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Advanced Topics in Numerical Solutions of PDEs

Luca Heltai, Gianluigi Rozza
Course Type: 
PhD Course
Academic Year: 
20 h
  • Isogeometric Analysis Techniques (LH)
  • Boundary Element Methods (LH)
  • Numerical Optimal Control of PDEs (GR)
  • Reduced Basis Methods in Computational Mechanics (GR)
  • Shape Optimization (optional)


Material will be provided during classes, a calendar with topics and organization will be given during the first lecture.

The part given by Prof. Gianluigi Rozza will be held as intensive module on April 28-29-30 at SISSA campus Miramare, room D and it will focus on Reduced basis methods for computational mechanics. Details of this part are the following:



Lecture 9:30am-11:00am, 11:30am-1:00pm, 2:30pm-4:00pm

Exercise 4:30pm-6:00pm


Lecture 9:30am-11:00am, 11:30am-1:00pm

Exercise 2:30pm-4:00pm, 4:30pm-6:00pm


Lecture 9:30am-11:00am

Seminar MHPC/mathLab 11:30am-1:00pm (Prof. K. Urban, Ulm.)


Learning outcomes/Objectives:

The module aims to provide the basic aspects of numerical approximation and efficient solution of parametrized PDEs for computational mechanics problem (heat and mass transfer, linear elasticity, viscous and potential flows).


Module Description :

In this course we present reduced basis (RB) approximation and associated a posteriori error estimation for rapid and reliable solution of parametrized partial differential equations (PDEs). The focus is on rapidly convergent Galerkin approximations on a subspace spanned by ``snapshots''; rigorous and sharp a posteriori error estimators for the outputs/quantities of interest; efficient selection of quasi-optimal samples in general parameter domains; and Offline-Online computational procedures for rapid calculation in the many-query and real-time contexts. We develop the RB methodology for a wide range of (coercive and non-coercive) elliptic and parabolic PDEs with several examples drawn from heat transfer, elasticity and fracture, acoustics, and fluid dynamics. We introduce the concept of affine and non-affine parametric dependence, some elements of approximation and algebraic stability. Finally, we consider application of RB techniques to parameter estimation, optimization, optimal control, and a comparison with other reduced order techniques, like Proper Orthogonal Decomposition.

Some tutorials are prepared for the course based on FEniCS and Python. Lecture notes, slides and reading material is provided during the classes.

A seminar by Prof. Karsten Urban (Ulm University, Germany) will be given on taught topics on April 30, 2015 (morning).



-Introduction to RB methods, offline-online computing, elliptic coercive affine problems

-Sampling, greedy algorithm, POD

-A posteriori error bounds

-Primal-Dual Approximation

-Time dependent problems: POD-greedy sampling

-Non-coercive problems

-Approximation of coercivity and inf-sup parametrized constants

-Geometrical parametrization

-Reference worked problems

-Examples of Applications in CFD

SISSA campus Miramare, room D

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