Additive domain decomposition operator splittings -- convergence analyses in a dissipative framework

TL;DR

This paper provides a rigorous convergence analysis of additive domain decomposition operator splitting schemes for parabolic systems within a dissipative framework, including error bounds for both linear and semilinear equations.

Contribution

It introduces a variational framework for analyzing domain decomposition schemes and derives optimal error estimates for common splitting methods, extending results to semilinear equations.

Findings

Optimal temporal error bounds for alternating direction implicit schemes.

Error analysis for first and second order additive splitting schemes.

Extension of error estimates to semilinear evolution equations with mild solutions.

Abstract

We analyze temporal approximation schemes based on overlapping domain decompositions. As such schemes enable computations on parallel and distributed hardware, they are commonly used when integrating large-scale parabolic systems. Our analysis is conducted by first casting the domain decomposition procedure into a variational framework based on weighted Sobolev spaces. The time integration of a parabolic system can then be interpreted as an operator splitting scheme applied to an abstract evolution equation governed by a maximal dissipative vector field. By utilizing this abstract setting, we derive an optimal temporal error analysis for the two most common choices of domain decomposition based integrators. Namely, alternating direction implicit schemes and additive splitting schemes of first and second order. For the standard first-order additive splitting scheme we also extend the error analysis to semilinear evolution equations, which may only have mild solutions.