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A stabilizer-free C^(0) weak Galerkin method for the biharmonic equations

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Science China Mathematics 2023年第3期66卷 627-646页

作者： Peng Zhu Shenglan Xie xiaoshen wang College of Data Science Jiaxing UniversityJiaxing 314001China College of Information Engineering Jiaxing Nanhu UniversityJiaxing 314001China Department of Mathematics University of Arkansas at Little RockLittle RockAR 72204USA

In this article, we present and analyze a stabilizer-free C^(0)weak Galerkin(SF-C^(0)WG) method for solving the biharmonic problem. The SF-C^(0)WG method is formulated in terms of cell unknowns which are C^(0)continuous piecewise polynomials of degree k + 2 with k≥0 and in terms of face unknowns which are discontinuous piecewise polynomials of degree k + 1. The formulation of this SF-C^(0)WG method is without the stabilized or penalty term and is as simple as the C1conforming finite element scheme of the biharmonic problem. Optimal order error estimates in a discrete H^(2)-like norm and the H^(1)norm for k≥0 are established for the corresponding WG finite element solutions. Error estimates in the L^(2)norm are also derived with an optimal order of convergence for k > 0 and sub-optimal order of convergence for k = 0. Numerical experiments are shown to confirm the theoretical results.

关键词： weak Galerkin finite element method weak Laplacian biharmonic equations

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A Posteriori Error Estimate of Weak Galerkin FEM for Stokes Problem Using Auxiliary Subspace Techniques

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Communications in Computational Physics 2023年第2期33卷 511-537页

作者： Jiachuan Zhang Ran Zhang xiaoshen wang School of Physical and Mathematical Sciences Nanjing Tech UniversityNanjing 211816China School of Mathematics Jilin UniversityChangchun 130012China Department of Mathematics and Statistics University of Arkansas at Little RockArkansas 72204USA.

Based on the auxiliary subspace techniques,a posteriori error estimator of nonconforming weak Galerkin finite element method(WGFEM)for Stokes problem in two and three dimensions is presented.Without saturation assumption,we prove that the WGFEM approximation error is bounded by the error estimator up to an oscillation term.The computational cost of the approximation and the error problems is considered in terms of size and sparsity of the system matrix.To reduce the computational cost of the error problem,an equivalent error problem is constructed by using diagonalization techniques,which needs to solve only two diagonal linear algebraic systems corresponding to the degree of freedom(d.o.f)to get the error estimator.Numerical experiments are provided to demonstrate the effectiveness and robustness of the a posteriori error estimator.

关键词： Auxiliary subspace techniques diagonalization techniques weak Galerkin A posteriori error estimate Stokes problem.

Polymeric aluminum porphyrin:Controllable synthesis of ultra-low molecular weight CO_(2)-based polyols

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中国化学快报:英文版 2023年第8期34卷 434-438页

作者： Chunwei Zhuo Han Cao xiaoshen wang Shunjie Liu Xianhong wang Key Laboratory of Polymer Ecomaterials Changchun Institute of Applied ChemistryChinese Academy of SciencesChangchun 130022China University of Science and Technology of China Hefei 230026China

Carbon dioxide-based polyols with ultra-low molecular weight(ULMW,Mn<1000 g/mol)are emergent polyurethane precursors with economic and environmental benefits.However,the lack of effective proton-tolerant catalytic systems limits the development of this field.In this work,the polymeric aluminum porphyrin catalyst(PAPC)system was applied to the copolymerization of CO_(2)and propylene oxide,where sebacic acid,bisphenol A,poly(ethylene glycol),and water were used as chain transfer agents to achieve the controlled synthesis of CO_(2)-polyols.The molecular weight of the resulting CO_(2)-polyols could be facilely regulated in the range of 400–930 g/mol at low catalyst loadings,fully demonstrating its catalytic advantages of high activity,high product selectivity,and excellent proton tolerance of PAPC.Meanwhile,the catalytic efficiency of PAPC could reach up to 2.1–5.2 kg/g under organic CTA conditions,even reaching 1.9 kg/g using water as the CTA.The cPC content could be controlled within 1.0 wt%under the optimized conditions,indicating the excellent controllability of the PAPC system.ULMW CO_(2)-polyols combines the advantages of low viscosity(∼3000 mPa s at 25°C),low glass transition temperature(∼−73°C),and high carbonate unit content(∼40%),which is important for the development of high-performance polyurethanes.

关键词： CO_(2)-polyols Polymeric catalyst Aluminum porphyrin Ultra-low molecular weight Proton tolerance

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A stabilizer-free C;weak Galerkin method for the biharmonic equations

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Science China Mathematics 2022年

作者： Peng Zhu Shenglan Xie xiaoshen wang College of Data Science Jiaxing University College of Information Engineering Jiaxing Nanhu University Department of Mathematics University of Arkansas at Little Rock

In this article,we present and analyze a stabilizer-free C;weak Galerkin （SF-C0WG） method for solving the biharmonic problem.The SF-C0WG method is formulated in terms of cell unknowns which are C;continuous piecewise polynomials of degree k+2 with k≥0 and in terms of face unknowns which are discontinuous piecewise polynomials of degree k+1.The formulation of this SF-C0WG method is without the stabilized or penalty term and is as simple as the C;conforming finite element scheme of the biharmonic problem.Optimal order error estimates in a discrete H;-like norm and the H;norm for k≥0 are established for the corresponding WG finite element solutions.Error estimates in the L;norm are also derived with an optimal order of convergence for k>0 and sub-optimal order of convergence for k=0.Numerical experiments are shown to confirm the theoretical results.

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On the superconvergence of a WG method for the elliptic problem with variable coefficients

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Science China Mathematics 2023年

作者： Junping wang xiaoshen wang Xiu Ye Shangyou Zhang Peng Zhu Department of Mathematics University of Arkansas at Little Rock Department of Mathematical Sciences University of Delaware Division of Mathematical Sciences National Science Foundation College of Data Science Jiaxing University

This article extends a recently developed superconvergence result for weak Galerkin (WG)approximations for modeling partial differential equations from constant coefficients to variable coefficients.This superconvergence features a rate that is two-order higher than the optimal-order error estimates in the usual energy and L2norms.The extension from constant to variable coefficients for the modeling equations is highly non-trivial.The underlying technical analysis is based on the use of a sequence of projections and decompositions.Numerical results are presented to confirm the superconvergence theory for second-order elliptic problems with variable coefficients.

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FINITE ELEMENT METHODS FOR THE NAVIER-STOKES EQUATIONS BY H(div)ELEMENTS

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Journal of Computational Mathematics 2008年第3期26卷 410-436页

作者： Junping wang xiaoshen wang Xiu Ye Division of Mathematical Sciences National Science Foundation Arlington VA 22230 USA Department of Mathematics University of Arkansas at Little Rock Little Rock AK 72204 USA

We derived and analyzed a new numerical scheme for the Navier-Stokes equations by using H（div） conforming finite elements. A great deal of effort was given to an establishment of some Sobolev-type inequalities for piecewise smooth functions. In particular, the newly derived Sobolev inequalities were employed to provide a mathematical theory for the H（div） finite element scheme. For example, it was proved that the new finite element scheme has solutions which admit a certain boundedness in terms of the input data. A solution uniqueness was also possible when the input data satisfies a certain smallness condition. Optimal-order error estimates for the corresponding finite element solutions were established in various Sobolev norms. The finite element solutions from the new scheme feature a full satisfaction of the continuity equation which is highly demanded in scientific computing.

关键词： Finite element methods Navier-Stokes equations CFD

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A MODIFIED WEAK GALERKIN FINITE ELEMENT METHOD FOR SOBOLEV EQUATION

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Journal of Computational Mathematics 2015年第3期33卷 307-322页

作者： Fuzheng Gao xiaoshen wang School of Mathematics Shandong University Jinan Shandong 250100 China School of Materials Science and Engineering Shandong University Jinan Shandong China Department of Mathematics University of Arkansas at Little Rock 2801 S. University Avenue Little Rock AR 72204 USA.

For Sobolev equation, we present a new numerical scheme based on a modified weak Galerkin finite element method, in which differential operators are approximated by weak forms through the usual integration by parts. In particular, the numerical method allows the use of discontinuous finite element functions and arbitrary shape of element. Optimal order error estimates in discrete H^1 and L^2 norms are established for the corresponding modified weak Galerkin finite element solutions. Finally, some numerical results are given to verify theoretical results.

关键词： Galerkin FEMs Sobolev equation Discrete weak gradient Modified weak Galerkin Error estimate

HOMOTOPY CONTINUATION METHODS FOR STOCHASTIC TWO-POINT BOUNDARY VALUE PROBLEMS DRIVEN BY ADDITIVE NOISES

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Journal of Computational Mathematics 2014年第6期32卷 630-642页

作者： Yanzhao Cao Pengwang xiaoshen wang Department of Mathematics ~ Statistics Auburn University Auburn AL 368~9-5168 USA and School of Mathematics Jilin University Changchun 130012 China School of Mathematics Jilin University Changchun 130012 China Department of Mathematics and Statistics University of Arkansas at Little Rock Little Rock AR 72204 USA

In this paper the homotopy continuation method for stochastic two-point boundary value problems driven by additive noises is studied. The existence of the solution of the homotopy equation is proved. Numerical schemes are constructed and error estimates are obtained. Numerical experiments demonstrate the effectiveness of the homotopy continu- ation method over other commonly used methods such as the shooting method.

关键词： Stochastic differential equations Homotopy method Shooting method.

Hybridized weak Galerkin finite element method for linear elasticity problem in mixed form

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Frontiers of Mathematics in China 2018年第5期13卷 1121-1140页

作者： Ruishu wang xiaoshen wang Kai ZHANG Qian ZHOU School of Mathematics Jilin University Changchun 130012 China Department of Mathematics and Statistics University of Arkansas at Little RockLittle Rock AR 72204 US

A hybridization technique is applied to the weak Galerkin finite element method （WGFEM） for solving the linear elasticity problem in mixed form. An auxiliary function, the Lagrange multiplier defined on the boundary of elements, is introduced in this method. Consequently, the computational costs are much lower than the standard WGFEM. Optimal order error estimates are presented for the approximation scheme. Numerical results are provided to verify the theoretical results.

关键词： Linear elasticity weak Galerkin （WG） hybridization technique mixed finite element method