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

Abstract

Summary 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(div)‐conforming and exactly divergence‐free, even on time‐dependent domains. Numerical examples demonstrate the convergence properties and performance of the method.

A Space-Time Hybridizable Discontinuous Galerkin Method for Linear Free-Surface Waves

“…Note that if ρ = ρ g , the last integral on the right hand side of eq. ( 9) disappears and the trilinear form reduces exactly to the trilinear form introduced in [24] for single-phase flow. The last integral on the right hand side of eq.…”

“…An exactly mass conserving space-time HDG discretization for the single-phase Navier-Stokes equations was introduced in [24,25]. Here we modify this discretization to take into account that the density and viscosity may be discontinuous across elements (see eq.…”

“…Following [24,25] we consider a tetrahedral space-time mesh which is constructed as follows. First, the triangular spatial mesh of Ω n is extruded to the new time level t n+1 according to the domain deformation prescribed by the wave height.…”

HDGlab: An Open-Source Implementation of the Hybridisable Discontinuous Galerkin Method in MATLAB

“…Besides classical approaches to steady and unsteady Navier-Stokes equations, HDG-based space-time formulations were studied for their ability to effectively handle moving and deforming domains. More precisely, stemming from the HDG formulation introduced in [222], H(div)-conforming hybridised DG [160] and EHDG [161] methods were proposed. Hybridised DG and HDG methods with arbitrary Lagrangian Eulerian (ALE) formulations were thus presented in [134,140] and the resulting HDG-ALE framework was applied to fluid-structure interaction (FSI) problems involving incompressible [248] and weakly-compressible flows [176].…”

An embedded discontinuous Galerkin method for the Oseen equations

“…Finally, we can see that these types of methods mentioned above has been applied to the incompressible Navier-Stokes equations [5,29]. In recent years, the space-time formulations of these types of methods have been developed for the incompressible Navier-Stokes equations on moving domains [4,9].…”