Approximation of generalized Stokes problems using dual‐mixed finite elements without enrichment

Howell, Jason S.

doi:10.1002/fld.2356

Cited by 4 publications

(5 citation statements)

References 35 publications

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“…In that contribution, the authors prove that the continuous and discrete formulations are well posed and derive the associated a priori error analysis and a posteriori error estimates based on local problems. In , the pseudostress and the trace‐free velocity gradient are introduced as auxiliary unknowns, and a pseudostress–velocity formulation is considered, for which existence, uniqueness, and error estimates are derived. More recently, dual‐mixed methods based on the velocity–pseudostress and pseudostress have been introduced in and , respectively, for the generalized Stokes problem.…”

Section: Introductionmentioning

confidence: 99%

An augmented velocity–vorticity–pressure formulation for the Brinkman equations

Anaya

Gatica

Mora

et al. 2015

Numerical Methods in Fluids

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SUMMARYThis paper deals with the analysis of a new augmented mixed finite element method in terms of vorticity, velocity and pressure, for the Brinkman problem with non-standard boundary conditions. The approach is based on the introduction of Galerkin least-squares terms arising from the constitutive equation relating the aforementioned unknowns, and from the incompressibility condition. We show that the resulting augmented bilinear form is continuous and elliptic which, thanks to the Lax-Milgram Theorem, and besides proving the well-posedness of the continuous formulation, ensures the solvability and stability of the Galerkin scheme with any finite element subspace of the continuous space. In particular, Raviart-Thomas elements of any order k ≥ 0 for the velocity field, and piecewise continuous polynomials of degree k + 1 for both the vorticity and the pressure, can be utilized. A priori error estimates and the corresponding rates of convergence are also given here. Next, we derive two reliable and efficient residual-based a posteriori error estimators for this problem. The ellipticity of the bilinear form together with the local approximation properties of the Clément interpolation operator are the main tools for showing the reliability. In turn, inverse inequalities and the localization technique based on triangle-bubble and edge-bubble functions are utilized to show the efficiency. Finally, several numerical results illustrating the good performance of the method, confirming the properties of the estimators, and showing the behaviour of the associated adaptive algorithms, are reported. Copyright c 2014 John Wiley & Sons, Ltd.Received . . . KEY WORDS: Generalized Stokes equations; Brinkman problem; vorticity-based mixed formulation; augmented scheme; finite element method; a priori error estimates; a posteriori error analysis.

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Section: Introductionmentioning

confidence: 99%

An augmented velocity–vorticity–pressure formulation for the Brinkman equations

Anaya

Gatica

Mora

et al. 2015

Numerical Methods in Fluids

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“…While velocity and pressure are the primary unknowns in the classical mixed method for the Stokes and Navier–Stokes equations, the dual‐mixed method for Stokes and Navier–Stokes approximates the fluid stress ( S ), velocity ( u ), and velocity gradient ( G ) directly. Specifically, the following is the variational problem: given f ∈( L 2 (Ω)) d and g ∈( H 1/2 (Γ)) d , find

false(G, u, S false) \in {false(L^{2} false(normalΩ false) false)}^{d \times d} \times {false(L^{2} false(normalΩ false) false)}^{d} \times double-struckS

satisfying

\begin{matrix} aligncenter \\ align-1 & ν (G, H) - (1 / 2) (u \otimes u, H) - (S, H) = 0, \forall H \in {(L^{2} (Ω))}^{d \times d}, \\ align-1 & (1 / 2) (G u, v) - (div (S), v) = (f, v), \forall v \in {(L^{2} (Ω))}^{d}, \\ align-1 & (G, T) + (u, div (T)) = \int_{normalΓ} u_{normalΓ} \cdot T n d Γ, \forall T \in S, \end{matrix}

where

S = \{T \in {(H (div, Ω))}^{d \times d} | \int_{normalΩ} tr (T\}

…”

Section: Applications Of the Methodsmentioning

confidence: 99%

“…Problems. While the velocity and pressure are the primary unknowns in the classical mixed method for the Stokes and Navier-Stokes equations, the dual-mixed method [40,41,42] for Stokes and Navier-Stokes approximates the fluid stress (S), velocity (u), and velocity gradient (G) directly. Specifically, the variational problem is: given f ∈ (L 2 (Ω)) d and g ∈ (H 1/2 (Γ)) d , find (G, u, S)…”

Section: Dual-mixed Stokesmentioning

confidence: 99%

Prestructuring sparse matrices with dense rows and columns via null space methods

Howell

2017

Numerical Linear Algebra App

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Abstract. Several applied problems may produce large sparse matrices with a small number of dense rows and/or columns, which can adversely affect the performance of commonly used direct solvers. By posing the problem as a saddle point system, an unconventional application of a null space method can be employed to eliminate dense rows and columns. The choice of null space basis is critical in retaining the overall sparse structure of the matrix. A one-sided application of the null space method is also presented to eliminate either dense rows or columns. These methods can be considered techniques that modify the nonzero structure of the matrix before employing a direct solver, and may result in improved direct solver performance.

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“…In [10], an augmented formulation using the H(div) conforming element method has been developed. Also, the dual-mixed method has been studied in [12,13]. In all these works, the H(div) element was used to approximate the stress or stress-type dual variables.…”

Section: Introductionmentioning

confidence: 99%

A Posteriori Error Estimation for an Interior Penalty Type Method Employing $H(\mathrm{div})$ Elements for the Stokes Equations

Wang¹,

Wang²,

Ye³

2011

SIAM J. Sci. Comput.

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This paper establishes a posteriori error analysis for the Stokes equations discretized by an interior penalty type method using H(div) finite elements. The a posteriori error estimator is then employed for designing two grid refinement strategies; one is locally based and the other is globally based. The locally based refinement technique is believed to be able to capture local singularities in the numerical solution. The numerical formulations for the Stokes problem make use of H(div) conforming elements of Raviart-Thomas type. Therefore, the finite element solution features a full satisfaction of the continuity equation (mass conservation). The result of this paper provides a rigorous analysis for the method's reliability and efficiency. In particular, an H 1 norm a posteriori error estimator is obtained, together with upper and lower bound estimates. Numerical results are presented to verify the new theory of a posteriori error estimators.

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Approximation of generalized Stokes problems using dual‐mixed finite elements without enrichment

Cited by 4 publications

References 35 publications

An augmented velocity–vorticity–pressure formulation for the Brinkman equations

An augmented velocity–vorticity–pressure formulation for the Brinkman equations

Prestructuring sparse matrices with dense rows and columns via null space methods

A Posteriori Error Estimation for an Interior Penalty Type Method Employing $H(\mathrm{div})$ Elements for the Stokes Equations

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