A low-order finite element method is constructed and analysed for an incompressible non-Newtonian flow problem with power-law rheology. The method is based on a continuous piecewise linear approximation of the velocity field and piecewise constant approximation of the pressure. Stabilisation, in the form of pressure jumps, is added to the formulation to compensate for the failure of the inf-sup condition, and using an appropriate lifting of the pressure jumps a divergence-free approximation to the velocity field is built and included in the discretisation of the convection term. This construction allows us to prove the convergence of the resulting finite element method for the entire range $r>\frac{2 d}{d+2}$ of the power-law index $r$ for which weak solutions to the model are known to exist in $d$ space dimensions, $d \in \{2,3\}$.

翻译：设计和分析一种低级有限元素法,用于处理电法风湿的无法压缩的非牛顿流问题。该方法基于速度场的连续片断线近近近和压力的片断常近近。在配方中添加了稳定化,即以压力跳跃的形式,以补偿降压状况的失灵,并使用适当的降压,将无差异的近近近点建于速度场,并纳入对流术语的分解中。这一构造使我们能够证明由此得出的整个范围(${frac{2 d ⁇ d+2})的有限元素法方法的趋同。电法指数的美元稳定化,因为已知该模型的薄弱解决方案存在于美元空间尺寸,$d $2,3 ⁇ 美元。