An Ultraweak Formulation Of The Kirchhoff-Love Plate Bending Model And Dpg Approximation

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(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(div div). They are essential to construct basis functions for an approximation of H(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. Keywords. Author Keywords:Kirchhoff-Love model

plate bending

biharmonic problem

fourth-order elliptic PDE

discontinuous Petrov-Galerkin method

optimal test functions