A Two-Level Method with Backtracking for the Navier--Stokes Equations

Layton, William; Tobiska, Lutz

doi:10.1137/s003614299630230x

Cited by 193 publications

(95 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…In [24], [23] several two-level methods are considered to approximate the steady Navier-Stokes equations. They require solving a nonlinear system over a coarse mesh and, depending on the algorithm chosen, one Stokes problem, one linear Oseen problem or one Newton step over the fine mesh.…”

Section: Introductionmentioning

confidence: 99%

Two-Grid Mixed Finite-Element Approximations to the Navier–Stokes Equations Based on a Newton-Type Step

Durango

Novo

2017

J Sci Comput

View full text Add to dashboard Cite

Esta es la versión de autor del artículo publicado en: This is an author produced version of a paper published in: El acceso a la versión del editor puede requerir la suscripción del recurso Access to the published version may require subscription Two-grid mixed finite-element approximations to the Navier-Stokes equations based on a Newton-type stepFrancisco Durango Julia Novo * February 21, 2017Abstract A two-grid scheme to approximate the evolutionary Navier-Stokes equations is introduced and analyzed. A standard mixed finite element approximation is first obtained over a coarse mesh of size H at any positive time T > 0. Then, the approximation is postprocessed by means of solving a steady problem based on one step of a Newton iteration over a finer mesh of size h < H. The method increases the rate of convergence of the standard Galerkin method in one unit in terms of H and equals the rate of convergence of the standard Galerkin method over the fine mesh h. However, the computational cost is essentially the cost of approaching the Navier-Stokes equations with the plain Galerkin method over the coarse mesh of size H since the cost of solving one single steady problem is negligible compared with the cost of computing the Galerkin approximation over the full time interval (0, T ]. For the analysis we take into account the loss of regularity at initial time of the solution of the Navier-Stokes equations in the absence of nonlocal compatibility conditions. Some numerical experiments are shown.

show abstract

Section: Introductionmentioning

confidence: 99%

Two-Grid Mixed Finite-Element Approximations to the Navier–Stokes Equations Based on a Newton-Type Step

Durango

Novo

2017

J Sci Comput

View full text Add to dashboard Cite

show abstract

“…There are also many other authors who have used this method for many different applications. See, for example, Axelsson et al [2,3,4], Girault and Lions [7], Layton et al [9,10,11], and Utnes [15].…”

Section: Introductionmentioning

confidence: 99%

A two-grid discretization method for decoupling systems of partial differential equations

Jin¹,

Shu²,

Xu³

2006

Math. Comp.

View full text Add to dashboard Cite

Abstract. In this paper, we propose a two-grid finite element method for solving coupled partial differential equations, e.g., the Schrödinger-type equation. With this method, the solution of the coupled equations on a fine grid is reduced to the solution of coupled equations on a much coarser grid together with the solution of decoupled equations on the fine grid. It is shown, both theoretically and numerically, that the resulting solution still achieves asymptotically optimal accuracy.

show abstract

“…The following results have been well established: a(·, ·) is bounded and coercive; b(·, ·) is bounded and satisfies the inf-sup condition [16,18,36]; and the nonlinear term satisfies the following estimates [22,29,30,33]. LEMMA 2.1.…”

mentioning

confidence: 95%

Some multilevel decoupled algorithms for a mixed navier-stokes/darcy model

Cai

Huang

2017

Adv Comput Math

View full text Add to dashboard Cite

Abstract. In this work, several multilevel decoupled algorithms are proposed for a mixed Navier-Stokes/Darcy model. These algorithms are based on either successively or parallelly solving two linear subdomain problems after solving a coupled nonlinear coarse grid problem. Error estimates are given to demonstrate the approximation accuracy of the algorithms. Experiments based on both the first order and the second order discretizations are presented to show the effectiveness of the decoupled algorithms.Key words. Fluid flow coupled with porous media flow, Darcy law, Navier-Stokes equations, Interface coupling, Multilevel algorithm, Decoupling, Linearization AMS subject classifications. 65F08, 65F10, 65N30, 65N551. Introduction. The coupling of incompressible fluid flow with porous media flow is an interesting but challenging topic. For describing the interactions of the fluid flow with the porous media flow, a coupled Stokes/Darcy or Naiver-Stokes/Darcy system is typically used as a macro-scale sharp interface model [2,3,6,8,9,12,13,14,16,17,18,19,20,21,23,24,25,34,35,37,40,41,42]. The coupled Navier-Stokes/Darcy model is composed of a nonlinear Navier-Stokes equations for fluid flow, a Darcy law equation for porous media flow, plus certain interface conditions for describing the interactions of the different types of flows. Numerical methods for this model [8,13,23,42] usually result in a coupled and nonlinear saddle point problem, for which numerical difficulties increase as the mesh size decreases.Let us consider a domain Ω ⊂ R d (d = 2 or 3), consisting of a fluid region Ω f and a porous media region Ω p separated by an interface Γ. As shown in Fig. 1.1, Ω = Ω f Ω p and Γ = Ω f Ω p . The interface Γ is assumed to be smooth enough [23].The fluid flow in Ω f is governed by the steady state Navier-Stokes equations:

show abstract

A Two-Level Method with Backtracking for the Navier--Stokes Equations

Cited by 193 publications

References 30 publications

Two-Grid Mixed Finite-Element Approximations to the Navier–Stokes Equations Based on a Newton-Type Step

Two-Grid Mixed Finite-Element Approximations to the Navier–Stokes Equations Based on a Newton-Type Step

A two-grid discretization method for decoupling systems of partial differential equations

Some multilevel decoupled algorithms for a mixed navier-stokes/darcy model

Contact Info

Product

Resources

About