A novel conforming discontinuous Galerkin(CDG)finite element method is introduced for the Poisson equation on rectangular meshes.This CDG method with discontinuous P_(k)(k≥1)elements converges to the true solution tw...A novel conforming discontinuous Galerkin(CDG)finite element method is introduced for the Poisson equation on rectangular meshes.This CDG method with discontinuous P_(k)(k≥1)elements converges to the true solution two orders above the continuous finite element counterpart.Superconvergence of order two for the CDG finite element solution is proved in an energy norm and the L^(2)norm.A local post-process is defined which lifts a P_(k)CDG solution to a discontinuous P_(k+2)solution.It is proved that the lifted P_(k+2)solution converges at the optimal order.The numerical tests illustrate the theoretic findings.展开更多
A family of conforming finite elements are designed on rectangular grids for solving the Reissner-Mindlin plate equation.The rotation is approximated by C^(1)-Q_(k+1)in one direction and C^(0)-Q_(k)in the other direct...A family of conforming finite elements are designed on rectangular grids for solving the Reissner-Mindlin plate equation.The rotation is approximated by C^(1)-Q_(k+1)in one direction and C^(0)-Q_(k)in the other direction finite elements.The displacement is approximated by C^(1)-Q_(k+1,k+1).The method is locking-free without using any projection/reduction operator.Theoretical proof and numerical confirmation are presented.展开更多
This paper is to study the convergence and superconvergence of rectangular finite elements under anisotropic meshes. By using of the orthogonal expansion method, an anisotropic Lagrange interpolation is presented. The...This paper is to study the convergence and superconvergence of rectangular finite elements under anisotropic meshes. By using of the orthogonal expansion method, an anisotropic Lagrange interpolation is presented. The family of Lagrange rectangular elements with all the possible shape function spaces are considered, which cover the Intermediate families, Tensor-product families and Serendipity families. It is shown that the anisotropic interpolation error estimates hold for any order Sobolev norm. We extend the convergence and superconvergence result of rectangular finite elements to arbitrary rectangular meshes in a unified way.展开更多
In this paper,several new energy identities of metamaterial Maxwell’s equations with the perfectly electric conducting(PEC)boundary condition are proposed and proved.These new energy identities are different from the...In this paper,several new energy identities of metamaterial Maxwell’s equations with the perfectly electric conducting(PEC)boundary condition are proposed and proved.These new energy identities are different from the Poynting theorem.By using these new energy identities,it is proved that the Yee scheme on non-uniform rectangular meshes is stable in the discrete L2 and H1 norms when the Courant-Friedrichs-Lewy(CFL)condition is satisfied.Numerical experiments in twodimension(2D)and 3D are carried out and confirm our analysis,and the superconvergence in the discrete H1 norm is found.展开更多
New superconvergent structures are proposed and analyzed for the finite volume element(FVE)method over tensorial meshes in general dimension d(for d≥2);we call these orthogonal superconvergent structures.In this fram...New superconvergent structures are proposed and analyzed for the finite volume element(FVE)method over tensorial meshes in general dimension d(for d≥2);we call these orthogonal superconvergent structures.In this framework,one has the freedom to choose the superconvergent points of tensorial k-order FVE schemes(for k≥3).This flexibility contrasts with the superconvergent points(such as Gauss points and Lobatto points)for current FE schemes and FVE schemes,which are fixed.The orthogonality condition and the modified M-decomposition(MMD)technique that are developed over tensorial meshes help in the construction of proper superclose functions for the FVE solutions and in ensuring the new superconvergence properties of the FVE schemes.Numerical experiments are provided to validate our theoretical results.展开更多
文摘A novel conforming discontinuous Galerkin(CDG)finite element method is introduced for the Poisson equation on rectangular meshes.This CDG method with discontinuous P_(k)(k≥1)elements converges to the true solution two orders above the continuous finite element counterpart.Superconvergence of order two for the CDG finite element solution is proved in an energy norm and the L^(2)norm.A local post-process is defined which lifts a P_(k)CDG solution to a discontinuous P_(k+2)solution.It is proved that the lifted P_(k+2)solution converges at the optimal order.The numerical tests illustrate the theoretic findings.
文摘A family of conforming finite elements are designed on rectangular grids for solving the Reissner-Mindlin plate equation.The rotation is approximated by C^(1)-Q_(k+1)in one direction and C^(0)-Q_(k)in the other direction finite elements.The displacement is approximated by C^(1)-Q_(k+1,k+1).The method is locking-free without using any projection/reduction operator.Theoretical proof and numerical confirmation are presented.
文摘This paper is to study the convergence and superconvergence of rectangular finite elements under anisotropic meshes. By using of the orthogonal expansion method, an anisotropic Lagrange interpolation is presented. The family of Lagrange rectangular elements with all the possible shape function spaces are considered, which cover the Intermediate families, Tensor-product families and Serendipity families. It is shown that the anisotropic interpolation error estimates hold for any order Sobolev norm. We extend the convergence and superconvergence result of rectangular finite elements to arbitrary rectangular meshes in a unified way.
基金supported by the National Natural Science Foundation of China Grant No.11671233 and the Shandong Provincial Science and Technology Development Program Grant No.2018GGX101036.
文摘In this paper,several new energy identities of metamaterial Maxwell’s equations with the perfectly electric conducting(PEC)boundary condition are proposed and proved.These new energy identities are different from the Poynting theorem.By using these new energy identities,it is proved that the Yee scheme on non-uniform rectangular meshes is stable in the discrete L2 and H1 norms when the Courant-Friedrichs-Lewy(CFL)condition is satisfied.Numerical experiments in twodimension(2D)and 3D are carried out and confirm our analysis,and the superconvergence in the discrete H1 norm is found.
基金This work is supported in part by the National Natural Science Foundation of China under grants 11701211,11871092,12131005the China Postdoctoral Science Foundation under grant 2021M690437。
文摘New superconvergent structures are proposed and analyzed for the finite volume element(FVE)method over tensorial meshes in general dimension d(for d≥2);we call these orthogonal superconvergent structures.In this framework,one has the freedom to choose the superconvergent points of tensorial k-order FVE schemes(for k≥3).This flexibility contrasts with the superconvergent points(such as Gauss points and Lobatto points)for current FE schemes and FVE schemes,which are fixed.The orthogonality condition and the modified M-decomposition(MMD)technique that are developed over tensorial meshes help in the construction of proper superclose functions for the FVE solutions and in ensuring the new superconvergence properties of the FVE schemes.Numerical experiments are provided to validate our theoretical results.
