An optimal piecewise cubic nonconforming finite element scheme for the planar biharmonic equation on general triangulations  

作　　者：Shuo Zhang 

机构地区：[1]LSEC,Institute of Computational Mathematics and Scientific/Engineering Computing,Academy of Mathematics and Systems Science,Chinese Academy of Sciences,Beijing 100190,China [2]School of Mathematical Sciences,University of Chinese Academy of Sciences,Beijing 100049,China

出　　处：《Science China Mathematics》2021年第11期2579-2602,共24页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China(Grant Nos.11871465 and 11471026);the Strategic Priority Research Program of Chinese Academy of Sciences(Grant No.XDB 41000000)。

摘　　要：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.

关 键 词：biharmonic equation optimal cubic finite element scheme general triangulation discretized Stokes complex discrete strengthened Miranda-Talenti estimate 

分 类 号：O241.82[理学—计算数学]