Energy stable and momentum conserving hybrid finite element method for the incompressible Navier-Stokes equations

TL;DR

This paper introduces a hybrid finite element method for the incompressible Navier-Stokes equations that ensures energy stability, momentum conservation, and local mass conservation, using continuous Lagrange multipliers and discontinuous pressure fields.

Contribution

It develops a novel hybrid finite element approach that guarantees energy stability and conservation properties for Navier-Stokes simulations, with efficient degrees of freedom and improved local mass conservation.

Findings

The method achieves energy stability without point-wise divergence-free velocity.

It ensures local mass and momentum conservation in numerical simulations.

Numerical results support the theoretical stability and conservation properties.

Abstract

A hybrid method for the incompressible Navier--Stokes equations is presented. The method inherits the attractive stabilizing mechanism of upwinded discontinuous Galerkin methods when momentum advection becomes significant, equal-order interpolations can be used for the velocity and pressure fields, and mass can be conserved locally. Using continuous Lagrange multiplier spaces to enforce flux continuity across cell facets, the number of global degrees of freedom is the same as for a continuous Galerkin method on the same mesh. Different from our earlier investigations on the approach for the Navier--Stokes equations, the pressure field in this work is discontinuous across cell boundaries. It is shown that this leads to very good local mass conservation and, for an appropriate choice of finite element spaces, momentum conservation. Also, a new form of the momentum transport terms for the method is constructed such that global energy stability is guaranteed, even in the absence of a point-wise solenoidal velocity field. Mass conservation, momentum conservation and global energy stability are proved for the time-continuous case, and for a fully discrete scheme. The presented analysis results are supported by a range of numerical simulations.