Stability of Petrov-Galerkin Discretizations: Application to the Space-Time Weak Formulation for Parabolic Evolution Problems

This paper is concerned with the stability of Petrov-Galerkin discretizations with application to parabolic evolution problems in space-time weak form. We will prove that the discrete inf-sup condition for an a priori fixed Petrov-Galerkin discretization is satisfied uniformly under standard approximation and smoothness conditions without any further coupling between the discrete trial and test spaces for sufficiently regular operators. It turns out that one needs to choose different discretization levels for the trial and test spaces in order to obtain a positive lower bound for the discrete inf-sup condition which is independent of the discretization levels. In particular, we state the required number of extra layers in order to guarantee uniform boundedness of the discrete inf-sup constants explicitly. This general result will be applied to the space-time weak formulation of parabolic evolution problems as an important model example. In this regard, we consider suitable hierarchical families of discrete spaces. The results apply, e.g., for finite element discretizations as well as for wavelet discretizations. Due to the Riesz basis property, wavelet discretizations allow for optimal preconditioning independently of the grid spacing. Moreover, our predictions on the stability, especially in view of the dependence on the refinement levels w.r.t. the test and trial spaces, are underlined by numerical results. Furthermore, it can be observed that choosing the same discretization levels would, indeed, lead to stability problems.