Parareal for diffusion problems with space- and time-dependent   coefficients
  For the time-parallel Parareal method, there exists both numerical and analytical proof that it converges very well for diffusive problems like the heat equation. Many applications, however, do not lead to simple homogeneous diffusive scenarios but feature strongly inhomogeneous and possibly anisotropic coefficients. The paper presents results from a numerical study of how space- and time-dependent coefficients in a diffusion setup affect Parareal's convergence behaviour. It is shown that, for the presented examples, non-constant diffusion coefficients have only marginal influence on how fast Parareal converges. Furthermore, an example is shown that illustrates how for linear problems the maximum singular value of the Parareal iteration matrix can be used to estimate convergence rates. 
Daniel Ruprecht and Robert Speck and Rolf Krause
math.NA