A stable finite element for the stokes equations

Arnold, Douglas N.; Brezzi, Franco; Fortin, Michel

doi:10.1007/bf02576171

Cited by 1,014 publications

(686 citation statements)

References 4 publications

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“…While these schemes were not developed as DG methods, they have now been brought into the unified DG framework [21]. More recently, many researchers [13,9,8,20,19,42] have applied DG methods to diffusive operators.…”

Section: Discontinuous Galerkin Methodsmentioning

confidence: 99%

“…Arnold et al [21] provides a unified analysis, including error estimates, of most of the DG methods available for elliptic operators.…”

Section: Discontinuous Galerkin Methodsmentioning

confidence: 99%

“…While the inviscid terms should make the discrete system non-singular, this result is clearly undesirable. Second, stability and optimal order of accuracy in L 2 cannot be proven [21]. In fact, Cockburn and Shu [20] have shown that for purely elliptic problems, odd order interpolants produce sub-optimal order of accuracy equal to O(h p ).…”

Section: Bassi and Rebay Discretizationmentioning

confidence: 99%

“…Unfortunately, as shown by numerous authors [20,21,42], this discretization is problematic for multiple reasons. First, on some meshes the viscous contribution to the Jacobian may be singular.…”

Section: Bassi and Rebay Discretizationmentioning

confidence: 99%

“…Furthermore, for purely elliptic problems with homogeneous Dirichlet boundary conditions, Arnold et al [21] proves optimal error convergence in the L 2 norm for p ≥ 1 when η f > 3 (note: the condition η f > 3 is required to prove stability). However, in Section 2.3…”

Section: Bassi and Rebay Discretizationmentioning

confidence: 99%

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Multigrid Solution for High-Order Discontinuous Galerkin Discretizations of the Compressible Navier-Stokes Equations

Oliver

Fidkowski

Darmofal

Computational Fluid Dynamics 2004

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Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. AbstractA high-order discontinuous Galerkin finite element discretization and p-multigrid solution procedure for the compressible Navier-Stokes equations are presented. The discretization has an element-compact stencil such that only elements sharing a face are coupled, regardless of the solution space. This limited coupling maximizes the effectiveness of the p-multigrid solver, which relies on an element-line Jacobi smoother. The element-line Jacobi smoother solves implicitly on lines of elements formed based on the coupling between elements in a p = 0 discretization of the scalar transport equation. Fourier analysis of 2-D scalar convection-diffusion shows that the element-line Jacobi smoother as well as the simpler element Jacobi smoother are stable independent of p and flow condition. Mesh refinement studies for simple problems with analytic solutions demonstrate that the discretization achieves optimal order of accuracy of O(h p+1 ). A subsonic, airfoil test case shows that the multigrid convergence rate is independent of p but weakly dependent on h. Finally, higher-order is shown to outperform grid refinement in terms of the time required to reach a desired accuracy level.

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Section: Discontinuous Galerkin Methodsmentioning

confidence: 99%

“…Arnold et al [21] provides a unified analysis, including error estimates, of most of the DG methods available for elliptic operators.…”

Section: Discontinuous Galerkin Methodsmentioning

confidence: 99%

Section: Bassi and Rebay Discretizationmentioning

confidence: 99%

Section: Bassi and Rebay Discretizationmentioning

confidence: 99%

Section: Bassi and Rebay Discretizationmentioning

confidence: 99%

See 3 more Smart Citations

Multigrid Solution for High-Order Discontinuous Galerkin Discretizations of the Compressible Navier-Stokes Equations

Oliver

Fidkowski

Darmofal

Computational Fluid Dynamics 2004

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A mixed-enhanced formulation tetrahedral finite elements

Taylor

2000

Int. J. Numer. Meth. Engng.

126

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This paper considers the solution of problems in three-dimensional solid mechanics using tetrahedral ÿnite elements. A formulation based on a mixed-enhanced treatment involving displacement, pressure and volume e ects is presented. The displacement and pressure are used as nodal quantities while volume e ects and enhanced modes belong to individual elements. Both small and ÿnite deformation problems are addressed and sample solutions are given to illustrate the performance of the formulation.

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Multigrid and Krylov Subspace Methods for the Discrete Stokes Equations

Elman

1996

Int. J. Numer. Meth. Fluids

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Discretization of the Stokes equations produces a symmetric inde nite system of linear equations. For stable discretizations, a variety of numerical methods have been proposed that have rates of convergence independent of the mesh size used in the discretization. In this paper, we compare the performance of four such methods: variants of the Uzawa, preconditioned conjugate gradient, preconditioned conjugate residual, and multigrid methods, for solving several two-dimensional model problems. The results indicate that where it is applicable, multigrid with smoothing based on incomplete factorizaton is more e cient than the other methods, but typically by no more than a factor of two. The conjugate residual method has the advantages of being both independent of iteration parameters and widely applicable.

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A stable finite element for the stokes equations

Cited by 1,014 publications

References 4 publications

Multigrid Solution for High-Order Discontinuous Galerkin Discretizations of the Compressible Navier-Stokes Equations

Multigrid Solution for High-Order Discontinuous Galerkin Discretizations of the Compressible Navier-Stokes Equations

A mixed-enhanced formulation tetrahedral finite elements

Multigrid and Krylov Subspace Methods for the Discrete Stokes Equations

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