Space-Time Reduced Basis Methods for Time-Periodic Parametric Differential Equations

We consider space-time methods for time-periodic problems and discuss well-posedness of the variational formulation as well as the use of Reduced Basis Methods (RBM) in a parameterized setting. We propose a possible discretization enforcing periodic boundary conditions in time and provide lower bounds for the space-time inf-sup constant. Rigorous RBM space-time a-posteriori error bounds for state and output are derived and it is shown that this approach yields results similar to those in stationary elliptic settings. This is contrasted with the more common time-stepping approach which requires fixed-point methods with often long transient phases to obtain periodicity. We discuss RBM in this context and derive the corresponding a-posteriori bounds. A convection-diffusion-reaction example is numerically investigated with regard to the inf-sup constant as well as the performance of both space-time and fixed-point RBM. We show the reliable representation of the stability by the space-time inf-sup constant and observe the advantage of space-time approaches in the online phase of the RBM.