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

Führer, Thomas; Heuer, Norbert; Niemi, Antti

doi:10.1090/mcom/3381

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