On a modified Hilbert transformation, the discrete inf-sup condition, and error estimates

In this paper, we analyze the discrete inf-sup condition and related error estimates for a modified Hilbert transformation as used in the space-time discretization of time-dependent partial differential equations. It turns out that the stability constant CS S depends linearly on the finite element mesh size h . While the ratio CS/h S / h decreases as 1/T T for T -> infinity , numerical results indicate a decay of CS/h S / h similar or equal to v - a for some a is an element of [ 41 , 13 ] in the polynomial degree v of the finite element basis functions. However, in most cases, we can show optimal convergence. We present a series of numerical experiments which illustrate the theoretical findings.