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

TL;DR

This paper introduces an ultraweak variational formulation for the Kirchhoff-Love plate model and develops a DPG discretization, proving well-posedness and convergence, with numerical validation in 2D and 3D.

Contribution

It presents a novel ultraweak formulation and a DPG scheme for the Kirchhoff-Love plate model, including new tools for trace and jump control in Sobolev spaces.

Findings

Proved well-posedness of the ultraweak formulation.

Established quasi-optimal convergence of the DPG scheme.

Numerical experiments confirm convergence on various meshes.

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.