Shangyou Zhang

A Modified weak Galerkin finite element method for the Maxwell equations on polyhedral meshes

Chunmei Wang, Xiu Ye, Shangyou Zhang

We introduce a new numerical method for solving time-harmonic Maxwell's equations via the modified weak Galerkin technique. The inter-element functions of the weak Galerkin finite…

Locking-free HDG methods for Reissner-Mindlin plates equations on polygonal meshes

Gang Chen, Lu Zhang, Shangyou Zhang

We present and analyze a new hybridizable discontinuous Galerkin method (HDG) for the Reissner-Mindlin plate bending system. Our method is based on the formulation utilizing Helmho…

High Order Morley Elements for Biharmonic Equations on Polytopal Partitions

Dan Li, Chunmei Wang, Junping Wang +1

This paper introduces an extension of the Morley element for approximating solutions to biharmonic equations. Traditionally limited to piecewise quadratic polynomials on triangular…

On a Sharp Estimate of Overlapping Schwarz Methods in $H (curl; Ω)$ and $H (div; Ω)$

Qigang Liang, Xuejun Xu, Shangyou Zhang

The previous proved-bound is $C (1 + \frac{H ^{2}}{δ ^{2}})$ for the condition number of the overlapping domain decomposition $H (curl; Ω)$ and $H (div; Ω)$ …

math.NA2022

Weak Galerkin Finite Element Methods for Quad-Curl Problems

Chunmei Wang, Junping Wang, Shangyou Zhang

This article introduces a weak Galerkin (WG) finite element method for quad-curl problems in three dimensions. It is proved that the proposed WG method is stable and accurate in an…

math.NA2022

A parallel iterative procedure for weak Galerkin methods for second order elliptic problems

Chunmei Wang, Junping Wang, Shangyou Zhang

A parallelizable iterative procedure based on domain decomposition is presented and analyzed for weak Galerkin finite element methods for second order elliptic equations. The conve…

math.NA2022

The nodal basis of $C^{m}$ - $P_{k}^{(3)}$ and $C^{m}$ - $P_{k}^{(4)}$ finite elements on tetrahedral and 4D simplicial grids

Shangyou Zhang

We construct the nodal basis of $C^{m}$ - $P_{k}^{(3)}$ ( $k \geq 2^{3} m + 1$ ) and $C^{m}$ - $P_{k}^{(4)}$ ( $k \geq 2^{4} m + 1$ ) finite elements on 3D tetrahedral and 4D simplicial grids, respectivel…

math.NA2021★ 1 cited

A Weak Galerkin Method for Elasticity Interface Problems

Chunmei Wang, Shangyou Zhang

This article introduces a weak Galerkin (WG) finite element method for linear elasticity interface problems on general polygonal/ployhedra partitions. The developed WG method has b…

math.NA2021

A weak Galerkin-mixed finite element method for the Stokes-Darcy problem

Hui Peng, Qilong Zhai, Ran Zhang +1

In this paper, we propose a new numerical scheme for the coupled Stokes-Darcy model with Beavers-Joseph-Saffman interface condition. We use the weak Galerkin method to discretize t…

math.NA2020★ 7 cited

The Weak Galerkin Finite Element Method for the Transport-Reaction Equation

Tie Zhang, Shangyou Zhang

We present and analyze a weak Galerkin finite element method for solving the transport-reaction equation in $d$ space dimensions. This method is highly flexible by allowing the use…

math.NA2020★ 3 cited

The Weak Galerkin Finite Element Method for the Symmetric Hyperbolic Systems

Tie Zhang, Shangyou Zhang

In this paper, we present and analyze a weak Galerkin finite element (WG) method for solving the symmetric hyperbolic systems. This method is highly flexible by allowing the use of…

A stabilizer free, pressure robust, and superconvergence weak Galerkin finite element method for the Stokes Equations on polytopal mesh

Lin Mu, Xiu Ye, Shangyou Zhang

In this paper, we propose a new stabilizer free and pressure robust WG method for the Stokes equations with super-convergence on polytopal mesh in the primary velocity-pressure for…

An interpolated Galerkin finite element method for the Poisson equation

Tatyana Sorokina, Shangyou Zhang

When solving the Poisson equation by the finite element method, we use one degree of freedom for interpolation by the given Laplacian - the right hand side function in the partial…

A $P_{k + 2}$ polynomial lifting operator on polygons and polyhedrons

Xiu Ye, Shangyou Zhang

A $P_{k + 2}$ polynomial lifting operator is defined on polygons and polyhedrons. It lifts discontinuous polynomials inside the polygon/polyhedron and on the faces to a one-piece $P_…

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

Xiu Ye, Shangyou Zhang

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 hig…

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

Xiu Ye, Shangyou Zhang

A weak Galerkin (WG) finite element method without stabilizers was introduced in [J. Comput. Appl. Math., 371 (2020). arXiv:1906.06634] on polytopal mesh. Then it was improved in […

A mixed finite element method on polytopal mesh

Yanping Lin, Xiu Ye, Shangyou Zhang

In this paper, we introduce new stable mixed finite elements of any order on polytopal mesh for solving second order elliptic problem. We establish optimal order error estimates fo…

math.NA2020★ 3 cited

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

Xiu Ye, Shangyou Zhang

A stabilizer free weak Galerkin (WG) finite element method on polytopal mesh has been introduced in Part I of this paper (J. Comput. Appl. Math, 371 (2020) 112699. arXiv:1906.06634…

math.NA2020★ 16 cited

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

Xiu Ye, Shangyou Zhang

A new discontinuous Galerkin finite element method for the Stokes equations is developed in the primary velocity-pressure formulation. This method employs discontinuous polynomials…

math.NA2020★ 2 cited

A stabilizer-free pressure-robust finite element method for the Stokes equations

Xiu Ye, Shangyou Zhang

In this paper, we introduce a new finite element method for solving the Stokes equations in the primary velocity-pressure formulation. This method employs $H (d i v)$ finite elements…

De Rham Complexes for Weak Galerkin Finite Element Spaces

Chunmei Wang, Junping Wang, Xiu Ye +1

Two de Rham complex sequences of the finite element spaces are introduced for weak finite element functions and weak derivatives developed in the weak Galerkin (WG) finite element…