A convergent post-processed discontinuous Galerkin method for incompressible flow with variable density

TL;DR

This paper introduces a stable, convergent finite element method combining discontinuous Galerkin and H1-conforming techniques for simulating incompressible variable-density flows, ensuring accuracy and stability in 3D domains.

Contribution

It presents a novel linearized semi-implicit, decoupled finite element approach with post-processed velocity for variable-density incompressible flows, proven to be unconditionally stable and convergent.

Findings

Method is unconditionally stable.

Achieves convergence for smooth solutions in 3D domains.

Combines discontinuous Galerkin with H1-conforming finite elements.

Abstract

We propose a linearized semi-implicit and decoupled finite element method for the incompressible Navier--Stokes equations with variable density. Our method is fully discrete and shown to be unconditionally stable. The velocity equation is solved by an H1-conforming finite element method, and an upwind discontinuous Galerkin finite element method with post-processed velocity is adopted for the density equation. The proposed method is proved to be convergent in approximating reasonably smooth solutions in three-dimensional convex polyhedral domains.