  We construct four variants of space-time finite element discretizations based
on linear tensor-product and simplex-type finite elements. The resulting
discretizations are continuous in space, and continuous or discontinuous in
time. In a first test run, all four methods are applied to a linear scalar
advection-diffusion model problem. Then, the convergence properties of the
time-discontinuous space-time finite element discretizations are studied in
numerical experiments. Advection velocity and diffusion coefficient are varied,
such that the parabolic case of pure diffusion (heat equation), as well as, the
hyperbolic case of pure advection (transport equation) are included in the
study. For each model parameter set, the L2 error at the final time is computed
for spatial and temporal element lengths ranging over several orders of
magnitude to allow for an individual evaluation of the methods' spatial,
temporal, and spacetime accuracy. In the parabolic case, particular attention
is paid to the influence of time-dependent boundary conditions. Key findings
include a spatial accuracy of second order and a temporal accuracy between
second and third order. The temporal accuracy tends towards third order
depending on how advection-dominated the test case is, on the choice of the
specific discretization method, and on the time-(in)dependence and treatment of
the boundary conditions. Additionally, the potential of time-continuous simplex
space-time finite elements for heat flux computations is demonstrated with a
piston ring pack test case.
