Reduced Basis Method for 3D Problems governed by Parametrized PDEs and Applications

In this thesis we will deal with the creation of a Reduced Basis (RB) approximation of parametrized Partial Differential Equations (PDE) for three-dimensional problems. The the idea behind RB is to decouple the generation and projection stages (Offline/Online computational proce- dures) of the approximation process in order to solve parametrized (PDE) in a fast, cheap and reliable way. The RB method, especially applied to 3D problems, allows great computational savings with respect to the clas- sical Galerkin Finite Element (FE) Method. The standard FE method is typically ill suited to (i) iterative contexts like in optimization, sensitivity analysis and many queries in general and (ii) real time evaluation. We consider both coercive and noncoercive PDEs. For each class we discuss the steps to set up a RB approximation, either from an analytical and a numerical point of view. Then we present the applications of the RB method to three different problems of engineering interest and applica- bility: (i) a steady thermal conductivity problem in heat transfer; (ii) a linear elasticity problem; (iii) Stokes flows with emphasis on geometrical and physical parameters.