Convergence analysis of domain decomposition based time integrators for degenerate parabolic equations
For researchers in numerical analysis and PDEs, this work offers a theoretical foundation for parallel time integration of degenerate parabolic problems, which is an incremental advance over existing operator splitting methods.
This paper provides a rigorous convergence analysis for domain decomposition based time integrators applied to degenerate parabolic equations, such as the p-Laplace and porous medium equations, without assuming restrictive regularity on solutions or domains.
Domain decomposition based time integrators allow the usage of parallel and distributed hardware, making them well-suited for the temporal discretization of parabolic systems, in general, and degenerate parabolic problems, in particular. The latter is due to the degenerate equations' finite speed of propagation. In this study, a rigours convergence analysis is given for such integrators without assuming any restrictive regularity on the solutions or the domains. The analysis is conducted by first deriving a new variational framework for the domain decomposition, which is applicable to the two standard degenerate examples. That is, the $p$-Laplace and the porous medium type vector fields. Secondly, the decomposed vector fields are restricted to the underlying pivot space and the time integration of the parabolic problem can then be interpreted as an operators splitting applied to a dissipative evolution equation. The convergence results then follow by employing elements of the approximation theory for nonlinear semigroups.