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An ultraweak formulation of the Kirchhoff-Love plate bending model and DPG approximation


Authors: Thomas Führer, Norbert Heuer and Antti H. Niemi
Journal: Math. Comp. 88 (2019), 1587-1619
MSC (2010): Primary 74S05, 74K20, 35J35; Secondary 65N30, 35J67
DOI: https://doi.org/10.1090/mcom/3381
Published electronically: October 5, 2018
MathSciNet review: 3925478
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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({\rm div}\,\, {\bf 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({\rm div}\,\, {\bf div}\!)$. They are essential to construct basis functions for an approximation of $ H({\rm div}\,\, {\bf 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
Email: tofuhrer@mat.uc.cl

Norbert Heuer
Affiliation: Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, Santiago, Chile
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

DOI: https://doi.org/10.1090/mcom/3381
Keywords: Kirchhoff--Love model, plate bending, biharmonic problem, fourth-order elliptic PDE, discontinuous Petrov--Galerkin method, optimal test functions
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.
Article copyright: © Copyright 2018 American Mathematical Society