A numerical study of the additive Schwarz preconditioned exact Newton method (ASPEN) as a nonlinear preconditioner for immiscible and compositional porous media flow

Domain decomposition methods are widely used as preconditioners for Krylov subspace linear solvers. In the simulation of porous media flow there has recently been a growing interest in nonlinear preconditioning methods for Newton’s method. In this work, we perform a numerical study of a spatial additive Schwarz preconditioned exact Newton (ASPEN) method as a nonlinear preconditioner for Newton’s method applied to both fully implicit or sequential implicit schemes for simulating immiscible and compositional multiphase flow. We first review the ASPEN method and discuss how the resulting linearized global equations can be recast so that one can use standard preconditioners developed for the underlying model equations. We observe that the local fully implicit or sequential implicit updates efficiently handle the local nonlinearities, whereas long-range interactions are resolved by the global ASPEN update. The combination of the two updates leads to a very competitive algorithm. We illustrate the behavior of the algorithm for conceptual one and two-dimensional cases, as well as realistic three dimensional models. A complexity analysis demonstrates that Newton’s method with a fully implicit scheme preconditioned by ASPEN is a very robust and scalable alternative to the well-established Newton’s method for fully implicit schemes.

An adaptively enriched coarse space for Schwarz preconditioners for P1 discontinuous Galerkin multiscale finite element problems

In this paper, we propose a two-level additive Schwarz domain decomposition preconditioner for the symmetric interior penalty Galerkin method for a second-order elliptic boundary value problem with highly heterogeneous coefficients. A specific feature of this preconditioner is that it is based on adaptively enriching its coarse space with functions created by solving generalized eigenvalue problems on thin patches covering the subdomain interfaces. It is shown that the condition number of the underlined preconditioned system is independent of the contrast if an adequate number of functions are used to enrich the coarse space. Numerical results are provided to confirm this claim.