A class of upwind finite volume element method based on tetrahedron partition is put forward for a nonlinear convection diffusion problem. Some techniques, such as calculus of variations, commutating operators and the...A class of upwind finite volume element method based on tetrahedron partition is put forward for a nonlinear convection diffusion problem. Some techniques, such as calculus of variations, commutating operators and the a priori estimate, are adopted. The a priori error estimate in L2-norm and H1-norm is derived to determine the error between the approximate solution and the true solution.展开更多
In this work,a modified weak Galerkin finite element method is proposed for solving second order linear parabolic singularly perturbed convection-diffusion equations.The key feature of the proposed method is to replac...In this work,a modified weak Galerkin finite element method is proposed for solving second order linear parabolic singularly perturbed convection-diffusion equations.The key feature of the proposed method is to replace the classical gradient and divergence operators by the modified weak gradient and modified divergence operators,respectively.We apply the backward finite difference method in time and the modified weak Galerkin finite element method in space on uniform mesh.The stability analyses are presented for both semi-discrete and fully-discrete modified weak Galerkin finite element methods.Optimal order of convergences are obtained in suitable norms.We have achieved the same accuracy with the weak Galerkin method while the degrees of freedom are reduced in our method.Various numerical examples are presented to support the theoretical results.It is theoretically and numerically shown that the method is quite stable.展开更多
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. I...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.展开更多
A weak Galerkin finite element method with stabilization term, which is symmetric, positive definite and parameter free, was proposed to solve parabolic equations by using weakly defined gradient operators over discon...A weak Galerkin finite element method with stabilization term, which is symmetric, positive definite and parameter free, was proposed to solve parabolic equations by using weakly defined gradient operators over discontinuous functions. In this paper, we derive the optimal order error estimate in L2 norm based on dual argument. Numerical experiment is conducted to confirm the theoretical results.展开更多
A conforming discontinuous Galerkinfinite element method was introduced by Ye and Zhang,on simplicial meshes and on polytopal meshes,which has theflexibility of using discontinuous approximation and an ultra simple form...A conforming discontinuous Galerkinfinite element method was introduced by Ye and Zhang,on simplicial meshes and on polytopal meshes,which has theflexibility of using discontinuous approximation and an ultra simple formulation.The main goal of this paper is to improve the above discontinuous Galerkinfinite element method so that it can handle nonhomogeneous Dirichlet boundary conditions effectively.In addition,the method has been generalized in terms of approximation of the weak gradient.Error estimates of optimal order are established for the correspond-ing discontinuousfinite element approximation in both a discrete H1 norm and the L2 norm.Numerical results are presented to confirm the theory.展开更多
文摘A class of upwind finite volume element method based on tetrahedron partition is put forward for a nonlinear convection diffusion problem. Some techniques, such as calculus of variations, commutating operators and the a priori estimate, are adopted. The a priori error estimate in L2-norm and H1-norm is derived to determine the error between the approximate solution and the true solution.
基金supported in part by National Natural Science Foundation of China (No.11871038).
文摘In this work,a modified weak Galerkin finite element method is proposed for solving second order linear parabolic singularly perturbed convection-diffusion equations.The key feature of the proposed method is to replace the classical gradient and divergence operators by the modified weak gradient and modified divergence operators,respectively.We apply the backward finite difference method in time and the modified weak Galerkin finite element method in space on uniform mesh.The stability analyses are presented for both semi-discrete and fully-discrete modified weak Galerkin finite element methods.Optimal order of convergences are obtained in suitable norms.We have achieved the same accuracy with the weak Galerkin method while the degrees of freedom are reduced in our method.Various numerical examples are presented to support the theoretical results.It is theoretically and numerically shown that the method is quite stable.
基金Acknowldgements. The authors would like to express their sincere thanks to the editor and referees for their very helpful comments and suggestions, which greatly improved the quality of this paper. We also would like to thank Dr. Xiu Ye for useful discussions. The first author＇s research is partially supported by the Natural Science Foundation of Shandong Province of China grant ZR2013AM023, the Project Funded by China Postdoctoral Science Foundation no. 2014M560547, the Fundamental Research Funds of Shandong University no. 2015JC019, and NSAF no. U1430101.
文摘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.
基金The authors thank the referees and the editor for their invaluable comments and suggestions which have helped to improved the paper greatly. Also, the authors would like to thank Prof. Xiu Ye for useful discussions and Dr. Paul Scott for helpful revision. This work is done when the first author is visiting Department of Mathematics and Statistics, University of Arkansas at Little Rock under supported by the State Scholarship Fund from the China Scholarship Council. The first author＇s research is partially supported by the Natural Science Foundation of Shandong Province of China grant ZR2013AM023, ZR2012AM019.
文摘A weak Galerkin finite element method with stabilization term, which is symmetric, positive definite and parameter free, was proposed to solve parabolic equations by using weakly defined gradient operators over discontinuous functions. In this paper, we derive the optimal order error estimate in L2 norm based on dual argument. Numerical experiment is conducted to confirm the theoretical results.
基金supported in part by National Natural Science Foundation of China(NSFC No.11871038)supported in part by National Science Foundation Grant DMS-1620016.
文摘A conforming discontinuous Galerkinfinite element method was introduced by Ye and Zhang,on simplicial meshes and on polytopal meshes,which has theflexibility of using discontinuous approximation and an ultra simple formulation.The main goal of this paper is to improve the above discontinuous Galerkinfinite element method so that it can handle nonhomogeneous Dirichlet boundary conditions effectively.In addition,the method has been generalized in terms of approximation of the weak gradient.Error estimates of optimal order are established for the correspond-ing discontinuousfinite element approximation in both a discrete H1 norm and the L2 norm.Numerical results are presented to confirm the theory.
