Weak Galerkin finite element methods for the biharmonic equation on polytopal meshes

Mu, Lin; Wang, Junping; Ye, Xiu

doi:10.1002/num.21855

Cited by 224 publications

(228 citation statements)

References 19 publications

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“…The method was first introduced in [18,19] for second order elliptic equations, and was later extended to other partial differential equations including the Stokes equations [20] and the biharmonic equation [11,17]. The current research indicates that the concept of discrete weak differential operators offers a new paradigm in numerical methods for partial differential equations.…”

Section: Introductionmentioning

confidence: 96%

A Weak Galerkin Finite Element Method for the Maxwell Equations

Wang

et al. 2014

J Sci Comput

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189

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This paper introduces a numerical scheme for the time-harmonic Maxwell equations by using weak Galerkin (WG) finite element methods. The WG finite element method is based on two operators: discrete weak curl and discrete weak gradient, with appropriately defined stabilizations that enforce a weak continuity of the approximating functions. This WG method is highly flexible by allowing the use of discontinuous approximating functions on arbitrary shape of polyhedra and, at the same time, is parameter free. Optimal-order of convergence is established for the WG approximations in various discrete norms which are either H 1 -like or L 2 and L 2 -like. An effective implementation of the WG method is developed through variable reduction by following a Schur-complement approach, yielding a system of linear equations involving unknowns associated with element boundaries only. Numerical results are presented to confirm the theory of convergence.

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Section: Introductionmentioning

confidence: 96%

A Weak Galerkin Finite Element Method for the Maxwell Equations

Wang

et al. 2014

J Sci Comput

Self Cite

189

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“…However, the WG finite element schemes in [1] are limited to the classical finite element partitions consisting of triangles (d = 2) or tetrahedra (d = 3). Following the stabilization approach of [2] for the mixed formulation, we developed a new class of WG finite element method in [3] for the primal formulation by allowing the use of finite elements of arbitrary shape. In particular, the WG finite element scheme in [3] is based on the polynomial combination of (P k (T ), P k (e), [P k−1 (T )] d ) so that the discrete weak gradient is computed in a polynomial space with degree lower than that of v 0 .…”

Section: Introductionmentioning

confidence: 99%

A weak Galerkin finite element method with polynomial reduction

Wang

2015

Journal of Computational and Applied Mathematics

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133

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a b s t r a c tThe weak Galerkin (WG) is a novel numerical method based on variational principles for weak functions and their weak partial derivatives defined as distributions. In the implementation, the weak partial derivatives and the weak functions are approximated by polynomials with various degrees of freedom. The accuracy and the computational complexity of the corresponding WG scheme is significantly impacted by the selection of such polynomials. This paper presents an optimal combination for the polynomial spaces that minimize the number of unknowns in the numerical scheme without compromising the accuracy of the numerical approximation. For illustrative purpose, the authors use the second order elliptic equation to demonstrate the basic ideas of polynomial reduction. Consequently, a new weak Galerkin finite element method is proposed and analyzed. 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. In addition, the paper presents some numerical results to demonstrate the power of the WG method in dealing with finite element partitions with arbitrary polygons in 2D or polyhedra in 3D. The numerical examples include various finite element partitions such as triangular mesh, quadrilateral mesh, honeycomb mesh in 2D and mesh with deformed cubes in 3D.

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“…WG finite element methods were first introduced in [6] for solving steady second order elliptic problem and later on in [8] for shape regular polytopal meshes. The method has been successfully applied to elliptic interface problems [10], Helmholtz equations [13], and biharmonic equations [11,12], in this paper, we applied the WG finite element methods for non steady diffusion convection problem and proved that the error estimate in 2 L -norm, the elliptic property, and the energy conservation law.…”

Section: Introductionmentioning

confidence: 99%

Full-Discrete Weak Galerkin Finite Element Method for Solving Diffusion-Convection Problem.

Hamdan

2017

JAM

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I S S N 2 3 4 7 -1 9 2 1 V o l u m e 1 3 N u m b e r 4 J o u r n a l o f A d v a n c e i n M a t h e m a t i c s 7333 | P a g e S e p t e m b e r 2 0 1 6 w w w . c i r w o r l d . hkashkool@yahoo.com asmaahammdan93@gmail.com ABSTRACT This paper applied and analyzes full-discrete weak Galerkin (WG) finite element method for non steady two dimensional convection-diffusion problems on conforming polygon. We approximate the time derivative by backward finite difference method and the elliptic form by WG finite element method. The main idea of WG finite element methods is the use of weak functions and their corresponding discrete weak derivatives in standard weak form of the model problem. The theoretical evidence proved that the error estimate in 2 L -norm, the properties of the bilinear form, (v-elliptic and continuity), stability, and the energy conservation law. KeywordsConvection -diffusion equation; full discrete weak Galerkin finite element method; error estimate; conservation law.

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Weak Galerkin finite element methods for the biharmonic equation on polytopal meshes

Cited by 224 publications

References 19 publications

A Weak Galerkin Finite Element Method for the Maxwell Equations

A Weak Galerkin Finite Element Method for the Maxwell Equations

A weak Galerkin finite element method with polynomial reduction

Full-Discrete Weak Galerkin Finite Element Method for Solving Diffusion-Convection Problem.

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