GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
Domain decomposition: parallel multilevel methods for elliptic partial differential equations
Domain decomposition: parallel multilevel methods for elliptic partial differential equations
Nonlinearly Preconditioned Inexact Newton Algorithms
SIAM Journal on Scientific Computing
On Convergence of the Additive Schwarz Preconditioned Inexact Newton Method
SIAM Journal on Numerical Analysis
Journal of Computational Physics
Potential-based reduced Newton algorithm for nonlinear multiphase flow in porous media
Journal of Computational Physics
Journal of Computational Physics
Hi-index | 31.45 |
Domain decomposition (DD) methods, such as the additive Schwarz method, are almost exclusively applied to linearized equations. In the context of nonlinear problems, these linear systems appear as part of a Newton iteration. However, applying DD methods directly to the original nonlinear problem has some attractive features, most notably that the Newton iterations now solve local problems, and thus are expected to be simpler. Furthermore, strong, local nonlinearities may to a less extent affect the numerical algorithm. For linear problems, DD can be applied both as an iterative solver or as a preconditioner. For nonlinear problems, it has until recently only been understood how to use DD as a solver. This article offers a systematic study of domain decomposition strategies in the context of nonlinear porous-medium flow problems. The study thus compares four different approaches, which represents DD applied both as a solver and preconditioner, to both the linearized and nonlinear equations. Our model equations are those obtained from a fully implicit discretization of immiscible two-phase flow in heterogeneous porous media. In particular we emphasize the case of nonlinear preconditioning, an algorithm that to our knowledge so far has not been studied nor implemented for flow in porous media. Our results show that the novel algorithm is up to 75% faster than the standard algorithm for the most challenging problems for a moderate number of subdomains.