A Systematic Study on Weak Galerkin Finite Element Methods for Second Order Elliptic Problems

Wang, Junping; Wang, Ruishu; Zhai, Qingfeng; Zhang, Ran

doi:10.1007/s10915-017-0496-6

Cited by 38 publications

(48 citation statements)

References 29 publications

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“…The concept of weak derivatives makes WG a widely applicable numerical technique for a large variety of of PDEs arising from the mathematical modeling of practical problems in science and engineering. There is an abundant literature on such PDEs; see, e.g., elliptic equation [18,20,21,22,27,36,37,39], parabolic equation [19,42,43,45], system of equations [23,25,26,28,29,31,35,40,44], interface problems [17,30,32]. One close relative of the WG finite element method is the hybridizable discontinuous Galerkin (HDG) method [10].…”

Section: Introductionmentioning

confidence: 99%

A Systematic Study on Weak Galerkin Finite Element Method for Second Order Parabolic Problems

Deka¹,

Kumar²

2021

Preprint

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A systematic numerical study on weak Galerkin (WG) finite element method for second order linear parabolic problems is presented by allowing polynomial approximations with various degrees for each local element. Convergence of both semidiscrete and fully discrete WG solutions are established in L ∞ (L 2 ) and L ∞ (H 1 ) norms for a general WG element (P k (K), Pj(∂K), P l (K)2 ), where k ≥ 1, j ≥ 0 and l ≥ 0 are arbitrary integers. The fully discrete space-time discretization is based on a first order in time Euler scheme. Our results are intended to extend the numerical analysis of WG methods for elliptic problems [J. Sci. Comput., 74 (2018), 1369-1396 to parabolic problems. Numerical experiments are reported to justify the robustness, reliability and accuracy of the WG finite element method.

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Section: Introductionmentioning

confidence: 99%

A Systematic Study on Weak Galerkin Finite Element Method for Second Order Parabolic Problems

Deka¹,

Kumar²

2021

Preprint

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“…Wang and Ye in 2011 proposed the weak Galerkin finite element for the second-order elliptic equations [19,29]. The method is applied to many problems, such as Stokes equations [17,18,[20][21][22]28], Brinkman problem [23,27], Biharmonic equations [30], eigenvalue problems [26] and Stochastic problems [6,33] and so on. The partition of the domain can be arbitrary polygonal or polyhedral.…”

Section: Introductionmentioning

confidence: 99%

The Weak Galerkin Finite Element Method for Solving the Time-Dependent Integro-Differential Equations

Wang¹

2020

AAMM

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In this paper, we solve linear parabolic integral differential equations using the weak Galerkin finite element method (WG) by adding a stabilizer. The semidiscrete and fully-discrete weak Galerkin finite element schemes are constructed. Optimal convergent orders of the solution of the WG in L 2 and H 1 norm are derived. Several computational results confirm the correctness and efficiency of the method.

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“…The goal of this paper is to overcome the aforementioned difficulties using the weak Galerkin method. The WG method was first proposed in [37], and further developed in [8,29,31,36,38,44,45,47,48]. Recently, the weak Galerkin method has been extended to elliptic interface problems [10], linear hyperbolic equations [43], Navier-Stokes equations [46,49], Helmholtz equations [11,32], and discrete maximum principles [18,39].…”

Section: Introductionmentioning

confidence: 99%

The Weak Galerkin Method for Elliptic Eigenvalue Problems

Zhai¹

2019

CICP

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This article is devoted to studying the application of the weak Galerkin (WG) finite element method to the elliptic eigenvalue problem with an emphasis on obtaining lower bounds. The WG method uses discontinuous polynomials on polygonal or polyhedral finite element partitions. The non-conforming finite element space of the WG method is the key of the lower bound property. It also makes the WG method more robust and flexible in solving eigenvalue problems. We demonstrate that the WG method can achieve arbitrary high convergence order. This is in contrast with existing nonconforming finite element methods which can provide lower bound approximations by linear finite elements. Numerical results are presented to demonstrate the efficiency and accuracy of the theoretical results.

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A Systematic Study on Weak Galerkin Finite Element Methods for Second Order Elliptic Problems

Cited by 38 publications

References 29 publications

A Systematic Study on Weak Galerkin Finite Element Method for Second Order Parabolic Problems

A Systematic Study on Weak Galerkin Finite Element Method for Second Order Parabolic Problems

The Weak Galerkin Finite Element Method for Solving the Time-Dependent Integro-Differential Equations

The Weak Galerkin Method for Elliptic Eigenvalue Problems

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