A locally conservative, discontinuous least‐squares finite element method for the Stokes equations

Bochev, Pavel B.; Lai, James; Olson, Luke N.

doi:10.1002/fld.2536

Cited by 33 publications

(27 citation statements)

References 36 publications

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“…Computational results in [1] confirm that the dS-VP formulation attains high mass conservation. Substitution of the stream function by a vector potential a h such that u h D r a h extends (24) to three dimensions.…”

Section: Discontinuous Stream Function Vorticity-pressure Least-squamentioning

confidence: 59%

“…The approach presented in is to consider discontinuous velocity fields in and and then to represent the velocity on each element by a curl of a discontinuous stream function. The resulting discontinuous stream function, continuous vorticity–pressure (dS‐VP) version of J ( h ) is given by

\begin{matrix} aligned \\ right J_{MathClass-open(h MathClass-close)}^{S} MathClass-open(ψ_{h}, ω_{h}, p_{h}; f MathClass-close) = & left {∥\nabla \times ω_{h} + \nabla p_{h} - f∥}_{MathClass-open(h MathClass-close)}^{2} + \sum_{κ \in K_{h} MathClass-open(Ω MathClass-close)} {∥\nabla \times \nabla \times ψ_{h} - ω_{h}∥}_{0, κ}^{2} & right \\ right & left + \sum_{ε \in E_{h, 0}} h^{- 1} {∥MathClass-open[\nabla \times ψ_{h} MathClass-close]∥}_{0, ε}^{2} + h^{- 3} {∥MathClass-open[ψ_{h} MathClass-close]∥}_{0, ε}^{2} + \sum_{} \end{matrix}

…”

Section: Quotation Of Resultsmentioning

confidence: 99%

“…For clarity, we develop the new dV‐VP formulation in three stages. The first stage reprises the approach of to relax the C 0 continuity for the velocity space only. Therefore, we change the approximating space from to or its equal‐order counterpart and modify J ( h ) in and to allow discontinuous velocity fields:

\begin{matrix} aligned \\ right J_{MathClass-open(h MathClass-close)}^{V} MathClass-open(u_{h}, ω_{h}, p_{h}; f MathClass-close) = & left {∥\nabla \times ω_{h} + \nabla p_{h} - f∥}_{MathClass-open(h MathClass-close)}^{2} + \sum_{κ \in K_{h} MathClass-open(Ω MathClass-close)} ({∥\nabla \times u_{h} - ω_{h}∥}_{0, κ}^{2} + {∥\nabla \cdot u_{h}∥}_{0, κ}^{2}) & right \\ right & left + \sum_{ϵ \in E_{h, 0}} h^{- 1} ∥MathClass-open[u_{h <...>}∥ \end{matrix}

…”

Section: Discontinuous Velocity Vorticity–pressure Least‐squares Methodsmentioning

confidence: 99%

“…The approach presented in [1] is to consider discontinuous velocity fields in (18) and (19) and then to represent the velocity on each element by a curl of a discontinuous stream function. The resulting discontinuous stream function, continuous vorticity-pressure (dS-VP) version of J .h/ is given by…”

Section: Discontinuous Stream Function Vorticity-pressure Least-squamentioning

confidence: 99%

“… to develop least‐squares methods, which improve mass conservation, while remaining straightforward to implement and solve using publicly available libraries such as the Trilinos packages Intrepid and ML . The discontinuous stream function, continuous vorticity, and pressure method (dS‐VP) uses a discontinuous stream function to obtain a locally divergence‐free finite element solution of the Stokes equations. The method achieves nearly perfect conservation of mass on a series of challenging test problems, yet requires the use of an additional stream function.…”

Section: Introductionmentioning

confidence: 99%

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A non‐conforming least‐squares finite element method for incompressible fluid flow problems

Bochev

Lai

Olson

2012

Numerical Methods in Fluids

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SUMMARYIn this paper, we develop least‐squares finite element methods (LSFEMs) for incompressible fluid flows with improved mass conservation. Specifically, we formulate a new locally conservative LSFEM for the velocity–vorticity–pressure Stokes system, which uses a piecewise divergence‐free basis for the velocity and standard C0 elements for the vorticity and the pressure. The new method, which we term dV‐VP improves upon our previous discontinuous stream‐function formulation in several ways. The use of a velocity basis, instead of a stream function, simplifies the imposition and implementation of the velocity boundary condition, and eliminates second‐order terms from the least‐squares functional. Moreover, the size of the resulting discrete problem is reduced because the piecewise solenoidal velocity element is approximately one‐half of the dimension of a stream‐function element of equal accuracy. In two dimensions, the discontinuous stream‐function LSFEM [1] motivates modification of our functional, which further improves the conservation of mass. We briefly discuss the extension of this modification to three dimensions. Computational studies demonstrate that the new formulation achieves optimal convergence rates and yields high conservation of mass. We also propose a simple diagonal preconditioner for the dV‐VP formulation, which significantly reduces the condition number of the LSFEM problem. Published 2012. This article is a US Government work and is in the public domain in the USA.

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Section: Discontinuous Stream Function Vorticity-pressure Least-squamentioning

confidence: 59%

\begin{matrix} aligned \\ right J_{MathClass-open(h MathClass-close)}^{S} MathClass-open(ψ_{h}, ω_{h}, p_{h}; f MathClass-close) = & left {∥\nabla \times ω_{h} + \nabla p_{h} - f∥}_{MathClass-open(h MathClass-close)}^{2} + \sum_{κ \in K_{h} MathClass-open(Ω MathClass-close)} {∥\nabla \times \nabla \times ψ_{h} - ω_{h}∥}_{0, κ}^{2} & right \\ right & left + \sum_{ε \in E_{h, 0}} h^{- 1} {∥MathClass-open[\nabla \times ψ_{h} MathClass-close]∥}_{0, ε}^{2} + h^{- 3} {∥MathClass-open[ψ_{h} MathClass-close]∥}_{0, ε}^{2} + \sum_{} \end{matrix}

…”

Section: Quotation Of Resultsmentioning

confidence: 99%

\begin{matrix} aligned \\ right J_{MathClass-open(h MathClass-close)}^{V} MathClass-open(u_{h}, ω_{h}, p_{h}; f MathClass-close) = & left {∥\nabla \times ω_{h} + \nabla p_{h} - f∥}_{MathClass-open(h MathClass-close)}^{2} + \sum_{κ \in K_{h} MathClass-open(Ω MathClass-close)} ({∥\nabla \times u_{h} - ω_{h}∥}_{0, κ}^{2} + {∥\nabla \cdot u_{h}∥}_{0, κ}^{2}) & right \\ right & left + \sum_{ϵ \in E_{h, 0}} h^{- 1} ∥MathClass-open[u_{h <...>}∥ \end{matrix}

…”

Section: Discontinuous Velocity Vorticity–pressure Least‐squares Methodsmentioning

confidence: 99%

Section: Discontinuous Stream Function Vorticity-pressure Least-squamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A non‐conforming least‐squares finite element method for incompressible fluid flow problems

Bochev

Lai

Olson

2012

Numerical Methods in Fluids

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Least‐squares virtual element method for the convection‐diffusion‐reaction problem

Wang

2021

Numerical Meth Engineering

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In this paper, we introduce a least‐squares virtual element method for the convection‐diffusion‐reaction problem in mixed form. We use the H(div) virtual element and continuous virtual element to approximate the flux and the primal variables, respectively. The method allows for the use of very general polygonal meshes. Optimal order a priori error estimates are established for the flux and the primal variables in suitable norms. The least‐squares method offers an efficient a posteriori error estimator without extra effort. Moreover, the hanging nodes are naturally treated in the virtual element method, which provides the high flexibility in mesh refinement because the local mesh postprocessing is never required. Both attractive features motivate us to develop the a posteriori error estimate of the method. Numerical experiments are shown to illustrate the accuracy of the theoretical analysis and demonstrate that the adaptive mesh refinement driven by the proposed estimator can efficiently capture the boundary and the interior layers.

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First‐Order system least squares for generalized‐Newtonian coupled Stokes‐Darcy flow

Münzenmaier

2014

Numerical Methods Partial

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The coupled problem for a generalized Newtonian Stokes flow in one domain and a generalized Newtonian Darcy flow in a porous medium is studied in this work. Both flows are treated as a first-order system in a stress-velocity formulation for the Stokes problem and a volumetric flux-hydraulic potential formulation for the Darcy problem. The coupling along an interface is done using the well-known Beavers-Joseph-Saffman interface condition. A least squares finite element method is used for the numerical approximation of the solution. It is shown that under some assumptions on the viscosity the error is bounded from above and below by the least squares functional. An adaptive refinement strategy is examined in several numerical examples where boundary singularities are present. Due to the nonlinearity of the problem a Gauss-Newton method is used to iteratively solve the problem. It is shown that the linear variational problems arising in the Gauss-Newton method are well posed.

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A locally conservative, discontinuous least‐squares finite element method for the Stokes equations

Cited by 33 publications

References 36 publications

A non‐conforming least‐squares finite element method for incompressible fluid flow problems

A non‐conforming least‐squares finite element method for incompressible fluid flow problems

Least‐squares virtual element method for the convection‐diffusion‐reaction problem

First‐Order system least squares for generalized‐Newtonian coupled Stokes‐Darcy flow

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