Parareal algorithm for coupled elliptic-parabolic problems

We present a convergence analysis of the parallel-in-time integration method known as the Parareal algorithm for degenerate differential-algebraic systems arising from quasi-static Biot models, which govern coupled flow and deformation in porous media. The underlying system exhibits a saddle-point structure and degeneracy due to the quasi-static assumption. We extend the Parareal algorithm to this setting and propose three coarse propagators: monolithic, fixed-stress, and multirate fixed-stress schemes. For each, we derive sufficient conditions for convergence and establish explicit time step restrictions that guarantee contractivity of the iteration matrix. Numerical experiments show computational savings accrued by using a parareal solver in multiphysics simulations involving poroelasticity and other coupled systems.