A Weak Galerkin Finite Element Method for the Stokes Equations

Junping Wang; Xiu Ye

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A Weak Galerkin Finite Element Method for the Stokes Equations

Junping Wang, Xiu Ye

Numerical Analysis · Submitted 12 Feb 2013

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The authors' main claim Optimal-order error estimates are established for the corresponding numerical approximation in various norms.

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This paper introduces a weak Galerkin (WG) finite element method for the Stokes equations in the primary velocity-pressure formulation. This WG method is equipped with stable finite elements consisting of usual polynomials of degree $k\ge 1$ for the velocity and polynomials of degree $k-1$ for the pressure, both are discontinuous. The velocity element is enhanced by polynomials of degree $k-1$ on the interface of the finite element partition. All the finite element functions are discontinuous for which the usual gradient and divergence operators are implemented as distributions in properly-defined spaces. Optimal-order error estimates are established for the corresponding numerical approximation in various norms. It must be emphasized that the WG finite element method is designed on finite element partitions consisting of arbitrary shape of polygons or polyhedra which are shape regular.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	This paper introduces a weak Galerkin (WG) finite element method for the Stokes equations in the primary velocity-pressure formulation.	contribution 1.00	0.92	0.15	asserted 1.00
s1	This WG method is equipped with stable finite elements consisting of usual polynomials of degree $k\ge 1$ for the velocity and polynomials of degree $k-1$ for the pressure, both are discontinuous.	method 0.91	0.83	0.07	asserted 1.00
s2	The velocity element is enhanced by polynomials of degree $k-1$ on the interface of the finite element partition.	method 0.96	0.71	0.04	asserted 1.00
s3	All the finite element functions are discontinuous for which the usual gradient and divergence operators are implemented as distributions in properly-defined spaces.	method 0.99	0.63	0.05	asserted 1.00
s4	Optimal-order error estimates are established for the corresponding numerical approximation in various norms.	result 0.95	0.93	0.49	asserted 1.00
s5	It must be emphasized that the WG finite element method is designed on finite element partitions consisting of arbitrary shape of polygons or polyhedra which are shape regular.	method 0.71	0.57	0.06	asserted 1.00

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Original abstract

This paper introduces a weak Galerkin (WG) finite element method for the Stokes equations in the primary velocity-pressure formulation. This WG method is equipped with stable finite elements consisting of usual polynomials of degree $k\ge 1$ for the velocity and polynomials of degree $k-1$ for the pressure, both are discontinuous. The velocity element is enhanced by polynomials of degree $k-1$ on the interface of the finite element partition. All the finite element functions are discontinuous for which the usual gradient and divergence operators are implemented as distributions in properly-defined spaces. Optimal-order error estimates are established for the corresponding numerical approximation in various norms. It must be emphasized that the WG finite element method is designed on finite element partitions consisting of arbitrary shape of polygons or polyhedra which are shape regular.

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18 arXiv papers cite this in 48 sentences

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Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States the same result 1 paper

Optimal-order error estimates are established for the corresponding WG finite element solutions in various norms.

A Weak Galerkin Finite Element Scheme for solving the stationary Stokes Equations · 0.97

Papers citing it · 29 sentences without a confident label

Uses its method or tool 5 papers

In the weak Galerkin approach, the differential operator $$ is discretized as [equation] where $∇_w$ is a discrete weak gradient [ref] and $∂_ji,w^2$ is a discrete weak second order partial derivative [ref] (also see Section [ref] for their definition).

A Primal-Dual Weak Galerkin Finite Element Method for Fokker-Planck Type Equations · Introduction · 0.88
The goal of this paper is to develop a new finite element method for the model problem ([ref]) by using the weak Galerkin strategy recently introduced in [ref] for partial differential equations.

A Primal-Dual Weak Galerkin Finite Element Method for Second Order Elliptic Equations in Non-Divergence Form · Introduction · 0.86
In this paper, we are concerned with the development of efficient new numerical methods for linear elasticity equations by using the weak Galerkin finite element method recently developed in [ref] .

A Locking-Free Weak Galerkin Finite Element Method for Elasticity Problems in the Primal Formulation · Introduction · 0.87

Compares with it 2 papers

In addition, if we choose high order polynomials to approximate the model and use Schur complement to reduce the interior DOF of the velocity and pressure by the boundary DOF, the total DOF of this new method is less than the scheme of [ref] .

A Weak Galerkin Finite Element Scheme for solving the stationary Stokes Equations · Introduction · 0.99
The spaces defined above are different from the spaces introduced in [ref] , which are designed for the Darcy flow and for the Stokes equations individually.

Weak Galerkin method for the coupled Darcy-Stokes flow · Weak Galerkin discretization · 0.89

Cites it as background 10 papers

Weak Galerkin (WG) [ref] is a generic finite element method for partial differential equations where the differential operators (e.g., gradient, divergence, curl, Laplacian etc) in the variational form are approximated by weak forms as generalized distributions.

A Hybridized Formulation for the Weak Galerkin Mixed Finite Element Method · Introduction · 0.99
A weak Galerkin finite element method was introduced in [ref] for the Stokes equations in the primal velocity-pressure formulation.

A Discrete Divergence-Free Weak Galerkin Finite Element Method for the Stokes Equations · Introduction · 0.86
This thought was first proposed in [ref] for a model second order elliptic problem, and this method was further developed in [ref] .

A Weak Galerkin Finite Element Scheme for the Biharmonic Equations by Using Polynomials of Reduced Order · Introduction · 0.87

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