  Optimized Schwarz Waveform Relaxation methods have been developed over the
last decade for the parallel solution of evolution problems. They are based on
a decomposition in space and an iteration, where only subproblems in space-time
need to be solved. Each subproblem can be simulated using an adapted numerical
method, for example with local time stepping, or one can even use a different
model in different subdomains, which makes these methods very suitable also
from a modeling point of view. For rapid convergence however, it is important
to use effective transmission conditions between the space-time subdomains, and
for best performance, these transmission conditions need to take the physics of
the underlying evolution problem into account. The optimization of these
transmission conditions leads to a mathematically hard best approximation
problem of homographic type. We study in this paper in detail this problem for
the case of linear advection reaction diffusion equations in two spatial
dimensions. We prove comprehensively best approximation results for
transmission conditions of Robin and Ventcel type. We give for each case closed
form asymptotic values for the parameters, which guarantee asymptotically best
performance of the iterative methods. We finally show extensive numerical
experiments, and we measure performance corresponding to our analysis
