A Second Order in Time Modified Lagrange--Galerkin Finite Element Method for the Incompressible Navier--Stokes Equations

Bermejo, Rodolfo; Sastre, Pedro Galán del; Saavedra, Laura

doi:10.1137/11085548x

Cited by 51 publications

(32 citation statements)

References 18 publications

(19 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Süli [27] develops a methodology based on mathematical induction on the index n to calculate the error estimates for LG-BDF1 methods, such an approach can be extended to the calculation of the error estimates for LG-BDF2 methods as well, see [5]. Recalling that m and m 1 are the degree of the polynomials of X h and M h respectively, one proves the error estimates by making the following assumptions:…”

Section: Error Bounds For Lg-bdf1 and Lg-bdf2 Methodsmentioning

confidence: 99%

“…The location of points inside the elements of a mesh is a trivial task in structured meshes, for instance, in meshes composed of squares or hexahedra, but if the mesh is unstructured the location of points is not that simple; hence, LG methods may become less efficient than they look at first. To partially overcome these drawbacks, some variations of conventional LG method, such as the area-weighting method for quadrilateral structured meshes [20], exact integration [22] for straight side triangular meshes with linear elements, and the modified LG methods [5,6], have been proposed. We do not consider such variations in this paper.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Lagrange–Galerkin methods for the incompressible Navier-Stokes equations: a review

Bermejo

Saavedra

2016

Communications in Applied and Industrial Mathematics

View full text Add to dashboard Cite

We review in this paper the development of Lagrange-Galerkin (LG) methods to integrate the incompressible Navier-Stokes equations (NSEs) for engineering applications. These methods were introduced in the computational fluid dynamics community in the early eighties of the past century, and at that time they were considered good methods for both their theoretical stability properties and the way of dealing with the nonlinear terms of the equations; however, the numerical experience gained with the application of LG methods to different problems has identified drawbacks of them, such as the calculation of specific integrals that arise in their formulation and the calculation of the flow trajectories, which somehow have hampered the applicability of LG methods. In this paper, we focus on these issues and summarize the convergence results of LG methods; furthermore, we shall briefly introduce a new stabilized LG method suitable for high Reynolds numbers.

show abstract

Section: Error Bounds For Lg-bdf1 and Lg-bdf2 Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Lagrange–Galerkin methods for the incompressible Navier-Stokes equations: a review

Bermejo

Saavedra

2016

Communications in Applied and Industrial Mathematics

View full text Add to dashboard Cite

show abstract

“…Lemma The Galerkin projection ( R h , Q h ) satisfies

∥ R_{h} ∥_{L^{\infty}} ⩽ C (∥ boldu ∥_{2} + ∥ p ∥_{1}),

∥ \nabla R_{h} ∥_{L^{\infty}} ⩽ C (∥ boldu ∥_{W^{1, \infty}} + ∥ p ∥_{L^{\infty}}) .

…”

Section: Numerical Analysismentioning

confidence: 99%

A semi‐discrete defect correction finite element method for unsteady incompressible magnetohydrodynamics equations

Liu

2017

Math Methods in App Sciences

View full text Add to dashboard Cite

In this report, we give a semi‐discrete defect correction finite element method for the unsteady incompressible magnetohydrodynamics equations. The defect correction method is an iterative improvement technique for increasing the accuracy of a numerical solution without applying a grid refinement. Firstly, the nonlinear magnetohydrodynamics equations is solved with an artificial viscosity term. Then, the numerical solutions are improved on the same grid by a linearized defect‐correction technique. Then, we give the numerical analysis including stability analysis and error analysis. The numerical analysis proves that our method is stable and has an optimal convergence rate. In order to show the effect of our method, some numerical results are shown. Copyright © 2017 John Wiley & Sons, Ltd.

show abstract

“…There have been a large number of works concentrated on the numerical solutions of Navier-Stokes equations. We refer the readers to monographs [1,2] for the theoretical and numerical analysis, [3][4][5][6] for finite difference methods, [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] for FEMs, [25][26][27] for characteristics FEMs, [28,29] for discontinuous Galerkin method. More precisely, ax fast finite difference method was proposed in [4] based on the vorticity streamfunction formulation.…”

Section: Introductionmentioning

confidence: 99%

Superconvergence Analysis of Low Order Nonconforming Mixed Finite Element Methods for Time-Dependent Navier-Stokes Equations

Yang¹

2021

JCM

View full text Add to dashboard Cite

In this paper, the superconvergence properties of the time-dependent Navier-Stokes equations are investigated by a low order nonconforming mixed finite element method (MFEM). In terms of the integral identity technique, the superclose error estimates for both the velocity in broken H 1-norm and the pressure in L 2-norm are first obtained, which play a key role to bound the numerical solution in L ∞-norm. Then the corresponding global superconvergence results are derived through a suitable interpolation postprocessing approach. Finally, some numerical results are provided to demonstrated the theoretical analysis.

show abstract

A Second Order in Time Modified Lagrange--Galerkin Finite Element Method for the Incompressible Navier--Stokes Equations

Cited by 51 publications

References 18 publications

Lagrange–Galerkin methods for the incompressible Navier-Stokes equations: a review

Lagrange–Galerkin methods for the incompressible Navier-Stokes equations: a review

A semi‐discrete defect correction finite element method for unsteady incompressible magnetohydrodynamics equations

Superconvergence Analysis of Low Order Nonconforming Mixed Finite Element Methods for Time-Dependent Navier-Stokes Equations

Contact Info

Product

Resources

About