The lowest-order stabilizer free weak Galerkin finite element method

Al‐Taweel, Ahmed; Wang, Xiaoshen

doi:10.1016/j.apnum.2020.06.012

Cited by 13 publications

(5 citation statements)

References 9 publications

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“…Various configurations of (P k (T ), P l (e), [P j (T )] d ) with l ≥ k and j ≥ k + 1 lead to different schemes, cf. [1][2][3][22][23][24][25][26]. Later on, the definition of weak gradient operator is modified by using the standard gradient of interior test functions instead of the divergence of trial functions [23].…”

Section: Introductionmentioning

confidence: 99%

“…In order to reach the optimal convergence order for approximating the secondorder elliptic equations, in many published literature on WG-FEM and SFWG-FEM, the solution is usually assumed to have at least H 2 -smoothness [1,23]. As a result, it is demonstrated that the convergence rates are at least O(h) in H 1 -norm and O(h 2 ) in L 2 -norm with mesh-size h, respectively.…”

Section: Introductionmentioning

confidence: 99%

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A Stabilizer Free Weak Galerkin Finite Element Method for Elliptic Equation with Lower Regularity

Wang,

Zou,

null

et al. 2024

NMTMA

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This paper presents error analysis of a stabilizer free weak Galerkin finite element method (SFWG-FEM) for second-order elliptic equations with low regularity solutions. The standard error analysis of SFWG-FEM requires additional regularity on solutions, such as H 2 -regularity for the second-order convergence. However, if the solutions are in H 1+s with 0 < s < 1, numerical experiments show that the SFWG-FEM is also effective and stable with the (1 + s)-order convergence rate, so we develop a theoretical analysis for it. We introduce a standard H 2 finite element approximation for the elliptic problem, and then we apply the SFWG-FEM to approach this smooth approximating finite element solution. Finally, we establish the error analysis for SFWG-FEM with low regularity in both discrete H 1 -norm and standard L 2 -norm. The (P k (T ), P k−1 (e), [P k+1 (T )] d ) elements with dimensions of space d = 2, 3 are employed and the numerical examples are tested to confirm the theory.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Stabilizer Free Weak Galerkin Finite Element Method for Elliptic Equation with Lower Regularity

Wang,

Zou,

null

et al. 2024

NMTMA

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“…Here j = k + n − 1 and n is the number of sides of polytopal element. The authors in [1,2] have relaxed the requirement of polynomial degree of approximation. Rational function Wachspress coordinates [4] are used in [5,6] to approximate weak gradient.…”

Section: Introductionmentioning

confidence: 99%

A stabilizer free weak Galerkin finite element method on polytopal mesh: Part III

Zhang

2021

Journal of Computational and Applied Mathematics

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on polytopal mesh. Then it was improved in [arXiv:2008.13631] with order one superconvergence. The goal of this paper is to develop a new stabilizer free WG method on polytopal mesh. This method has convergence rates two orders higher than the optimal convergence rates for the corresponding WG solution in both an energy norm and the L 2 norm. The numerical examples are tested for low and high order elements in two and three dimensional spaces.

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“…In [11,12], gradient is approximated by a polynomial of order j = k + n − 1 where n is the number of sides of polygonal element. This result has been improved in [1,2] by reducing the degree of polynomial j. Recently, new stabilizer free WG methods have been developed in [13,14] for second order elliptic equations on polytopal mesh, which have superconvergence.…”

Section: Introductionmentioning

confidence: 99%

A stabilizer free WG Method for the Stokes Equations with order two superconvergence on polytopal mesh

Zhang

2020

Preprint

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A stabilizer free WG method is introduced for the Stokes equations with superconvergence on polytopal mesh in primary velocity-pressure formulation. Convergence rates two order higher than the optimal-order for velocity of the WG approximation is proved in both an energy norm and the L 2 norm. Optimal order error estimate for pressure in the L 2 norm is also established.The numerical examples cover low and high order approximations, and 2D and 3D cases.

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The lowest-order stabilizer free weak Galerkin finite element method

Cited by 13 publications

References 9 publications

A Stabilizer Free Weak Galerkin Finite Element Method for Elliptic Equation with Lower Regularity

A Stabilizer Free Weak Galerkin Finite Element Method for Elliptic Equation with Lower Regularity

A stabilizer free weak Galerkin finite element method on polytopal mesh: Part III

A stabilizer free WG Method for the Stokes Equations with order two superconvergence on polytopal mesh

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