Abstract
In this note we will explore some applications of the recently constructed piecewise affine, H1-conforming element that fits in a discrete de Rham complex (Christiansen and Hu, 2018). In particular we show how the element leads to locking free methods for incompressible elasticity and viscosity robust methods for the Brinkman model.
Highlights
It is well known that standard finite element methods are not in general well-suited for the approximation of nearly incompressible elasticity or incompressible flow problems
In particular low order approximation spaces often suffer from locking in the incompressible limit [1]
Drawing on pioneering work by Scott and Vogelius in the mid-eighties [6], recently some new results on H 1-conforming piecewise polynomial approximation spaces compatible with the de Rham complex have been published [7,8,9,10,11,12]
Summary
It is well known that standard finite element methods are not in general well-suited for the approximation of nearly incompressible elasticity or incompressible flow problems. In particular low order approximation spaces often suffer from locking in the incompressible limit [1] They may typically exhibit instability when Darcy flow is considered if the element was designed for Stokes’ problem [2]. Drawing on pioneering work by Scott and Vogelius in the mid-eighties [6], recently some new results on H 1-conforming piecewise polynomial approximation spaces compatible with the de Rham complex have been published [7,8,9,10,11,12] Such elements are interesting, since they provide a tool for the robust approximation of models in mechanics where a divergence constraint is present.
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