An ultraweak formulation of the Kirchhoff–Love plate bending model and DPG approximation
Thomas Führer, Norbert Heuer, and Antti H. Niemi
Math. Comp. 88 (318), 1587–1619, 2019.
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 H²
(standard Sobolev space of scalar functions) and H(div Div)
(symmetric tensor functions with L²-components whose twice
iterated divergence is in L², 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.