Globalization strategies for Newton–Krylov methods for stabilized FEM discretization of Navier–Stokes equations

Bellavia, Stefania; Berrone, Stefano

doi:10.1016/j.jcp.2007.07.021

Cited by 11 publications

(4 citation statements)

References 28 publications

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“…This approach allows the use of implicit schemes for nonlinear equations at a reasonable cost by linearizing the discretization of the nonlinear convective term. Hence, to get W * := W n + δW * we employ an inexact Newton algorithm combined with a Krylov subspace method for the solution of linear systems, see also [8][9][10] and references therein. The objective is thus to find the root of the vector function…”

Section: Implicit Transport-diffusion Stagementioning

confidence: 99%

An Implicit Staggered Hybrid Finite Volume/Finite Element Solver for the Incompressible Navier-Stokes Equations

Lucca,

null,

Dumbser

2023

EAJAM

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We present a novel fully implicit hybrid finite volume/finite element method for incompressible flows. Following previous works on semi-implicit hybrid FV/FE schemes, the incompressible Navier-Stokes equations are split into a pressure and a transportdiffusion subsystem. The first of them can be seen as a Poisson type problem and is thus solved efficiently using classical continuous Lagrange finite elements. On the other hand, finite volume methods are employed to solve the convective subsystem, in combination with Crouzeix-Raviart finite elements for the discretization of the viscous stress tensor. For some applications, the related CFL condition, even if depending only in the bulk velocity, may yield a severe time restriction in case explicit schemes are used. To overcome this issue an implicit approach is proposed. The system obtained from the implicit discretization of the transport-diffusion operator is solved using an inexact Newton-Krylov method, based either on the BiCStab or the GMRES algorithm. To improve the convergence properties of the linear solver a symmetric Gauss-Seidel (SGS) preconditioner is employed, together with a simple but efficient approach for the reordering of the grid elements that is compatible with MPI parallelization. Besides, considering the Ducros flux for the nonlinear convective terms we can prove that the discrete advection scheme is kinetic energy stable. The methodology is carefully assessed through a set of classical benchmarks for fluid mechanics. A last test shows the potential applicability of the method in the context of blood flow simulation in realistic vessel geometries.

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Section: Implicit Transport-diffusion Stagementioning

confidence: 99%

An Implicit Staggered Hybrid Finite Volume/Finite Element Solver for the Incompressible Navier-Stokes Equations

Lucca,

null,

Dumbser

2023

EAJAM

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“…By contrast, a damped Newton procedure using a line search that satisfies the Armijo-Wolfe criteria is guaranteed to converge on nondegenerate, if unrealistic, input data. The line search method that we used here is one way to achieve global convergence of Newton's method, but there are other approaches to achieve global convergence -for example, homotopy continuation (Tezaur et al, 2015) or the trust region method (Bellavia and Berrone, 2007).…”

Section: Convex Optimizationmentioning

confidence: 99%

icepack: a new glacier flow modeling package in Python, version 1.0

et al. 2021

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Abstract. We introduce a new software package called “icepack” for modeling the flow of glaciers and ice sheets. The icepack package is built on the finite element modeling library Firedrake, which uses the Unified Form Language (UFL), a domain-specific language embedded into Python for describing weak forms of partial differential equations. The diagnostic models in icepack are formulated through action principles that are specified in UFL. The components of each action functional can be substituted for different forms of the user's choosing, which makes it easy to experiment with the model physics. The action functional itself can be used to define a solver convergence criterion that is independent of the mesh and requires little tuning on the part of the user. The icepack package includes the 2D shallow ice and shallow stream models. We have also defined a 3D hybrid model based on spectral semi-discretization of the Blatter–Pattyn equations. Finally, icepack includes a Gauss–Newton solver for inverse problems that runs substantially faster than the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method often used in the glaciological literature. The overall design philosophy of icepack is to be as usable as possible for a wide a swath of the glaciological community, including both experts and novices in computational science.

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“…The nonlinear system (11)-(12) can be solved rather easily with an inexact Newton-Krylov method [3,4], based on the GMRES algorithm of Saad and Schultz [33]. To improve convergence of the GMRES method, a simple block-diagonal scaling is used as preconditioner throughout this paper.…”

Section: Space-time Ldg Schemementioning

confidence: 99%