Convergence analysis of domain decomposition based time integrators for degenerate parabolic equations

TL;DR

This paper provides a rigorous convergence analysis of domain decomposition based time integrators for degenerate parabolic equations, applicable to problems like the p-Laplace and porous medium equations, without restrictive regularity assumptions.

Contribution

It introduces a new variational framework for domain decomposition applicable to degenerate parabolic problems and proves convergence using nonlinear semigroup approximation theory.

Findings

Convergence of the proposed integrators is established for degenerate parabolic equations.

The framework applies to p-Laplace and porous medium equations.

No restrictive regularity assumptions are needed for the analysis.

Abstract

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.