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How to Prove the Discrete Reliability for Nonconforming Finite Element Methods

How to Prove the Discrete Reliability for Nonconforming Finite Element Methods
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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.

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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.

Author Details

Carsten Carstensen

Sophie Puttkammer

  1. Nonconforming Virtual Elements for the Biharmonic Equation with Morley Degrees of Freedom on Polygonal Meshes

    Carstensen, Carsten | Khot, Rekha | Pani, Amiya K.

    SIAM Journal on Numerical Analysis, Vol. 61 (2023), Iss. 5 P.2460

    https://doi.org/10.1137/22M1496761 [Citations: 2]

  2. Adaptive hybrid high-order method for guaranteed lower eigenvalue bounds

    Carstensen, Carsten | Gräßle, Benedikt | Tran, Ngoc Tien

    Numerische Mathematik, Vol. 156 (2024), Iss. 3 P.813

    https://doi.org/10.1007/s00211-024-01407-w [Citations: 1]

  3. Convergent adaptive hybrid higher-order schemes for convex minimization

    Carstensen, Carsten | Tran, Ngoc Tien

    Numerische Mathematik, Vol. 151 (2022), Iss. 2 P.329

    https://doi.org/10.1007/s00211-022-01284-1 [Citations: 1]

  4. Unified a priori analysis of four second-order FEM for fourth-order quadratic semilinear problems

    Carstensen, Carsten | Nataraj, Neela | Remesan, Gopikrishnan C. | Shylaja, Devika

    Numerische Mathematik, Vol. 154 (2023), Iss. 3-4 P.323

    https://doi.org/10.1007/s00211-023-01356-w [Citations: 2]

  5. A Priori and a Posteriori Error Analysis of the Crouzeix–Raviart and Morley FEM with Original and Modified Right-Hand Sides

    Carstensen, Carsten | Nataraj, Neela

    Computational Methods in Applied Mathematics, Vol. 21 (2021), Iss. 2 P.289

    https://doi.org/10.1515/cmam-2021-0029 [Citations: 12]

  6. Convergence of Adaptive Crouzeix–Raviart and Morley FEM for Distributed Optimal Control Problems

    Dond, Asha K. | Nataraj, Neela | Nayak, Subham

    Computational Methods in Applied Mathematics, Vol. 24 (2024), Iss. 3 P.599

    https://doi.org/10.1515/cmam-2023-0083 [Citations: 1]

  7. Adaptive Morley FEM for the von Kármán Equations with Optimal Convergence Rates

    Carstensen, Carsten | Nataraj, Neela

    SIAM Journal on Numerical Analysis, Vol. 59 (2021), Iss. 2 P.696

    https://doi.org/10.1137/20M1335613 [Citations: 10]

  8. Morley finite element method for the von Kármán obstacle problem

    Carstensen, Carsten | Gaddam, Sharat | Nataraj, Neela | Pani, Amiya K. | Shylaja, Devika

    ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 55 (2021), Iss. 5 P.1873

    https://doi.org/10.1051/m2an/2021042 [Citations: 1]

  9. Minimal residual methods in negative or fractional Sobolev norms

    Monsuur, Harald | Stevenson, Rob | Storn, Johannes

    Mathematics of Computation, Vol. 93 (2023), Iss. 347 P.1027

    https://doi.org/10.1090/mcom/3904 [Citations: 4]

  10. Direct Guaranteed Lower Eigenvalue Bounds with Optimal a Priori Convergence Rates for the Bi-Laplacian

    Carstensen, Carsten | Puttkammer, Sophie

    SIAM Journal on Numerical Analysis, Vol. 61 (2023), Iss. 2 P.812

    https://doi.org/10.1137/21M139921X [Citations: 5]

  11. A posteriori error analysis for a distributed optimal control problem governed by the von Kármán equations

    Chowdhury, Sudipto | Dond, Asha K. | Nataraj, Neela | Shylaja, Devika

    ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 56 (2022), Iss. 5 P.1655

    https://doi.org/10.1051/m2an/2022040 [Citations: 1]

  12. Unifying a posteriori error analysis of five piecewise quadratic discretisations for the biharmonic equation

    Carstensen, Carsten | Gräßle, Benedikt | Nataraj, Neela

    Journal of Numerical Mathematics, Vol. 0 (2023), Iss. 0

    https://doi.org/10.1515/jnma-2022-0092 [Citations: 0]

  13. Conforming and nonconforming finite element methods for biharmonic inverse source problem

    Nair, M Thamban | Shylaja, Devika

    Inverse Problems, Vol. 38 (2022), Iss. 2 P.025001

    https://doi.org/10.1088/1361-6420/ac3ec5 [Citations: 1]

  14. Adaptive guaranteed lower eigenvalue bounds with optimal convergence rates

    Carstensen, Carsten | Puttkammer, Sophie

    Numerische Mathematik, Vol. 156 (2024), Iss. 1 P.1

    https://doi.org/10.1007/s00211-023-01382-8 [Citations: 0]

  15. Morley FEM for a Distributed Optimal Control Problem Governed by the von Kármán Equations

    Chowdhury, Sudipto | Nataraj, Neela | Shylaja, Devika

    Computational Methods in Applied Mathematics, Vol. 21 (2021), Iss. 1 P.233

    https://doi.org/10.1515/cmam-2020-0030 [Citations: 5]

  16. Lowest-order equivalent nonstandard finite element methods for biharmonic plates

    Carstensen, Carsten | Nataraj, Neela

    ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 56 (2022), Iss. 1 P.41

    https://doi.org/10.1051/m2an/2021085 [Citations: 7]

  17. A Posteriori Error Control for Fourth-Order Semilinear Problems with Quadratic Nonlinearity

    Carstensen, Carsten | Gräßle, Benedikt | Nataraj, Neela

    SIAM Journal on Numerical Analysis, Vol. 62 (2024), Iss. 2 P.919

    https://doi.org/10.1137/23M1589852 [Citations: 0]