Analysis of a Stabilised Finite Element Method for Power-Law Fluids

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 > 2d/d+2 of the power-law index r for which weak solutions to the model are known to exist in d space dimensions, d is an element of {2, 3}.

Automated summary

This paper presents a low-order finite element method for the incompressible non-Newtonian flow problem with power-law rheology. The method uses a linear approximation of the velocity field and a constant approximation of the pressure. To compensate for the failure of the inf-sup condition, pressure jumps are added to the formulation, and a divergence-free approximation of the velocity field is included in the discretization. The convergence of the method is proven for a range of power-law index values.