A divergence free $C^0$-RIPG stream function formulation of the incompressible Stokes system with variable viscosity

TL;DR

This paper introduces a novel divergence-free $C^0$-RIPG stream function formulation for the incompressible Stokes system with variable viscosity, offering a stable, convergent, and efficient method for 2D flow simulations.

Contribution

The paper develops a new $C^0$-interior penalty Galerkin discretization for the Stokes system that preserves divergence-free velocity fields and handles variable viscosity without standard biharmonic problems.

Findings

The proposed method achieves optimal convergence rates.

Numerical experiments validate stability and convergence.

Comparisons demonstrate effectiveness in mantle convection benchmarks.

Abstract

Pointwise divergence free velocity field approximations of the Stokes system are gaining popularity due to their necessity in precise modelling of physical flow phenomena. Several methods have been designed to satisfy this requirement; however, these typically come at a greater cost when compared with standard conforming methods, for example, because of the complex implementation and development of specialized finite element bases. Motivated by the desire to mitigate these issues for 2D simulations, we present a $C^0$-interior penalty Galerkin (IPG) discretization of the Stokes system in the stream function formulation. In order to preserve a spatially varying viscosity this approach does not yield the standard and well known biharmonic problem. We further employ the so-called robust interior penalty Galerkin (RIPG) method; stability and convergence analysis of the proposed scheme is undertaken. The former, which involves deriving a bound on the interior penalty parameter is particularly useful to address the $\mathcal{O}(h^{-4})$ growth in the condition number of the discretized operator. Numerical experiments confirming the optimal convergence of the proposed method are undertaken. Comparisons with thermally driven buoyancy mantle convection model benchmarks are presented.