Alternative Subcell Discretisations for Viscoelastic Flow: Velocity Gradient Approximation Alternative Subcell Discretisations for Viscoelastic Flow: Velocity Gradient Approximation

Under subcell discretisation for viscoelastic flow, we have given further consideration to the compatibility of function spaces for stress/velocity-gradient approximation (see [JNNFM, special issue AERC 2006). This has been conducted through the three scheme discretisations (quad-fe(par), fe(sc) and fe/fv(sc)). In this companion study, we have extended the application of the original implementation for velocity gradient approximation, which was of localised superconvergent recovered form, continuous and quadratic on the parent fe-triangular element. This has led to the consideration of both localised (pointwise) and global (Galerkin weighted-residual) approximations for velocity gradients, highlighting some of their advantages and disadvantages. Each representation is based on linear/quadratic order upon parent or subcell element stencils. We consider Oldroyd modelling and the contraction flow benchmark, covering abrupt and rounded-corner planar geometries. The localised super-convergent quadratic velocitygradient treatment affords strong stability and accuracy properties for the three scheme variants. Through associated analysis, we have successfully linked global approximations to their localised counterparts, depicting the inadequacy of inaccurate but stable versions through their corresponding solution features. The inaccuracy of the global treatment can be repaired through an increase in mass iteration number. The efficiency of localised schemes is particularly attractive over their global alternatives, being less restrictive to choice of spatial-order. Such schemes come into their own when chosen to represent strongly localised solution features, such as arise in non-smooth flows.