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.
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 approzimations 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.
Lecture notes, slides and reading material will be provided (please send an email in advance to gianluigi.rozza@sissa.it to register).
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
- Parameters space exploration, sampling, Greedy algorithm, POD
- Residual based a posteriori error bounds and stability factors
- 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 and flow control
- Tutorials