A locally conservative and energy-stable finite element for the Navier--Stokes problem on time-dependent domains
This work provides a novel numerical method for simulating incompressible flows in moving domains, addressing key stability and conservation properties for computational fluid dynamics.
The authors developed a finite element method for the Navier-Stokes problem on time-dependent domains that is locally conservative, energy-stable, and pressure-robust, achieving exactly divergence-free velocity fields. Numerical examples confirm convergence and performance.
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 $H({\rm div})$-conforming and exactly divergence-free, even on time-dependent domains. Numerical examples demonstrate the convergence properties and performance of the method.