Convergence Analysis for  the Multiplicative Schwarz Preconditioned Inexact Newton Algorithm

Liu, Lulu; Keyes, David E.

doi:10.1137/15m1028182

Cited by 11 publications

(8 citation statements)

References 17 publications

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“…where β > 0 is a preselected constant. Then, the subspaces V k b and V k g corresponding to (22) and (23) are defined as…”

Section: Discussion On Different Elimination Strategiesmentioning

confidence: 99%

“…The subspaces V k b and V k g are defined the same as (22) and (23). Following the idea of restricted elimination, we define the restricted bad subset I k,ε b by excluding some mesh points near the interface, such that I k,ε…”

Section: Discussion On Different Elimination Strategiesmentioning

confidence: 99%

“…For the rest of the paper, we refer to this step as a nonlinear elimination (NE) preconditioner and it is applied as a subspace correction step within a Newton iteration. This can be viewed as a right nonlinear preconditioning method [3,7], as opposed to the left nonlinear preconditioning method [6,[22][23][24]. A high level description of the INB-NE method for solving (15) at each time step is presented in Algorithm 1.…”

Section: The Nonlinear Elimination Preconditionermentioning

confidence: 99%

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A nonlinear elimination preconditioned inexact Newton method for blood flow problems in human artery with stenosis

Luo

Shiu

Chen

et al. 2019

Journal of Computational Physics

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Blood flow problem Navier-Stokes equations with resistive boundary conditions Finite element Inexact Newton method Nonlinear preconditioning Parallel computing Simulation of blood flows in the human artery is a promising tool for understanding the hemodynamics. The blood flow is often smooth in a healthy artery, but may become locally chaotic in a diseased artery with stenosis, and as a result, a traditional solver may take many iterations to converge or does not converge at all. To overcome the problem, we develop a nonlinearly preconditioned Newton method in which the variables associated with the stenosis are iteratively eliminated and then a global Newton method is applied to the smooth part of the system. More specifically, we model the blood flow in a patientspecific artery based on the unsteady incompressible Navier-Stokes equations with resistive boundary conditions discretized by a fully implicit finite element method. The resulting nonlinear system at each time step is solved by using an inexact Newton method with a domain decomposition based Jacobian solver. To improve the convergence and robustness of the Newton method for arteries with stenosis, we develop an adaptive restricted region-based nonlinear elimination preconditioner which performs subspace correction to remove the local high nonlinearities. Numerical experiments for several cerebral arteries are presented to demonstrate the superiority of the proposed algorithm over the classical method with respect to some physical and numerical parameters. We also report the parallel scalability of the proposed algorithm on a supercomputer with thousands of processor cores.

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“…where β > 0 is a preselected constant. Then, the subspaces V k b and V k g corresponding to (22) and (23) are defined as…”

Section: Discussion On Different Elimination Strategiesmentioning

confidence: 99%

Section: Discussion On Different Elimination Strategiesmentioning

confidence: 99%

Section: The Nonlinear Elimination Preconditionermentioning

confidence: 99%

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A nonlinear elimination preconditioned inexact Newton method for blood flow problems in human artery with stenosis

Luo

Shiu

Chen

et al. 2019

Journal of Computational Physics

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“…A strategy based on the field-splitting of pressure and saturation. This strategy is a right version of left multiplicative nonlinear preconditioning, namely, the multiplicative Schwarz preconditioned inexact Newton (MSPIN) method [33,35], where the subproblems with respect to different physical variables are solved sequentially, similar to a Gauss-Seidel iteration. In the NE context, we perform the subspace correction step (Step 3 in Algorithm 3) in two stages to eliminate alternatively the pressure components and the saturation components.…”

Section: 2mentioning

confidence: 99%

“…The additive Schwarz preconditioned inexact Newton algorithm (ASPIN) [7,23,24,25] belongs to this class, where solutions of nonlinear subdomain problems are solved and then combined together using an additive Schwarz framework. A multiplicative Schwarz version of ASPIN, named MSPIN, was introduced in [33,35] for two-dimensional (2D) incompressible flows and two-phase flows in porous media [36].…”

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confidence: 99%

Nonlinear Preconditioning Strategies for Two-Phase Flows in Porous Media Discretized by a Fully Implicit Discontinuous Galerkin Method

Luo¹,

Cai²,

Keyes³

2021

SIAM J. Sci. Comput.

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We consider numerical simulation of two-phase flows in porous media using implicit methods. Because of the complex features involving heterogeneous permeability and nonlinear capillary effects the nonlinear algebraic systems arising from the discretization are very difficult to solve. The traditional Newton method suffers from slow convergence in the form of a long stagnation or sometimes does not converge at all. In the paper, we develop some nonlinear preconditioning strategies for the system of two-phase flows discretized by a fully implicit discontinuous Galerkin method. The preconditioners identify and approximately eliminate the local high nonlinearities that cause the Newton method to take small updates. Specifically, we propose two elimination strategies, one is based on exploring the unbalanced nonlinearities of the pressure and the saturation fields, and the other is based on identifying certain elements of the finite element space that have much higher nonlinearities than the rest of the elements. We compare the performance and robustness of the proposed algorithms with an existing single-field elimination approach and the classical inexact Newton method with respect to some physical and numerical parameters. Experiments on three-dimensional porous media applications show that the proposed algorithms are superior to the other methods in terms of the robustness and parallel efficiency.

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An adaptive nonlinear elimination preconditioned inexact Newton algorithm for highly local nonlinear multicomponent PDE systems

Yang

Hwang

2018

Applied Numerical Mathematics

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Convergence Analysis for the Multiplicative Schwarz Preconditioned Inexact Newton Algorithm

Cited by 11 publications

References 17 publications

A nonlinear elimination preconditioned inexact Newton method for blood flow problems in human artery with stenosis

A nonlinear elimination preconditioned inexact Newton method for blood flow problems in human artery with stenosis

Nonlinear Preconditioning Strategies for Two-Phase Flows in Porous Media Discretized by a Fully Implicit Discontinuous Galerkin Method

An adaptive nonlinear elimination preconditioned inexact Newton algorithm for highly local nonlinear multicomponent PDE systems

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