The Continuous Galerkin Method Is Locally Conservative

Hughes, Thomas J.R.; Engel, Gerald; Mazzei, Luca; Larson, Mats G.

doi:10.1006/jcph.2000.6577

Cited by 221 publications

(153 citation statements)

References 9 publications

(11 reference statements)

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“…The integrals on P can, therefore, be expressed as a sum of integrals on the sub-volumes P ′ and P . This leads to the following decomposition of convective and diffusive integrals: We will now focus our attention on the behavior of the integrals in (38) and (39) as x n → 0. First it is easy to see that P ′ tends to P as x n tends to zero…”

Section: Relationship Between the Find Derivatives And The Auxiliary mentioning

confidence: 99%

“…is the normal derivative of the velocity computed at node P. Alternatively, an FE problem for a boundary node with a Dirichlet condition can also be expressed using of the auxiliary flux as proposed by Hughes et al [38] in the form:…”

Section: Relationship Between the Find Derivatives And The Auxiliary mentioning

confidence: 99%

“…A finite difference between the solution at the additional point and the solution at the boundary node yields the normal derivative at the boundary. The method is shown to provide nodal gradients that are consistent with the auxiliary fluxes [38] on Dirichlet boundary nodes. However, unlike the mixed methods based on the auxiliary flux, which provide the boundary flux or force, the FiND method computes the derivatives of each component of the velocity.…”

mentioning

confidence: 99%

“…An extension to internal nodes is proposed, which can be applied for one-dimensional problems, but which could not be extended to unstructured multidimensional meshes. Evaluation of boundary fluxes was also investigated by Hughes et al [38] in the context of global conservation. They introduce a mixed formulation with an auxiliary field, which amounts to the flux on the Dirichlet portion of the boundary.…”

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confidence: 99%

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Nodal velocity derivatives of finite element solutions: The FiND method for incompressible Navier–Stokes equations

Ilinca

Pelletier

2008

Numerical Meth Engineering

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SUMMARYThis paper presents the development and application of the finite node displacement (FiND) method to the incompressible Navier-Stokes equations. The method computes high-accuracy nodal derivatives of the finite element solutions. The approach imposes a small displacement to individual mesh nodes and solves a very small problem on the patch of elements surrounding the node. The only unknown is the value of the solution (u, p) at the displaced node. A finite difference between the original and the perturbed values provides the directional derivative. Verification by grid refinement studies is shown for two-dimensional problems possessing a closed-form solution: a Poiseuille flow and a flow mimicking a boundary layer. For internal nodes, the method yields accuracy slightly superior to that of the superconvergent patch recovery (SPR) technique of Zienkiewicz and Zhu (ZZ). We also present a variant of the method to treat boundary nodes. The local discretization is enriched by inserting an additional mesh point very close to the boundary node of interest. Computations show that the resulting nodal derivatives are much more accurate than those obtained by the ZZ SPR technique.

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Section: Relationship Between the Find Derivatives And The Auxiliary mentioning

confidence: 99%

Section: Relationship Between the Find Derivatives And The Auxiliary mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Nodal velocity derivatives of finite element solutions: The FiND method for incompressible Navier–Stokes equations

Ilinca

Pelletier

2008

Numerical Meth Engineering

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“…The mass error used to evaluate local conservation in these applications to date has typically been computed element-wise using a volumetric approach by which the discrete form of the primitive conservation law is formulated using physical arguments, see [8] for example. While conceptually satisfying, this approach is inconsistent with the finite element basis and equations used to solve the GWCE itself as brought to light by Berger and Howington [3] and Hughes et al [6]. In their work, Berger and Howington [31 show how to define partial fluxes in such a way as to be consistent with the local CG conservation statement.…”

Section: Introductionmentioning

confidence: 99%

In search of a consistent and conservative mass flux for the GWCE

Massey

Blain

2006

Computer Methods in Applied Mechanics and Engineering

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Multiscale and Stabilized Methods

Hughes¹,

Scovazzi²,

Franca³

2004

Encyclopedia of Computational Mechanics

125

179

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This article presents an introduction to multiscale and stabilized methods, which represent unified approaches to modeling and numerical solution of fluid dynamic phenomena. Finite element applications are emphasized but the ideas are general and apply to other numerical methods as well. (They have been used in the development of finite difference, finite volume, and spectral methods, in addition to finite element methods.) The analytical ideas are first illustrated for time‐harmonic wave‐propagation problems in unbounded fluid domains governed by the Helmholtz equation. This leads to the well‐known Dirichlet‐to‐Neumann formulation. A general treatment of the variational multiscale method in the context of an abstract Dirichlet problem is then presented, which is applicable to advective–diffusive processes and other processes of physical interest. It is shown how the exact theory represents a paradigm for subgrid‐scale models and a posteriori error estimation. Hierarchical p ‐methods and bubble function methods are examined in order to understand and ultimately approximate the ‘fine‐scale Green's function’ that appears in the theory. Relationships among so‐called residual‐free bubbles, element Green's functions, and stabilized methods are exhibited. These ideas are then generalized to a class of nonsymmetric, linear evolution operators formulated in space–time. The variational multiscale method also provides guidelines and inspiration for the development of stabilized methods (e.g. SUPG, GLS, etc.) that have attracted considerable interest and have been extensively utilized in engineering and the physical sciences. An overview of stabilized methods for advective–diffusive equations is presented. A variational multiscale treatment of incompressible viscous flows, including turbulence is also described. This represents an alternative formulation of Large Eddy Simulation, which provides a simplified theoretical framework of LES with potential for improved modeling.

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The Continuous Galerkin Method Is Locally Conservative

Cited by 221 publications

References 9 publications

Nodal velocity derivatives of finite element solutions: The FiND method for incompressible Navier–Stokes equations

Nodal velocity derivatives of finite element solutions: The FiND method for incompressible Navier–Stokes equations

In search of a consistent and conservative mass flux for the GWCE

Multiscale and Stabilized Methods

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