A note on the optimal degree of the weak gradient of the stabilizer free weak Galerkin finite element method

Al‐Taweel, Ahmed; Wang, Xiaoshen

doi:10.1016/j.apnum.2019.10.009

Cited by 41 publications

(29 citation statements)

References 11 publications

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“…It is easy to see that ‖ v ‖ 1, h defines a norm in V h . But ||| · ||| also defines a norm in V h , by the following norm equivalence which has been proved in References 10 and 27, to each component of v .

C_{1} ‖ v ‖_{1, h} \leq | | | v | | | \leq C_{2} ‖ v ‖_{1, h} \forall v \in V_{h} .

…”

Section: Well Posednessmentioning

confidence: 90%

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A conforming discontinuous Galerkin finite element method for the Stokes problem on polytopal meshes

Zhang

2021

Numerical Methods in Fluids

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A new discontinuous Galerkin finite element method for the Stokes equations is developed in the primary velocity‐pressure formulation. This method employs discontinuous polynomials for both velocity and pressure on general polygonal/polyhedral meshes. Most finite element methods with discontinuous approximation have one or more stabilizing terms for velocity and for pressure to guarantee stability and convergence. This new finite element method has the standard conforming finite element formulation, without any velocity or pressure stabilizers. Optimal‐order error estimates are established for the corresponding numerical approximation in various norms. The numerical examples are tested for low and high order elements up to the degree four in 2D and 3D spaces.

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C_{1} ‖ v ‖_{1, h} \leq | | | v | | | \leq C_{2} ‖ v ‖_{1, h} \forall v \in V_{h} .

…”

Section: Well Posednessmentioning

confidence: 90%

“…Remark The choice of j in (14) depends on the number of sides/faces of polygon/polyhedron. For triangular mesh, we can choose j = k + 1 27 . In general, j = n + k − 1, where n is the number of edges of polygon 10 …”

Section: Finite Element Methodsmentioning

confidence: 99%

A conforming discontinuous Galerkin finite element method for the Stokes problem on polytopal meshes

Zhang

2021

Numerical Methods in Fluids

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“…Removing stabilizers from WG finite element methods will simplify formulations and reduce programming complexity significantly. Stabilizer free WG finite element methods have been studied in [15][16][17]. The idea is to increase the connectivity of a weak function cross element boundary by raising the degree of polynomials for computing weak derivatives.…”

Section: Introductionmentioning

confidence: 99%

A stabilizer free weak Galerkin finite element method with supercloseness of order two

Al‐Taweel

Wang

et al. 2020

Numerical Methods Partial

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The weak Galerkin (WG) finite element method is an effective and flexible general numerical technique for solving partial differential equations. A simple WG finite element method is introduced for second‐order elliptic problems. First we have proved that stabilizers are no longer needed for this WG element. Then we have proved the supercloseness of order two for the WG finite element solution. The numerical results confirm the theory.

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“…Removing stabilizers from discontinuous finite element methods simplifies finite element formulations and reduces programming complexity. Stabilizer free WG finite element methods have been studied in [1,19,22]. The idea is increasing the connectivity of a weak function across element boundary by raising the degree of polynomials for computing weak derivatives.…”

mentioning

confidence: 99%

“…In [19], we have proved that a stabilizer can be removed from the WG finite element formulation for the WG element (P k (T ), P k (e), [P j (T )] d ) if j ≥ k + n − 1, where n is the number of edges/faces of an element. The condition j ≥ k + n − 1 has been relaxed in [1]. Stabilizer free DG methods have also been developed in [20,21].…”

mentioning

confidence: 99%

A new weak gradient for the stabilizer free weak Galerkin method with polynomial reduction

Ye¹,

Zhang²

2021

DCDS-B

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The weak Galerkin (WG) finite element method is an effective and flexible general numerical technique for solving partial differential equations. It is a natural extension of the classic conforming finite element method for discontinuous approximations, which maintains simple finite element formulation. Stabilizer free weak Galerkin methods further simplify the WG methods and reduce computational complexity. This paper explores the possibility of optimal combination of polynomial spaces that minimize the number of unknowns in the stabilizer free WG schemes without compromising the accuracy of the numerical approximation. A new stabilizer free weak Galerkin finite element method is proposed and analyzed with polynomial degree reduction. To achieve such a goal, a new definition of weak gradient is introduced. Error estimates of optimal order are established for the corresponding WG approximations in both a discrete H 1 norm and the standard L 2 norm. The numerical examples are tested on various meshes and confirm the theory.

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A note on the optimal degree of the weak gradient of the stabilizer free weak Galerkin finite element method

Cited by 41 publications

References 11 publications

A conforming discontinuous Galerkin finite element method for the Stokes problem on polytopal meshes

A conforming discontinuous Galerkin finite element method for the Stokes problem on polytopal meshes

A stabilizer free weak Galerkin finite element method with supercloseness of order two

A new weak gradient for the stabilizer free weak Galerkin method with polynomial reduction

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