Variable viscosity arises in many flow scenarios, often imposing numerical challenges. Yet, discretisation methods designed specifically for non-constant viscosity are few, and their analysis is even scarcer. In finite element methods for incompressible flows, the most popular approach to allow equal-order velocity-pressure interpolation are residual-based stabilisations. For low-order elements, however, the viscous part of that residual cannot be approximated, often compromising accuracy. Assuming slightly more regularity 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 two variants of this 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.

翻译：变粘度问题广泛存在于多种流动场景中，通常给数值模拟带来挑战。然而，专门为非恒定粘度设计的离散方法为数不多，其理论分析更为匮乏。在不可压缩流动的有限元方法中，实现等阶速度-压力插值最常用的途径是基于残差的稳定化方法。但对于低阶单元，该方法中粘性部分的残差往往无法近似，从而影响精度。若假设粘度场具有稍强的正则性，我们可以构建能够完全近似残差的稳定化方法，且不受有限元空间多项式阶次的限制。本文以广义Stokes系统为模型问题，分析了这种完全一致方法的两种变体。我们证明了解的唯一存在性，推导了稳定化参数的表达式，推广了恒定粘度情形下的若干经典结论。数值结果表明，我们的方法完全消除了低阶类PSPG稳定化通常引发的虚假压力边界层。