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Robust Falk-Neilan Finite Elements for the Reissner-Mindlin Plate

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Communications on Applied Mathematics and Computation 2023年第4期5卷 1697-1712页

作者： shangyou zhang Department of Mathematical Sciences University of DelawareNewarkDE19716USA

The family of Falk-Neilan P_(k)finite elements,combined with the Argyris P_(k+1)finite elements,solves the Reissner-Mindlin plate equation quasi-optimally and locking-free,on triangular *** method is truly conforming ... 详细信息

关键词： Locking free Reissner-Mindlin equation Finite element Triangular grids

AN ERROR ANALYSIS METHOD SPP-BEAM AND A CONSTRUCTION GUIDELINE OF NONCONFORMING FINITE ELEMENTS FOR FOURTH ORDER ELLIPTIC PROBLEMS

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Journal of Computational Mathematics 2020年第1期38卷 195-222页

作者： Jun Hu shangyou zhang LMAM and School of Mathematical Sciences Peking UniversityBeijing 100871China Department of Mathematical Sciences University of DelawareNewarkDE 19716USA

Under two hypotheses of nonconforming finite elements of fourth order elliptic problems,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 *** addition,it is universal enough to admit ***,we propose a sufficient condition for these hypotheses by imposing a set of in some sense necessary degrees of freedom of the shape function *** 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.

关键词： Nonconforming finite element A priori error analysis Biharmonic equation

Constructing Order Two Superconvergent WG Finite Elements on Rectangular Meshes

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Numerical Mathematics(Theory,Methods and Applications) 2023年第1期16卷 230-241页

作者： Xiu Ye shangyou zhang Department of Mathematics University of Arkansas at Little RockLittle RockAR 72204USA Department of Mathematical Sciences University of DelawareNewarkDE 19716USA

In this paper,we introduce a stabilizer free weak Galerkin(SFWG)finite element method for second order elliptic problems on rectangular *** a special weak Gradient space,an order two superconvergence for the SFWG finite element solution is obtained,in both L 2 and H1 norms.A local post-process lifts such a Pk weak Galerkin solution to an optimal order Pk+2 *** numerical results confirm the theory.

关键词： Finite element weak Galerkin method stabilizer free rectangular mesh

On the Full C_(1)-Q_(k) Finite Element Spaces on Rectangles and Cuboids

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Advances in Applied Mathematics and Mechanics 2010年第6期2卷 701-721页

作者： shangyou zhang Department of Mathematical Sciences University of DelawareNewarkDelaware 19716-2553USA

We study the extensions of the Bogner-Fox-Schmit element to the whole family of Q_(k) continuously differentiable finite elements on rectangular grids,for all k≥3,in 2D and *** show that the newly defined C_(1) spaces are maximal in the sense that they contain all C_(1)-Q_(k) functions of piecewise *** give examples of other extensions of C_(1)-Q_(k) *** result is consistent with the Strang’s conjecture(restricted to the quadrilateral grids in 2D and 3D).Some numerical results are provided on the family of C_(1) elements solving the biharmonic equation.

关键词： Differentiable finite element biharmonic equation Bogner-Fox-Schmit rectangle quadrilateral element hexahedral element Strang’s conjecture

Four-Order Superconvergent Weak Galerkin Methods for the Biharmonic Equation on Triangular Meshes

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Communications on Applied Mathematics and Computation 2023年第4期5卷 1323-1338页

作者： Xiu Ye shangyou zhang Department of Mathematics University of Arkansas at Little RockLittle RockAR72204USA Department of Mathematical Sciences University of DelawareNewarkDE19716USA

A stabilizer-free weak Galerkin(SFWG)finite element method was introduced and analyzed in Ye and zhang(SIAM ***.58:2572–2588,2020)for the biharmonic equation,which has an ultra simple finite element *** work is a continuation of our investigation of the SFWG method for the biharmonic *** new SFWG method is highly accurate with a convergence rate of four orders higher than the optimal order of convergence in both the energy norm and the L^(2)norm on triangular *** new method also keeps the formulation that is symmetric,positive definite,and ***-order superconvergence error estimates are proved for the corresponding SFWG finite element solutions in a discrete H^(2)*** of four orders in the L^(2)norm is also derived for k≥3,where k is the degree of the approximation *** postprocessing is proved to lift a P_(k)SFWG solution to a P_(k+4)solution elementwise which converges at the optimal *** examples are tested to verify the theor ies.

关键词： Finite element Weak Hessian Weak Galerkin(WG) Biharmonic equation Triangular mesh

A Cubic H3-Nonconforming Finite Element

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Communications on Applied Mathematics and Computation 2019年第1期1卷 81-100页

作者： Jun Hu shangyou zhang LMAM and School of Mathematical Sciences Peking UniversityBeijing 100871.China Department of Mathematical Sciences University of DelawareNewarkDE 19716USA

The lowest degree of polynomial for a finite element to solve a 2^th-order elliptic equation is *** Morley element is such a finite element,of polynomial degree 2,for solving a fourth-order biharmonic *** design a cubic H3-nonconforming macro-element on two-dimensional triangular grids,solving a sixth-order tri-harmonic *** also write down explicitly the 12 basis functions on each macro-element.A convergence theory is established and verified by numerical tests.

关键词： Nonconforming macro-element Minimum element Tri-harmonic equation

ON THE OPTIMAL CONVERGENCE RATE OF A ROBIN-ROBIN DOMAIN DECOMPOSITION METHOD

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Journal of Computational Mathematics 2014年第4期32卷 456-475页

作者： Wenbin Chen Xuejun Xu shangyou zhang School of Mathematical Sciences Fudan University Shanghai 200337 China LSEC ICMSEC Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing 100190 China Department of Mathematical Sciences University of Delaware Newark Delaware 18716 U.S.A.

In this work, we solve a long-standing open problem： Is it true that the convergence rate of the Lions＇ Robin-Robin nonoverlapping domain decomposition （DD） method can be constant, independent of the mesh size h？... 详细信息

关键词： Finite element Robin-Robin domain decomposition Convergence rate.

A family of 3D H^2-nonconforming tetrahedral finite elements for the biharmonic equation

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Science China Mathematics 2020年第8期63卷 1505-1522页

作者： Jun Hu Shudan Tian shangyou zhang LMAM and School of Mathematical Sciences Peking UniversityBeijing 100871China Department of Mathematical Sciences University of DelawareNewarkDE 19716USA

In this article,a family of H^2-nonconforming finite elements on tetrahedral grids is constructed for solving the biharmonic equation in 3 *** 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 ***,the result is improved for two low order cases by using P6 and P7 polynomials to enrich P4 and P5 polynomial spaces,*** error estimate is *** numerical results *** to confirm the theoretical findings.

关键词： H^2-nonconforming element finite element method biharmonic problem tetrahedral grid

A weak Galerkin-mixed finite element method for the Stokes-Darcy problem

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Science China Mathematics 2021年第10期64卷 2357-2380页

作者： Hui Peng Qilong Zhai Ran zhang shangyou zhang School of Mathematics Jilin UniversityChangchun 130012China Department of Mathematical Sciences University of DelawareNewarkDE 19716USA

In this paper,we propose a new numerical scheme for the coupled Stokes-Darcy model with the Beavers-Joseph-Saffman interface *** use the weak Galerkin method to discretize the Stokes equation and the mixed finite element method to discretize the Darcy equation.A discrete inf-sup condition is proved and the optimal error estimates are also *** experiments validate the theoretical analysis.

关键词： weak Galerkin finite element methods mixed finite element methods weak gradient coupled Stokes-Darcy problems