  This thesis is concerned with global properties of those cosmological
solutions of Einstein's field equations which obey accelerated expansion into
the future driven by a non-vanishing cosmological constant, as suggested by
current cosmological observations. In particular, the strong cosmic censorship
and BKL conjecture are investigated in special classes of spacetimes. For this,
a new numerical technique based on spectral methods is introduced to be able to
deal with situations where the spatial topology is non-trivial from the
numerical point of view, such as the 3-sphere. Therefor, an explicit spectral
regularization at the - in single patch approaches inevitable - coordinate
singularities is worked out. After having collected the necessary background
material, including a description of the problems of interest, we discuss the
method and its implementation. Then we perform various tests, in particular
also in the presence of cosmological singularities. After that, the numerical
method is applied to obtain insights into three open issues in mathematical
cosmology: first, we make an attempt to describe the singularity structure in
Gowdy spacetimes with positive cosmological constants with spatial 3-sphere and
those with spatial 3-torus topology. Second, we perform a stability analysis of
Gowdy symmetry within the class of U(1)-symmetric spacetimes for spatial
3-sphere topology. Third, we make a preliminary analysis of the behavior of
certain $\lambda$-Taub-NUT Cauchy horizons under Gowdy perturbations. Finally,
we summarize the current status and open problems, and give an outlook.
