A full space-time convergence order analysis of operator splittings for linear dissipative evolution equations

TL;DR

This paper establishes optimal convergence orders in both space and time for operator splitting methods applied to linear dissipative evolution equations, supported by theoretical proofs and numerical experiments.

Contribution

It provides a rigorous analysis of full space-time convergence orders for Douglas--Rachford and Peaceman--Rachford splitting methods combined with spatial discretizations.

Findings

Optimal convergence orders proven in theory

Numerical experiments confirm theoretical results

Application to 2D diffusion with finite element method

Abstract

The Douglas--Rachford and Peaceman--Rachford splitting methods are common choices for temporal discretizations of evolution equations. In this paper we combine these methods with spatial discretizations fulfilling some easily verifiable criteria. In the setting of linear dissipative evolution equations we prove optimal convergence orders, simultaneously in time and space. We apply our abstract results to dimension splitting of a 2D diffusion problem, where a finite element method is used for spatial discretization. To conclude, the convergence results are illustrated with numerical experiments.