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Numerical Approximation of the Solution of an Obstacle Problem Modelling the Displacement of Elliptic Membrane Shells via the Penalty Method

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Abstract

In this paper we establish the convergence of a numerical scheme based, on the Finite Element Method, for a time-independent problem modelling the deformation of a linearly elastic elliptic membrane shell subjected to remaining confined in a half space. Instead of approximating the original variational inequalities governing this obstacle problem, we approximate the penalized version of the problem under consideration. A suitable coupling between the penalty parameter and the mesh size will then lead us to establish the convergence of the solution of the discrete penalized problem to the solution of the original variational inequalities. We also establish the convergence of the Brezis–Sibony scheme for the problem under consideration. Thanks to this iterative method, we can approximate the solution of the discrete penalized problem without having to resort to nonlinear optimization tools. Finally, we present numerical simulations validating our new theoretical results.

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Funding

A.M. and P.P. were partly supported by the Research Fund of Indiana University and by the National Science Foundation under Grant Number DMS-2051032.

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  1. Department of Mathematics, The Ohio State University, 100 Math Tower, 231 West 18th Avenue, Columbus, OH, USA

    Aaron Meixner

  2. Department of Mathematics and Institute for Scientific Computing and Applied Mathematics, Indiana University Bloomington, 729 East Third Street, Bloomington, IN, USA

    Paolo Piersanti

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  1. Aaron Meixner
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Meixner, A., Piersanti, P. Numerical Approximation of the Solution of an Obstacle Problem Modelling the Displacement of Elliptic Membrane Shells via the Penalty Method. Appl Math Optim 89, 45 (2024). https://doi.org/10.1007/s00245-024-10112-x

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