A simple finite element method for the Stokes equations
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TL;DR: A basis for a divergence free subspace of the velocity field is constructed so that the original saddle point problem can be reduced to a symmetric and positive definite system with much fewer unknowns.
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Abstract: The goal of this paper is to introduce a simple finite element method to solve the Stokes equations. This method is in primal velocity-pressure formulation and is so simple such that both velocity and pressure are approximated by piecewise constant functions. Implementation issues as well as error analysis are investigated. A basis for a divergence free subspace of the velocity field is constructed so that the original saddle point problem can be reduced to a symmetric and positive definite system with much fewer unknowns. The numerical experiments indicate that the method is accurate.
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Citations
Finite element methods for viscous incompressible flows: A guide to theory, practice, and algorithms
Max D. Gunzburger
- 01 Jan 1989
TL;DR: In this paper, the authors considered the Finite Element Problem and the Div-St abi lity Condition, and proposed a method to solve the Finiteness Problem in finite element spaces.
147
A Uniformly Robust H(DIV) Weak Galerkin Finite Element Methods for Brinkman Problems
TL;DR: A uniform robust weak Galerkin finite element scheme for Brinkman equations and the major idea for achieving uniform energy-error estimate is to use a divergence preserving v...
21
A family of quadratic finite volume method for solving the Stokes equation
TL;DR: In this article , a family of stable quadratic finite volume methods for solving the Stokes equation over triangular meshes is presented, which have the optimal H 1 -norm error estimate for velocity without any restriction on mesh geometry of the primary partitions.
9
A posteriori error estimates for weak Galerkin methods for second order elliptic problems on polygonal meshes
TL;DR: A posteriori error estimates for the Weak Galerkin finite element methods (WG-FEMs) for second order elliptic problems are derived in terms of an H 1 − equivalent energy norm, and the error analysis of these methods is proved to be valid for polygonal meshes.
8
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A Pressure-Robust Weak Galerkin Finite Element Method for Navier-Stokes Equations.
TL;DR: A novel numerical scheme for the steady incompressible Navier-Stokes equations by the weak Galerkin methods that can achieve pressure-robust, which means, the velocity error is independent of the pressure and the irrotational body force.
References
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Vivette Girault,Pierre-Arnaud Raviart +1 more
- 19 Jun 1986
TL;DR: This paper presents the results of an analysis of the "Stream Function-Vorticity-Pressure" Method for the Stokes Problem in Two Dimensions and its applications to Mixed Approximation and Homogeneous Stokes Equations.
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Mixed and Hybrid Finite Element Methods
Franco Brezzi,Michel Fortin +1 more
- 23 Nov 2011
TL;DR: Variational Formulations and Finite Element Methods for Elliptic Problems, Incompressible Materials and Flow Problems, and Other Applications.
5.3K
Conforming and nonconforming finite element methods for solving the stationary Stokes equations I
M. Crouzeix,P.-A. Raviart +1 more
- 01 Jan 1973
TL;DR: Both conforming and nonconforming finite element methods are studied and various examples of simplicial éléments well suited for the numerical treatment of the incompressibility condition are given.
A numerical solution of the Navier-Stokes equations using the finite element technique
C. Taylor,Paul Hood +1 more
TL;DR: In this paper, two methods of finite element discretisation are presented, and a comparison of the effeciency of the methods associated with the solution of particular problems is made.
1.3K