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 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.
Highlights
We refer to the monographs [19,21,24] for an in-depth treatment of this approach. Another possibility is to apply the domain decomposition method to the full space-time domain Ω × (0, T ), which leads to an iterative procedure over parabolic problems that can be parallelized both in space and time; see, e.g., [12,13,15]
On these subdomains we introduce the partition of unity {χ }s=1 and the operator decomposition, or splitting, Domain decomposition time integrators
In contrast to the earlier domain decomposition based schemes, where an iterative procedure is required with possibly many instances of boundary communications, one time step of either splitting scheme only needs the solution of s elliptic equations together with the communication of the data related to the overlaps
Summary
We refer to the monographs [19,21,24] for an in-depth treatment of this approach Another possibility is to apply the domain decomposition method to the full space-time domain Ω × (0, T ), which leads to an iterative procedure over parabolic problems that can be parallelized both in space and time; see, e.g., [12,13,15]. In contrast to the earlier domain decomposition based schemes, where an iterative procedure is required with possibly many instances of boundary communications, one time step of either splitting scheme only needs the solution of s elliptic equations together with the communication of the data related to the overlaps. The main idea of the convergence analysis is to introduce the nonlinear Friedrich extensions of the operators f and f , via our new abstract energetic framework, and to employ a Lax-type result from the nonlinear semigroup theory [5]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
