On inf-sup stabilized finite element methods for transient problems

We consider the behavior of inf-sup stabilization in the context of transient problems with multiple time scales. Our motivation for studying this setting is provided by reacting flows problems for which small time steps are necessary in the integration process. We show that for algorithms defined through a process wherein spatial and temporal discretizations are separated, the coupling of implicit time integration with spatial inf-sup stabilization may lead to anomalous pressure behavior, including the onset of spurious oscillations, for very small time steps. Effectively, this coupling introduces a stability criterion resulting in a dependence between the spatial grid size and the time step. We illustrate our theoretical results by numerical examples that demonstrate the stability criterion.