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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Locking-free finite element methods for shells
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by Douglas N. Arnold and Franco Brezzi PDF
Math. Comp. 66 (1997), 1-14 Request permission


We propose a new family of finite element methods for the Naghdi shell model, one method associated with each nonnegative integer $k$. The methods are based on a nonstandard mixed formulation, and the $k$th method employs triangular Lagrange finite elements of degree $k+2$ augmented by bubble functions of degree $k+3$ for both the displacement and rotation variables, and discontinuous piecewise polynomials of degree $k$ for the shear and membrane stresses. This method can be implemented in terms of the displacement and rotation variables alone, as the minimization of an altered energy functional over the space mentioned. The alteration consists of the introduction of a weighted local projection into part, but not all, of the shear and membrane energy terms of the usual Naghdi energy. The relative error in the method, measured in a norm which combines the $H^{1}$ norm of the displacement and rotation fields and an appropriate norm of the shear and membrane stress fields, converges to zero with order $k+1$ uniformly with respect to the shell thickness for smooth solutions, at least under the assumption that certain geometrical coefficients in the Nagdhi model are replaced by piecewise constants.
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Additional Information
  • Douglas N. Arnold
  • Affiliation: Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802
  • MR Author ID: 27240
  • Email:
  • Franco Brezzi
  • Affiliation: Istituto di Analisi Numerica del C.N.R., Università di Pavia, 27100 Pavia, Italy
  • Email:
  • Received by editor(s): December 2, 1993
  • Received by editor(s) in revised form: April 3, 1995, and November 13, 1995
  • Additional Notes: The work of the first author was supported by National Science Foundation grants DMS-9205300 and DMS-9500672. The work of the second author was partially supported by the HCM Program on Shells, contract number ERBCHRXCT 940536.
  • © Copyright 1997 American Mathematical Society
  • Journal: Math. Comp. 66 (1997), 1-14
  • MSC (1991): Primary 65N30, 73K15, 73V05
  • DOI:
  • MathSciNet review: 1370847