A locally conservative and energy-stable finite element for the Navier--Stokes problem on time-dependent domains

Abstract

We present a finite element method for the incompressible Navier--Stokes problem that is locally conservative, energy-stable and pressure-robust on time-dependent domains. To achieve this, the space--time formulation of the Navier--Stokes problem is considered. The space--time domain is partitioned into space--time slabs which in turn are partitioned into space--time simplices. A combined discontinuous Galerkin method across space--time slabs, and space--time hybridized discontinuous Galerkin method within a space--time slab, results in an approximate velocity field that is $H(\mathrm{div})$-conforming and exactly divergence-free, even on time-dependent domains. Numerical examples demonstrate the convergence properties and performance of the method.

References in corpus (2)

• A hybridizable discontinuous Galerkin method for the Navier--Stokes equations with pointwise divergence-free velocity field
• Analysis of a hybridized/interface stabilized finite element method for the Stokes equations

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