A full space-time convergence order analysis of operator splittings for linear dissipative evolution equations
Provides rigorous convergence guarantees for operator splitting methods in numerical PDEs, but the results are incremental as they extend known theory to a specific class of problems.
The paper proves optimal simultaneous space-time convergence orders for Douglas-Rachford and Peaceman-Rachford splittings combined with spatial discretizations for linear dissipative evolution equations, validated with numerical experiments on a 2D diffusion problem.
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.