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

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 Modified Tikhonov Regularization Method for a Cauchy Problem of the biharmonic Equation

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Journal of Mathematical Research with Applications 2024年第3期44卷 359-386页

作者： Fan YANG Jianming XU Xiaoxiao LI School of Science Lanzhou University of Technology

In this paper, the Cauchy problem of biharmonic equation is considered. This problem is ill-posed, i.e., the solution(if exists) does not depend on the measurable data. Firstly, we give the conditional stability result under the a priori bound assumption for the exact solution.Secondly, a modified Tikhonov regularization method is used to solve this ill-posed problem.Under the a priori and the a posteriori regularization parameter choice rule, the error estimates between the regularization solutions and the exact solution are obtained. Finally, some numerical examples are presented to verify that our method is effective.

关键词： biharmonic equations inverse problem Cauchy problem Tikhonov regularization method

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Mathematic Model and Analytic Solution for a Cylinder Subject to Exponential Function

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Chinese Journal of Mechanical Engineering 2009年第4期22卷 587-593页

作者： LIU Wen SHAN Rui College of Science Yanshan University Qinhuangdao 066004 China

Hollow cylinders are widely used in spacecraft, rockets, weapons, metallurgy, materials, and mechanical manufacturing industries, and so on, hydraulic bulging roll cylinder and hydraulic press work all belong to hollow cylinders. However, up till now, the solution of the cylinder subjected to the pressures in the three-dimensional space is still at the stage of the analytical solution to the normal pressure or the approximate solution to the variable pressure by numerical method. The analytical solution to the variable pressure of the cylinder has not yet made any breakthrough in theory and can not meet accurate theoretical analysis and calculation requirements of the cylindrical in Engineering. In view of their importance, the precision calculation and theoretical analysis are required to investigate on engineering. A stress function which meets both the biharmonic equations and boundary conditions is constructed in the three-dimensional space. Furthermore, the analytic solution of a hollow cylinder subjected to exponential function distributed variable pressure on its inner and outer surfaces is deduced. By controlling the pressure subject to exponential function distributed variable pressure in the hydraulic bulging roller without any rolling load, using a static tester to record the strain supported hydraulic bulging roll, and comparing with the theoretical calculation, the experimental test result has a higher degree of agreement with the theoretical calculation. Simultaneously, the famous Lam6 solution can be deduced when given the unlimited length of cylinder along the axis. The analytic solution paves the way for the mathematic building and solution of hollow cylinder with randomly uneven pressure.

关键词： cylinder analytic solution distributed pressure in the form of exponential function stress function biharmonic equations

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A Modified Weak Galerkin Finite Element Method for the biharmonic Equation on Polytopal Meshes

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Communications on Applied Mathematics and Computation 2021年第1期3卷 91-105页

作者： Ming Cui Xiu Ye Shangyou Zhang College of Applied Science Beijing University of TechnologyBeijingChina Department of Mathematics University of Arkansas at Little RockLittle RockAR 72204USA Department of Mathematical Sciences University of DelawareNewarkDE 19716USA

A modified weak Galerkin(MWG)finite element method is developed for solving the biharmonic equation.This method uses the same finite element space as that of the discontinuous Galerkin method,the space of discontinuous polynomials on polytopal meshes.But its formulation is simple,symmetric,positive definite,and parameter independent,without any of six inter-element face-integral terms in the formulation of the discontinuous Galerkin method.Optimal order error estimates in a discrete H2 norm are established for the corresponding finite element solutions.Error estimates in the L^(2)norm are also derived with a sub-optimal order of convergence for the lowest-order element and an optimal order of convergence for all high-order of elements.The numerical results are presented to confirm the theory of convergence.

关键词： Finite element methods Weak Laplacian biharmonic equations Polytopal meshes

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