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 c S depends linearly on the finite element mesh size h. While the ratio c S / h decreases as 1 / T for T → ∞ , numerical results indicate a decay of c S / h ≃ ν − α for some α ∈ [ 1 4 , 1 3 ] in the polynomial degree ν 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. [ABSTRACT FROM AUTHOR]