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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

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.

A LEAST SQUARE BASED WEAK GALERKIN FINITE ELEMENT METHOD FOR SECOND ORDER ELLIPTIC EQUATIONS IN NON-DIVERGENCE FORM

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Acta Mathematica Scientia 2020年第5期40卷 1553-1562页

作者： Peng ZHU xiaoshen wang College of Mathematics Physics and Information EngineeringJiaxing UniversityJiaxing 314001China Department of Mathematics and Statistics University of Arkansas at Little RockLittle RockAR 72204USA

This article is devoted to establishing a least square based weak Galerkin method for second order elliptic equations in non-divergence form using a discrete weak Hessian ***,the resulting linear system is symmetric and positive definite,and thus the algorithm is easy to implement and *** analysis in the H2 equivalent norm is established on an arbitrary shape regular polygonal mesh.A superconvergence result is proved when the coefficient matrix is constant or piecewise *** examples are performed which not only verify the theoretical results but also reveal some unexpected superconvergence phenomena.

关键词： least square based weak Galerkin method non-divergence form weak Hessian operator polygonal mesh

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

A hybridized weak Galerkin finite element scheme for the Stokes equations

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Science China Mathematics 2015年第11期58卷 2455-2472页

作者： ZHAI QiLong ZHANG Ran wang xiaoshen Department of Mathematics Jilin University Department of Mathematics University of Arkansas at Little Rock

In this paper a hybridized weak Galerkin(HWG) finite element method for solving the Stokes equations in the primary velocity-pressure formulation is *** WG method uses weak functions and their weak derivatives which are defined as *** functions and weak derivatives can be approximated by piecewise polynomials with various *** combination of polynomial spaces leads to different WG finite element methods,which makes WG methods highly flexible and efficient in practical computation.A Lagrange multiplier is introduced to provide a numerical approximation for certain derivatives of the exact *** this new feature,the HWG method can be used to deal with jumps of the functions and their flux *** order error estimates are established for the corresponding HWG finite element approximations for both primal variables and the Lagrange multiplier.A Schur complement formulation of the HWG method is derived for implementation *** validity of the theoretical results is demonstrated in numerical tests.

关键词： hybridized weak Galerkin finite element methods weak gradient weak divergence Stokes equation

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

An alternating direction method of multipliers for elliptic equation constrained optimization problem

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Science China Mathematics 2017年第2期60卷 361-378页

作者： ZHANGKai LI JingShi SONG YongCun wang xiaoshen Department of Mathematics Jilin University Department of Mathematics and Statistics University of Arkansas at Little Rock

We propose an alternating direction method of multipliers(ADMM)for solving the state constrained optimization problems governed by elliptic *** unconstrained as well as box-constrained cases of the Dirichlet boundary control,Robin boundary control,and right-hand side control problems are considered *** continuous optimization problems are transformed into discrete optimization problems by the finite element method discretization,then are solved by *** ADMM is an efficient first order algorithm with global convergence,which combines the decomposability of dual ascent with the superior convergence properties of the method of *** shall present exhaustive convergence analysis of ADMM for these different type optimization *** numerical experiments are performed to verify the efficiency of the method.

关键词： ADMM optimal control problems elliptic equation constrained

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

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Science China Mathematics 2024年第8期67卷 1899-1910页

作者： Junping wang xiaoshen wang Xiu Ye Shangyou Zhang Peng Zhu Division of Mathematical Sciences National Science FoundationAlexandriaVA 22314USA Department of Mathematics University of Arkansas at Little RockLittle RockAR 72204USA Department of Mathematical Sciences University of DelawareNewarkDE 19716USA College of Data Science Jiaxing UniversityJiaxing 314001China

This article extends a recently developed superconvergence result for weak Galerkin(WG)approximations for modeling partial differential equations from constant coefficients to variable *** superconvergence features a rate that is two orders higher than the optimal-order error estimates in the usual energy and L^(2)*** extension from constant to variable coefficients for the modeling equations is highly *** underlying technical analysis is based on a sequence of projections and *** results confirm the superconvergence theory for second-order elliptic problems with variable coefficients.

关键词： weak Galerkin finite element methods superconvergence second-order elliptic problems stabilizerfree

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

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Chinese Chemical Letters 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 ***,the lack of effective proton-tolerant catalytic systems limits the development of this *** 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)-*** 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 ***,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 *** cPC content could be controlled within 1.0 wt%under the optimized conditions,indicating the excellent controllability of the PAPC *** 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