Führer, Thomas; Heuer, Norbert; Niemi, Antti H. (2019) An ultraweak formulation of the Kirchhoff-Love plate bending model and DPG approximation. Math. Comp. 88, 1587-1619. https://doi.org/10.1090/mcom/3381

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

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Author: Führer, Thomas1; Heuer, Norbert1; Niemi, Antti H.2
Organizations: 1Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, Santiago, Chile
2Structures and Construction Technology Research Unit, Faculty of Technology, University of Oulu, Erkki Koiso-Kanttilan katu 5, Linnanmaa, 90570 Oulu, Finland
Format: article
Version: accepted version
Access: open
Online Access: PDF Full Text (PDF, 0.7 MB)
Language: English
Published: American Mathematical Society, 2019
Publish Date: 2020-03-10
Description:

# 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 $$H(\operatorname {div}\,\, \mathbf{div}\!)$$ (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 $$H(\operatorname {div}\,\, \mathbf{div}\!)$$. They are essential to construct basis functions for an approximation of $$H(\operatorname {div}\,\, \mathbf{div}\!)$$.

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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Series: Mathematics of computation
ISSN: 1088-6842
ISSN-E: 1088-6842
ISSN-L: 1088-6842
Volume: 88
Pages: 1587 - 1619
DOI: 10.1090/mcom/3381