Stabilized reduced basis method for parametrized advection-diffusion PDEs

The aim of this master thesis is to study and develop a stabilized reduced basis method suitable for the approximation of the solution of parametrized advection-diffusion PDEs with high Péclet number, that is, roughly, the ratio between the advection coefficient and the diffusion one. Advection-diffusion equations are very important in many engineering applications, because they are used to model, for example, heat transfer phenomena or the diffusion of pollutants in the atmosphere. In such applications, we often need very fast evaluations of the approximated solution, depending on some input parameters. This happens, for example, in the case of real-time simulation. Moreover, we need rapid evaluations also if we have to perform repeated approximation of the solution, for different input parameters. An important case of this many-query situation is represented by some optimization problems, in which the objective function to optimize depends on the parameters through the solution of a PDE. The reduced basis (RB) method meets our need for rapidity and it is also able to guarantee the reliability of the solution, thanks to sharp a posteriori error bounds. We can find in literature many works about the application of the RB method to advection-diffusion problems but they mainly deal with equations in which the Péclet number is low. The need for stabilization arise from the fact that the finite element (FE) approximated solution - that the RB method aims to recover - shows strong instability problems that have to be fixed. In this work we want to go further in the study of the stabilization of the RB method for advection dominated problem in both steady and unsteady case. As regards the steady case, we first compare two possible stabilization strategies, by testing them on some test problems, in order to design an efficient stabilized reduced basis method. We will then test this method using the piecewise quadratic FE approximation as reference solution, instead of the usual piecewise linear one. We extend the method designed for the steady case to the time dependent case and we will carry out some numerical tests.