An exactly mass conserving space-time embedded-hybridized discontinuous Galerkin method for the Navier-Stokes equations on moving domains

TL;DR

This paper introduces a novel space-time embedded-hybridized discontinuous Galerkin (EHDG) method for solving Navier-Stokes equations on moving domains, emphasizing divergence-free velocity fields and energy stability.

Contribution

It develops a new EHDG method with continuous velocity trace unknowns, offering fewer degrees of freedom and exact divergence-free velocity fields on moving domains.

Findings

EHDG and HDG methods produce exactly divergence-free velocity fields.

EHDG method has fewer global degrees-of-freedom than HDG.

EDG method requires skew-symmetric formulation for energy stability.

Abstract

This paper presents a space-time embedded-hybridized discontinuous Galerkin (EHDG) method for the Navier--Stokes equations on moving domains. This method uses a different hybridization compared to the space-time hybridized discontinuous Galerkin (HDG) method we presented previously in (Int. J. Numer. Meth. Fluids 89: 519--532, 2019). In the space-time EHDG method the velocity trace unknown is continuous while the pressure trace unknown is discontinuous across facets. In the space-time HDG method, all trace unknowns are discontinuous across facets. Alternatively, we present also a space-time embedded discontinuous Galerkin (EDG) method in which all trace unknowns are continuous across facets. The advantage of continuous trace unknowns is that the formulation has fewer global degrees-of-freedom for a given mesh than when using discontinuous trace unknowns. Nevertheless, the discrete velocity field obtained by the space-time EHDG and EDG methods, like the space-time HDG method, is exactly divergence-free, even on moving domains. However, only the space-time EHDG and HDG methods result in divergence-conforming velocity fields. An immediate consequence of this is that the space-time EHDG and HDG discretizations of the conservative form of the Navier--Stokes equations are energy stable. The space-time EDG method, on the other hand, requires a skew-symmetric formulation of the momentum advection term to be energy-stable. Numerical examples will demonstrate the differences in solution obtained by the space-time EHDG, EDG, and HDG methods.