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 wellposedness 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.

An ultraweak formulation of the Reissner–Mindlin plate bending model and DPG approximation

“…Note that, setting t = 0, (5a)-(5d) turns into the Kirchhoff-Love plate bending model whose ultraweak setting was proposed and analyzed in [13]. Though, setting t = 0 in our variational formulation (to be developed), we recover the Kirchhoff-Love model from [13] without θ = ∇u as independent variable. This is due to the fact that the appropriate weighting of (5c) is by the factor t, just like in (5a).…”

“…In the following we collect the definitions and properties of spaces, norms and traces from this section in the limit t = 0, which is the Kirchhoff-Love case. For the clamped plate, the corresponding results are taken from [13], whereas for the simply supported plate we have to introduce spaces that reflect this boundary condition. Let us start collecting spaces and norms (the defined terms are those from [13] in the notation introduced there).…”

“…For the clamped plate, the corresponding results are taken from [13], whereas for the simply supported plate we have to introduce spaces that reflect this boundary condition. Let us start collecting spaces and norms (the defined terms are those from [13] in the notation introduced there). For any T ∈ T we have the space…”

“…Therefore, we define U c (0) ∶= H 2 0 (Ω) × H(div div, Ω) needed in the case that a = c but, in order to define the space U s (Ω) corresponding to a = s, we have to give n ⋅ Θn = 0 on Γ a meaning when Θ ∈ H(div div, Ω). To consider the setting for t = 0 we recall the following trace operators from [13],…”

“…As in [13], our focus is to develop a formulation that requires minimum regularity, only subject to the L 2 -regularity of the vertical load. This condition is owed to the discontinuity of test functions of DPG schemes.…”