Superconvergence Analysis of C^(m)Finite Element Methods for Fourth-Order Elliptic Equations I:One Dimensional Case  

作　　者：Waixiang Cao Lueling Jia Zhimin Zhang 

机构地区：[1]School of Mathematical Sciences,Beijing Normal University,Beijing 100875,China [2]School of Mathematics and Statistics,Shandong Normal University,Jinan,250014,China [3]Beijing Computational Science Research Center,Beijing 100193,China [4]Department of Mathematics,Wayne State University,Detroit,MI 48202,USA

出　　处：《Communications in Computational Physics》2023年第5期1466-1508,共43页计算物理通讯（英文）

基　　金：This work is supported in part by the National Natural Science Foundation of China under grants No.12271049,12101035,12131005,U1930402.

摘　　要：In this paper,we study three families of C^(m)(m=0,1,2)finite element methods for one dimensional fourth-order equations.They include C^(0)and C1 Galerkin methods and a C^(2)-C^(0)Petrov-Galerkin method.Existence,uniqueness and optimal error estimates of the numerical solution are established.A unified approach is proposed to study the superconvergence property of these methods.We prove that,for kth-order elements,the C^(0)and C1 finite element solutions and their derivative are superconvergent with rate h2k−2(k≥3)at all mesh nodes;while the solution of the C^(2)-C^(0)Petrov-Galerkin method and its first-and second-order derivatives are superconvergent with rate h^(2k−4)(k≥5)at all mesh nodes.Furthermore,interior superconvergence points for the l-th(0≤l≤m+1)derivate approximations are also discovered,which are identified as roots of special Jacobi polynomials,Lobatto points,and Gauss points.As a by-product,we prove that the C^(m)finite element solution is superconvergent towards a particular Jacobi projection of the exact solution in the Hl(0≤l≤m+1)norms.All theoretical findings are confirmed by numerical experiments.

关 键 词：C^(m)finite element methods SUPERCONVERGENCE fourth-order elliptic equations 

分 类 号：O241.82[理学—计算数学]