Newton-Schwarz Optimised Waveform Relaxation Krylov Accelerators for Nonlinear Reactive Transport

Abstract

Krylov-type methods are widely used in order to accelerate the convergence of Schwarz-type methods in the linear case. Authors in [2] have shown that they accelerate without overhead cost the convergence speed of Schwarz methods for different types of transmission conditions. In the nonlinear context, the well-known class of Newton-Krylov-Schwarz methods (cf. [5]) for steady-state problems or timedependent problems uses the following strategy: time-dependent problems are discretised uniformly in time first and then one proceeds as for steady-state problems, i.e. the nonlinear problem is solved by a Newton method where the linear system at each iteration is solved by a Krylov-type method preconditioned by an algebraic Schwarz method. The major limitation is that NKS methods do not allow different time discretisations in the subdomains since the problem is discretised in time uniformly up from the beginning.