Fully consistent lowest-order finite element methods for generalised Stokes flows with variable viscosity

In finite element methods for incompressible flows, the most popular approach to allow equal-order velocity-pressure pairs are residual-based stabilisations. When using first-order elements, however, the viscous part of the residual cannot be approximated, which often degrades accuracy. For constant viscosity, or by assuming a Lipschitz condition on the viscosity field, we can construct stabilisation methods that fully approximate the residual, regardless of the polynomial order of the finite element spaces. This work analyses and tests two variants of such a fully consistent approach, with the generalised Stokes system as a model problem. We prove unique solvability and derive expressions for the stabilisation parameter, generalising some classical results for constant viscosity. Numerical results illustrate how our method completely eliminates the spurious pressure boundary layers typically induced by low-order PSPG-like stabilisations.