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,uniq...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.展开更多
基金This work is supported in part by the National Natural Science Foundation of China under grants No.12271049,12101035,12131005,U1930402.
文摘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.
