  We consider a space-time variational formulation of parabolic
initial-boundary value problems in anisotropic Sobolev spaces in combination
with a Hilbert-type transformation. This variational setting is the starting
point for the space-time Galerkin finite element discretization that leads to a
large global linear system of algebraic equations. We propose and investigate
new efficient direct solvers for this system. In particular, we use a
tensor-product approach with piecewise polynomial, globally continuous ansatz
and test functions. The developed solvers are based on the Bartels-Stewart
method and on the Fast Diagonalization method, which result in solving a
sequence of spatial subproblems. The solver based on the Fast Diagonalization
method allows to solve these spatial subproblems in parallel leading to a full
parallelization in time. We analyze the complexity of the proposed algorithms,
and give numerical examples for a two-dimensional spatial domain, where sparse
direct solvers for the spatial subproblems are used.
