Superconvergence of numerical gradient for weak Galerkin finite element methods on nonuniform Cartesian partitions in three dimensions

Li, Dan; Nie, Yufeng; Wang, Chunmei

doi:10.1016/j.camwa.2019.03.010

Cited by 11 publications

(6 citation statements)

References 48 publications

(74 reference statements)

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“…Observe that a superconvergence theory in the H 1 ‐norm has been developed in for the diffusion equation on rectangular partitions; a slight modification of the analysis there will yield a superconvergence of order

O (h^{2})

for the SWG solutions of the full convection–diffusion equation (1.1) and (1.2).…”

Section: Swg On Polymeshmentioning

confidence: 99%

“…From (2.3), the linear extension of u b ∈ W ( T ) can be represented as (see Lem. 6.1 in for details)

s (u_{b}) = γ_{0} + γ_{1} (x - x_{T}) + γ_{2} (y - y_{T}),

where

\{\begin{matrix} left \\ γ_{0} = \frac{| e_{1} | (u_{b, 1} + u_{b, 2}) + | e_{3} | (u_{b, 3} + u_{b, 4})}{2 | e_{1} | + 2 | e_{3} |}, \\ γ_{1} = (u_{b, 2} - u_{b, 1}) / | e_{3} |, \\ γ_{2} = (u_{b, 4} - u_{b, 3}) / | e_{1} | . \end{matrix}

It follows that

\begin{array}{l} ((), u_{b} - frakturs ((), u_{b})) ((), M_{1}) & = ((), u_{b} - frakturs ((), u_{b})) ((), M_{2}) \\ = \frac{|e_{3}|}{2 ((), (||, e_{1}) + (||, e_{3}))} ((), u_{b, 1} + u_{b, 2} - u_{b, 3} - u_{b, 4}), \end{array}

…”

Section: Discrete Maximum Principlementioning

confidence: 99%

“…A simplified formulation of the WG finite element method has been developed in for second order elliptic equations in conjunction with the study of superconvergence and error estimates. This simplification idea was further applied to the Stokes equation in for the development of a superconvergence theory on nonuniform rectangular partitions for both the velocity and the pressure approximations.…”

Section: Introductionmentioning

confidence: 99%

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A discrete maximum principle for the weak Galerkin finite element method on nonuniform rectangular partitions

Liu

Wang

2019

Numerical Methods Partial

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This article establishes a discrete maximum principle (DMP) for the approximate solution of convection–diffusion–reaction problems obtained from the weak Galerkin (WG) finite element method on nonuniform rectangular partitions. The DMP analysis is based on a simplified formulation of the WG involving only the approximating functions defined on the boundary of each element. The simplified weak Galerkin (SWG) method has a reduced computational complexity over the usual WG, and indeed provides a discretization scheme different from the WG when the reaction terms are present. An application of the SWG on uniform rectangular partitions yields some 5‐ and 7‐point finite difference schemes for the second order elliptic equation. Numerical experiments are presented to verify the DMP and the accuracy of the scheme, particularly the finite difference scheme.

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O (h^{2})

for the SWG solutions of the full convection–diffusion equation (1.1) and (1.2).…”

Section: Swg On Polymeshmentioning

confidence: 99%

“…From (2.3), the linear extension of u b ∈ W ( T ) can be represented as (see Lem. 6.1 in for details)

s (u_{b}) = γ_{0} + γ_{1} (x - x_{T}) + γ_{2} (y - y_{T}),

where

\{\begin{matrix} left \\ γ_{0} = \frac{| e_{1} | (u_{b, 1} + u_{b, 2}) + | e_{3} | (u_{b, 3} + u_{b, 4})}{2 | e_{1} | + 2 | e_{3} |}, \\ γ_{1} = (u_{b, 2} - u_{b, 1}) / | e_{3} |, \\ γ_{2} = (u_{b, 4} - u_{b, 3}) / | e_{1} | . \end{matrix}

It follows that

\begin{array}{l} ((), u_{b} - frakturs ((), u_{b})) ((), M_{1}) & = ((), u_{b} - frakturs ((), u_{b})) ((), M_{2}) \\ = \frac{|e_{3}|}{2 ((), (||, e_{1}) + (||, e_{3}))} ((), u_{b, 1} + u_{b, 2} - u_{b, 3} - u_{b, 4}), \end{array}

…”

Section: Discrete Maximum Principlementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A discrete maximum principle for the weak Galerkin finite element method on nonuniform rectangular partitions

Liu

Wang

2019

Numerical Methods Partial

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show abstract

“…The usual regularity of the approximating functions is compensated by carefully-designed stabilizers. This WG method has been investigated for solving numerous model PDEs; see a limited list of references and references therein [12,18,19,23,37,34,31,32,33,13,14,35,36]. The research results indicate that the WG method has shown its great potential as a powerful numerical tool/technique in scientific computing.…”

mentioning

confidence: 99%

Curved Elements in Weak Galerkin Finite Element Methods

Li¹,

Wang²,

Wang³

2022

Preprint

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A mathematical analysis is established for the weak Galerkin finite element methods for the Poisson equation with Dirichlet boundary value when the curved elements are involved on the interior edges of the finite element partition or/and on the boundary of the whole domain in two dimensions. The optimal orders of error estimates for the weak Galerkin approximations in both the H 1 -norm and the L 2 -norm are established. Numerical results are reported to demonstrate the performance of the weak Galerkin methods on general curved polygonal partitions.

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“…So far all weak Galerkin finite element methods with stabilizers do not have superconvergence in both an energy norm and the L 2 norm. A order one superconvergence in an energy norm only is derived in [4,3]. The new method reduces the computation cost greatly of the other WG methods.…”

mentioning

confidence: 99%

A stabilizer free WG method for the Stokes equations with order two superconvergence on polytopal mesh

Zhang

2021

era

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<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula> norm. Optimal order error estimate for pressure in the <inline-formula><tex-math id="M2">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula> norm is also established. The numerical examples cover low and high order approximations, and 2D and 3D cases.</p>

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Superconvergence of numerical gradient for weak Galerkin finite element methods on nonuniform Cartesian partitions in three dimensions

Cited by 11 publications

References 48 publications

A discrete maximum principle for the weak Galerkin finite element method on nonuniform rectangular partitions

A discrete maximum principle for the weak Galerkin finite element method on nonuniform rectangular partitions

Curved Elements in Weak Galerkin Finite Element Methods

A stabilizer free WG method for the Stokes equations with order two superconvergence on polytopal mesh

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