The course 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) using reduced order methods.

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 within the new educational library RBniCS (open Source). Lecture notes, slides and reading material is provided during the classes.

Lectures will cover the material in the book : J. Hesthaven, G. Rozza, B. Stamm 'Certified reduced basis methods and a posteriori error bounds for parametrized PDEs', Springer 2015.

Link to educational software: http://mathlab.sissa.it/rbnics

Topics/Syllabus

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

Tutorials