The DPG methodology applied to different variational formulations of linear elasticity

Abstract

The flexibility of the DPG methodology is exposed by solving the linear elasticity equations under different variational formulations, including some with non-symmetric functional settings (different infinite-dimensional trial and test spaces). The family of formulations presented are proved to be mutually ill or well-posed when using traditional energy spaces on the whole domain. Moreover, they are shown to remain well-posed when using broken energy spaces and interface variables. Four variational formulations are solved in 3D using the DPG methodology. Numerical evidence is given for both smooth and singular solutions and the expected convergence rates are observed.

Discrete least-squares finite element methods

“…There appear to be no published manuscripts which derive or consider the possible advantages inherent in solving the overdetermined system of equations. We note that in specific instances where the test space is L 2 in at least one component-and therefore, for most alternative variational formulations [19,48]-the implementation of DPG permits some ambiguity. For instance, as developed in [48] and contrary to the DLS approach, one can choose not to discretize the L 2 -Riesz map and so construct a least-squares (strong variational formulations) or least-squares hybrid (all other variational formulations) discretization.…”

“…We argue to adopt a convention to distinguish between these approaches. That is, the discretization advocated for here should aptly be called the discrete least-squares discretization, and the least-squares and hybrid discretizations developed in [48] should be identified as such.…”

“…Likewise, (·, ·) T = K∈T (·, ·) K and similarly with ·, · ∂T . This second pairing, ·, · ∂T , can be understood, intuitively, as a mesh-boundary integral, however, the inquisitive reader may wish to examine [19,48] for further detail. The lift u lift ∈ H 1 (Ω) in (5.5) is identical to that from the least-squares setting presented before.…”

“…In order to explore less trivial variational formulations, consider the broken primal formulation of the Poisson equation in (5.3) (with α = 0) [28]. In this setting, we seek a [19,48] for definitions of mesh-trace Sobolev spaces) such that…”

“…Furthermore, Bubnov-Galerkin stiffness matrices for most difficult problems will not even be Hermitian/symmetric, so they can not admit any such factorization, A BG = M * M for any matrix M. Therefore, this observation is of extremely limited practical interest. [19,48] for definitions of mesh-trace Sobolev spaces) such that…”