SUMMARYIn this paper the development and behaviour of a new finite element algorithm for viscous incompressible flow is presented. The stability and background theory are discussed and the numerical performance is considered for some benchmark problems. The Taylor-Galerkin approach naturally leads to a timestepping algorithm which is shown to perform well for a wide range of Reynolds numbers (1

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Hybrid finite volume/element methods are investigated within the context of transient viscoelastic flows. A finite volume algorithm is proposed for the hyperbolic constitutive equation, of Oldroyd-form, whilst the continuity/momentum balance is accommodated through a TaylorGalerkin finite element method. Various finite volume combinations are considered to derive accurate and stable implementations. Consistency of formulation is key, embracing fluctuation distribution and median-dual-cell constructs, within a cell-vertex discretisation on triangles. In addition, we investigate the effect of treating the time-term in a finite element fashion, using mass-matrix iteration instead of the standard finite volume mass-lumping approach. We devise an accurate transient scheme that captures the analytical solution at short and long time, both in core flow and near shear boundaries. In this respect, some difficulties are highlighted. A new method emerges, with the Low Diffusion B (LDB, with or without mass-matrix iteration) as the optimal choice. We progress to a complex flow application and demonstrate some provocative features due to the influence of true transient boundary conditions on evolutionary flow-structure in a 4:1 start-up rounded-corner contraction problem. c (Year) John Wiley & Sons, Inc.