Error Estimations, Error Computations, and Convergence Rates in FEM for BVPs
Error Estimations, Error Computations, and Convergence Rates in FEM for BVPs作者机构：Department of Mechanical Engineering Univeristy of Kansas Lawrence KS USA Department of Mechanical Engineering Texas A&M University College Station TX USA
出 版 物：《Applied Mathematics》 (应用数学（英文）)
年 卷 期：2016年第7卷第12期
页 面：1359-1407页
学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学]
主 题：Finite Element Error Estimation Convergence Rate A Priori A Posteriori BVP Variationally Consistent Integral Form Variationally Inconsistent Integral Form Differential Operator Classification Self-Adjoint Non-Self-Adjoint Nonlinear
摘 要：This paper presents derivation of a priori error estimates and convergence rates of finite element processes for boundary value problems (BVPs) described by self adjoint, non-self adjoint, and nonlinear differential operators. A posteriori error estimates are discussed in context with local approximations in higher order scalar product spaces. A posteriori error computational framework (without the knowledge of theoretical solution) is presented for all BVPs regardless of the method of approximation employed in constructing the integral form. This enables computations of local errors as well as the global errors in the computed finite element solutions. The two most significant and essential aspects of the research presented in this paper that enable all of the features described above are: 1) ensuring variational consistency of the integral form(s) resulting from the methods of approximation for self adjoint, non-self adjoint, and nonlinear differential operators and 2) choosing local approximations for the elements of a discretization in a subspace of a higher order scalar product space that is minimally conforming, hence ensuring desired global differentiability of the approximations over the discretizations. It is shown that when the theoretical solution of a BVP is analytic, the a priori error estimate (in the asymptotic range, discussed in a later section of the paper) is independent of the method of approximation or the nature of the differential operator provided the resulting integral form is variationally consistent. Thus, the finite element processes utilizing integral forms based on different methods of approximation but resulting in VC integral forms result in the same a priori error estimate and convergence rate. It is shown that a variationally consistent (VC) integral form has best approximation property in some norm, conversely an integral form with best approximation property in some norm is variationally consistent. That is best approximation prop
维普期刊数据库