An ultraweak formulation of the Kirchhoff–Love plate bending model and DPG approximation
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- by Thomas Führer, Norbert Heuer and Antti H. Niemi HTML | PDF
- Math. Comp. 88 (2019), 1587-1619 Request permission
Abstract:
We develop and analyze an ultraweak variational formulation for a variant of the Kirchhoff–Love plate bending model. Based on this formulation, we introduce a discretization of the discontinuous Petrov–Galerkin type with optimal test functions (DPG). We prove well-posedness of the ultraweak formulation and quasi-optimal convergence of the DPG scheme.
The variational formulation and its analysis require tools that control traces and jumps in $H^2$ (standard Sobolev space of scalar functions) and $H(\textrm {div} \textbf {div}\!)$ (symmetric tensor functions with $L_2$-components whose twice iterated divergence is in $L_2$), and their dualities. These tools are developed in two and three spatial dimensions. One specific result concerns localized traces in a dense subspace of $H(\textrm {div} \textbf {div}\!)$. They are essential to construct basis functions for an approximation of $H(\textrm {div} \textbf {div}\!)$.
To illustrate the theory we construct basis functions of the lowest order and perform numerical experiments for a smooth and a singular model solution. They confirm the expected convergence behavior of the DPG method both for uniform and adaptively refined meshes.
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Additional Information
- Thomas Führer
- Affiliation: Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, Santiago, Chile
- MR Author ID: 1017746
- Email: tofuhrer@mat.uc.cl
- Norbert Heuer
- Affiliation: Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, Santiago, Chile
- MR Author ID: 314970
- Email: nheuer@mat.uc.cl
- Antti H. Niemi
- Affiliation: Structures and Construction Technology Research Unit, Faculty of Technology, University of Oulu, Erkki Koiso-Kanttilan katu 5, Linnanmaa, 90570 Oulu, Finland
- Email: antti.niemi@oulu.fi
- Received by editor(s): September 22, 2017
- Received by editor(s) in revised form: March 23, 2018, and May 11, 2018
- Published electronically: October 5, 2018
- Additional Notes: This research was supported by CONICYT through FONDECYT projects 1150056, 11170050, The Magnus Ehrnrooth Foundation, and by Oulun rakennustekniikan säätiö.
The second author is the corresponding author. - © Copyright 2018 American Mathematical Society
- Journal: Math. Comp. 88 (2019), 1587-1619
- MSC (2010): Primary 74S05, 74K20, 35J35; Secondary 65N30, 35J67
- DOI: https://doi.org/10.1090/mcom/3381
- MathSciNet review: 3925478