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Mathematical Analysis, Modelling, and Applications

The activity in mathematical analysis is mainly focussed on ordinary and partial differential equations, on dynamical systems, on the calculus of variations, and on control theory. Connections of these topics with differential geometry are also developed.The activity in mathematical modelling is oriented to subjects for which the main technical tools come from mathematical analysis. The present themes are multiscale analysismechanics of materialsmicromagneticsmodelling of biological systems, and problems related to control theory.The applications of mathematics developed in this course are related to the numerical analysis of partial differential equations and of control problems. This activity is organized in collaboration with MathLab for the study of problems coming from the real world, from industrial applications, and from complex systems.

Models and applications in CFD

The course refers to the use of computational fluid dynamics techniques to address advanced applications in environmental, cardiovascular and industrial contexts. Each topic will be corroborated by a set of numerical examples to be performed within the open source C++ finite volume library OpenFOAM. Please contact mgirfogl@sissa.it if you need more info.

Semigroup theory and applications

 

Program

Bochner integral; Pettis and Bochner theorems; vector valued distributions and Sobolev functions.

Elements on unbounded operators: closed, dissipative and maximal dissipative operators.

Semigroups and their generators.

Cauchy problem for abstract equations, Duhamel formula.

Hille-Yosida, Lumer-Phillips and Stone theorems, construction of (semi)groups associated to Heat, Wave, Klein Gordon and Schrödinger equations. 

Advanced FEM techniques

Note: all lessons are in room 133, except for the one on the 28/05 which is in room 136, and the one on the 02/07 which is in room 134.
 
This an advanced monographic course on the numerical analysis of finite element techniques. Each year, the state-of-the-art of a research level topic is selected and presented with strong interaction with the students. Past topics have included: Virtual Element Methods (VEM), Nonconforming FEM, Discontinuous Galerkin Methods.

 

Topics in computational fluid dynamics

Topics/Syllabus

  • Introduction to CFD, examples.
  • Constitutive laws
  • Incompressible flows.
  • Numerical methods for potential and thermal flows
  • Boundary layer theory
  • Thermodynamics effects, energy equation, enthalpy and entropy

Computational mechanics by reduced order methods

Mathematics Area, PhD in Mathematical Analysis, Modelling and Applications (AMMA)
Master in High Performance Computing (MHPC)
Lectures Prof Gianluigi Rozza, Tutorials coordinated by Dr Michele Girfoglio, Dr Federico Pichi and Dr Ivan Prusak.

Learning outcomes and objectives

Numerical solution of PDEs

 

Room 128: 8/3, 14/3,  21/3,  5/4, 12/4, 19/4

Room 139: 15/3, 22/3 

Room 133: 4/4, 11/4, 18/4, 25/4, 2/5, 9/5, 10/5, 16/5, 17/5

Room 134: 3/5

 

Advanced analysis - A

Rooms:
Lectures from 26/09 to 24/10 are in room 005
Lectures from 26/10 to 19/12 are in room 133

Program of the course Advanced Analysis –A (2023-2024)

Limit analysis of variational problems

The aim of this course is an introduction to the analysis of variational problems depending on a parameter. Such problems appear in different ways, and the parameter may be constitutive, a geometric inhomogeneity scale, or a coefficient of a perturbative term. It may have different effects favoring oscillations, concentration, topological singularities, dimension-reduction, etc., some times a combination of these.

Topics in continuum mechanics

This is a 60-hours introductory course on continuum mechanics and its applications. The aim is to provide first year students with a solid understanding of the fundamental principles of the subject.

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