Year: 2020
Author: Carsten Carstensen, Sophie Puttkammer
Journal of Computational Mathematics, Vol. 38 (2020), Iss. 1 : pp. 142–175
Abstract
Optimal convergence rates of adaptive finite element methods are well understood in terms of the axioms of adaptivity. One key ingredient is the discrete reliability of a residual-based a posteriori error estimator, which controls the error of two discrete finite element solutions based on two nested triangulations. In the error analysis of nonconforming finite element methods, like the Crouzeix-Raviart or Morley finite element schemes, the difference of the piecewise derivatives of discontinuous approximations to the distributional gradients of global Sobolev functions plays a dominant role and is the object of this paper. The nonconforming interpolation operator, which comes naturally with the definition of the aforementioned nonconforming finite element in the sense of Ciarlet, allows for stability and approximation properties that enable direct proofs of the reliability for the residual that monitors the equilibrium condition. The novel approach of this paper is the suggestion of a right-inverse of this interpolation operator in conforming piecewise polynomials to design a nonconforming approximation of a given coarse-grid approximation on a refined triangulation. The results of this paper allow for simple proofs of the discrete reliability in any space dimension and multiply connected domains on general shape-regular triangulations beyond newest-vertex bisection of simplices. Particular attention is on optimal constants in some standard discrete estimates listed in the appendices.
You do not have full access to this article.
Already a Subscriber? Sign in as an individual or via your institution
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jcm.1908-m2018-0174
Journal of Computational Mathematics, Vol. 38 (2020), Iss. 1 : pp. 142–175
Published online: 2020-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 34
Keywords: Discrete reliability Nonconforming finite element method Conforming companion Morley Crouzeix-Raviart Explicit constants Axioms of adaptivity.