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arXiv · 1805.07835

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

Thomas Führer·Norbert Heuer·Antti H. Niemi
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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 $H(\mathrm{div\,Div})$ (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 $H(\mathrm{div\,Div})$. They are essential to construct basis functions for an approximation of $H(\mathrm{div\,Div})$. 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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Thomas Führer, Norbert Heuer, Antti H. Niemi. 2018-05-20. An ultraweak formulation of the Kirchhoff-Love plate bending model and DPG approximation. https://arxiv.org/abs/1805.07835

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