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

Abstract

<div lang="en" class="abs"><p class="abs_header">Abstract</p><p>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.</p><p>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 \(H(\operatorname {div}\,\, \mathbf{div}\!)\).</p><p>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.</p></div>