Local Mass Conservation of Stokes Finite Elements

Boffi, Daniele; Cavallini, Nicola; Gardini, Francesca; Gastaldi, Lucia

doi:10.1007/s10915-011-9549-4

Cited by 60 publications

(53 citation statements)

References 17 publications

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“…Finally, we remark that a similar divergence-free condition is obtained for conforming finite elements [5].…”

Section: Postprocessing and Local Conservationsupporting

confidence: 71%

A Staggered Discontinuous Galerkin Method for the Stokes System

Kim¹,

Chung²,

Lee³

2013

SIAM J. Numer. Anal.

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Discontinuous Galerkin (DG) methods are a class of efficient tools for solving fluid flow problems. There are in the literature many greatly successful DG methods. In this paper, a new staggered DG method for the Stokes system is developed and analyzed. The key feature of our method is that the discrete system preserves the structures of the continuous problem, which results from the use of our new staggered DG spaces. This also provides local and global conservation properties, which are desirable for fluid flow applications. The method is based on the first order mixed formulation involving pressure, velocity, and velocity gradient. The velocity and velocity gradient are approximated by polynomials of the same degree while the choice of polynomial degree for pressure is flexible, namely, the approximation degree for pressure can be chosen as either that of velocity or one degree lower than that of velocity. In any case, stability and optimal convergence of the method are proved. Moreover, a superconvergence result with respect to a discrete H 1 -norm for the velocity is proved. Furthermore, a local postprocessing technique is proposed to improve the divergence free property of the velocity approximation and it is proved that the postprocessed velocity retains the original accuracy and is weakly divergence free with respect to pressure test functions. Numerical results are included to validate our theoretical estimates and to present the ability of our method for capturing singular solutions. Introduction.Discontinuous Galerkin (DG) methods are gaining popularity in solving fluid flow problems. For example, in [11], the local DG methods were developed for the Stokes system. These are stabilized mixed methods and a nonstandard inf-sup condition was proved. The methods give more flexibility in the choice of approximation spaces for the velocity and pressure. In particular, experimental results there showed that if polynomials of degree k and k − 1 are used to approximate both velocity and pressure, respectively, then the velocity converges with order k + 1 in L 2 while the pressure converges with order k. When polynomials of the same degree k are used to approximate both velocity and pressure, the numerical solution still converges, but the order of convergence for the pressure does not improve because the L 2 error of the pressure also depends on the energy error of the velocity.Due to their discontinuous nature, DG methods are also well suited for hpadaptivity; see [26] for a study on the mixed hp-DGFEM with the Q k −Q k−1 elements and see also [27] in which corner singularities are treated and the authors' derived ex-

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“…Finally, we remark that a similar divergence-free condition is obtained for conforming finite elements [5].…”

Section: Postprocessing and Local Conservationsupporting

confidence: 71%

A Staggered Discontinuous Galerkin Method for the Stokes System

Kim¹,

Chung²,

Lee³

2013

SIAM J. Numer. Anal.

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“…Then, for d = 2, possible pairs

((), {double-struckU}^{m}, {double-struckP}^{m})

that satisfy are P2–P0 and P2–(P1+P0), that is, we set

{double-struckU}^{m} = {([], S_{2}^{m})}^{d} \cap double-struckU

and either

{double-struckP}^{m} = S_{0}^{m}

S_{1}^{m} + S_{0}^{m}

. We note that the choice P2–(P1+P0) requires the weak constraint that all simplices have a vertex in Ω . For d = 3, pairs of spaces that satisfy

S_{0}^{m} \subset {double-struckP}^{m}

and are the P3–(P2+P0) element or stabilized spaces such as P1 face bubble –P0, , Remark 8.7.1], which is also called the SMALL element.…”

Section: Methodsmentioning

confidence: 99%

Fitted finite element discretization of two‐phase Stokes flow

Agnese

Nürnberg

2016

Numerical Methods in Fluids

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We propose a novel fitted finite element method for two-phase Stokes flow problems that uses piecewise linear finite elements to approximate the moving interface. The method can be shown to be unconditionally stable. Moreover, spherical stationary solutions are captured exactly by the numerical approximation. In addition, the meshes describing the discrete interface in general do not deteriorate in time, which means that in numerical simulations a smoothing or a remeshing of the interface mesh is not necessary. We present several numerical experiments for our numerical method, which demonstrate the accuracy and robustness of the proposed algorithm.

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“…We note that a similar assumption is required to carry out the analysis of the Taylor-Hood elements (cf. [20,6]). Besides the standard shape regularity assumption, this is the only assumption we make on the triangulation; quasi-uniform meshes are not required to carry out our analysis.…”

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confidence: 99%