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| 1 | A family of 3D H^2-nonconforming tetrahedral finite elements for the biharmonic equation显示文摘In this article,a family of H^2-nonconforming finite elements on tetrahedral grids is constructed for solving the biharmonic equation in 3 D.In the family,the Pl polynomial space is enriched by some high order polynomials for all l≥3 and the corresponding finite element solution converges at the order l-1 in H2 norm.Moreover,the result is improved for two low order cases by using P6 and P7 polynomials to enrich P4 and P5 polynomial spaces,respectively.The error estimate is proved.The numerical results are.provided to confirm the theoretical findings. | Jun Hu Shudan Tian Shangyou Zhang | 2020 | Science China Mathematics2020，63，8： | 2 |
| 2 | AN ERROR ANALYSIS METHOD SPP-BEAM AND A CONSTRUCTION GUIDELINE OF NONCONFORMING FINITE ELEMENTS FOR FOURTH ORDER ELLIPTIC PROBLEMS显示文摘Under two hypotheses of nonconforming finite elements of fourth order elliptic prob lems,we present a side-patchwise projection based error analysis method(SPP-BEAM for short).Such a method is able to avoid both the regularity condition of exact solutions in the classical error analysis method and the complicated bubble function technique in the recent medius error analysis method.In addition,it is universal enough to admit generalizations.Then,we propose a sufficient condition for these hypotheses by imposing a set of in some sense necessary degrees of freedom of the shape function spaces.As an application,we use the theory to design a P3 second order triangular H2 non-conforming element by enriching two P4 bubble functions and,another P4 second order triangular H2 nonconforming finite element,and a P3 second order tetrahedral H2 non-conforming element by enriching eight P4 bubble functions,adding some more degrees of freedom. | Jun Hu Shangyou Zhang | 2020 | Journal of Computational Mathematics2020，38，1： | 2 |
| 3 | A family of mixed finite elements for the biharmonic equations on triangular and tetrahedral grids显示文摘This paper introduces a new family of mixed finite elements for solving a mixed formulation of the biharmonic equations in two and three dimensions.The symmetric stress σ=−∇^(2)u is sought in the Sobolev space H(divdiv,Ω;S)simultaneously with the displacement u in L^(2)(Ω).By stemming from the structure of H(div,Ω;S)conforming elements for the linear elasticity problems proposed by Hu and Zhang(2014),the H(divdiv,Ω;S)conforming finite element spaces are constructed by imposing the normal continuity of divσ on the H(div,Ω;S)conforming spaces of P_(k) symmetric tensors.The inheritance makes the basis functions easy to compute.The discrete spaces for u are composed of the piecewise P_(k−2) polynomials without requiring any continuity.Such mixed finite elements are inf-sup stable on both triangular and tetrahedral grids for k≥3,and the optimal order of convergence is achieved.Besides,the superconvergence and the postprocessing results are displayed.Some numerical experiments are provided to demonstrate the theoretical analysis. | Jun Hu Rui Ma Min Zhang | 2021 | Science China Mathematics2021，64，12： | 0 |
| 4 | Convergence analysis of the rectangular Morley element scheme for second order problem in arbitrary dimensions显示文摘We present the convergence analysis of the rectangular Morley element scheme utilised on the second order problem in arbitrary dimensions. Specifically, we prove that the convergence of the scheme is of O(h) order in energy norm and of O(h^2) order in L^2 norm on general d-rectangular triangulations. Moreover, when the triangulation is uniform, the convergence rate can be of O(h^2) order in energy norm, and the convergence rate in L^2 norm is still of O(h^2) order, which cannot be improved. Numerical examples are presented to demonstrate our theoretical results. | MENG XiangYun YANG XueQin ZHANG Shuo | 2016 | Science China Mathematics2016，59，11： | 0 |
| 5 | An optimal piecewise cubic nonconforming finite element scheme for the planar biharmonic equation on general triangulations显示文摘This paper presents a nonconforming finite element scheme for the planar biharmonic equation,which applies piecewise cubic polynomials(P_(3))and possesses O(h^(2))convergence rate for smooth solutions in the energy norm on general shape-regular triangulations.Both Dirichlet and Navier type boundary value problems are studied.The basis for the scheme is a piecewise cubic polynomial space,which can approximate the H^(4) functions with O(h^(2))accuracy in the broken H^(2) norm.Besides,a discrete strengthened Miranda-Talenti estimate(▽^(2)_(h)·,▽^(2)_(h)·)=(Δh·,Δh·),which is usually not true for nonconforming finite element spaces,is proved.The finite element space does not correspond to a finite element defined with Ciarlet’s triple;however,it admits a set of locally supported basis functions and can thus be implemented by the usual routine.The notion of the finite element Stokes complex plays an important role in the analysis as well as the construction of the basis functions. | Shuo Zhang | 2021 | Science China Mathematics2021，64，11： | 0 |
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