIN2003

• 1. Project name: Modellistica Numerica per il Calcolo Scientifico e Applicazioni Avanzate.

  2. P.I.: Alfio Quarteroni, Politecnico di Milano

  3. Start period: November 20, 2003

  4. Period (months): 24

9. MIUR/PRIN2001

• 1. Project name: Metodi numerici avanzati per equazioni alle derivate parziali di interesse applicativo

  2. P.I.: Franco Brezzi, Università di Pavia

  3. Start period: December 12, 2001

  4. Period (months): 24

10. MIUR/PRIN2000

• 1. Project name: Scientific Computing: Innovative Models and Numerical Methods

  2. P.I.: Claudio Verdi, Università di Milano

  3. Start period: December 20, 2000

  4. Period (months): 24

Strategic collaborations

• • Alfio QUARTERONI, MOX, Dipartimento di Matematica, Politecnico di Milano (Italy)

  • Claudio CANUTO, Dipartimento di Matematica, Politecnico di Torino (Italy)

  • Daniele BOFFI, Dipartimento di Matematica, Università di Pavia (Italy)

  • Marco DISCACCIATI, Department of Mathematical Sciences, Loughborough University (UK)

  • Simone DEPARIS, MNS, Institute of Mathematics, Ecole Polytechnique Fédérale de Lausanne (CH)

  • Luca DEDE', MOX, Dipartimento di Matematica, Politecnico di Milano (Italy)

  • Pablo BLANCO, LNCC, Petrópolis, RJ (Brazil)

  • Valery I. AGOSHKOV, Russian Academy of Sciences, Moscow (Russia)

  • Davide FORTI, MNS, Institute of Mathematics, Ecole Polytechnique Fédérale de Lausanne (CH)