The construction of stable and div-free finite elements via Stokes complexes

26 January 2021
In this talk, we describe the methodology for constructing a divergence-free and stable pair of finite element spaces for the Stokes problem on cubical meshes of arbitrary dimension. We use the Stokes complex as a guiding tool. We state and exemplify the general procedure for deriving a divergence-free and stable finite element discretization from a Stokes complex. However, we develop a new strategy to prove the necessary inf-sup stability condition due to the lack of a Fortin operator. In particular, we first derive a local inf-sup condition with imposed boundary conditions and then translate this result to the global level by exploiting the element's degrees of freedom. Furthermore, we derive reduced finite elements with less global degrees of freedom. We show that the optimal order of convergence is achieved via both the original and reduced finite elements for the velocity approximation, and the pressure approximation is of optimal order when the reduced finite elements are used.
Ref. Stokes elements on cubic meshes yielding divergence-free approximations, M. Neilan and D. Sap, Calcolo, 53(3):263-283, 2016. 

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  • Numerical Analysis Group Internal Seminar