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In this work, a new high‐order finite volume element method with good spatial resolution characteristics is presented. The method is based on a functional representation of the unknowns based on the finite element method, a balance of physical quantities in weak formulation obtained by using the finite volume method, and an implicit reduction of some of the unknowns obtained by enforcing functional relations between some of them. Applications to hyperbolic and elliptic operators in 1D and 2D, as well as to Navier–Stokes equations for incompressible flows are presented. Copyright © 2002 John Wiley & Sons, Ltd.

In this work different regularization approaches for the solution of the div–curl problems arising from the treatment of the unsteady incompressible Navier–Stokes model are presented. Indeed it is shown how some popular methods for the resolution of the Navier–Stokes problem can be seen as different forms of the same regularization approach. The use of different discretization layouts is illustrated together with the advantages deriving from the employment of this technique. Finally, a strategy for the reduction of the number of the discretized unknowns of the div–curl problem by means of the Singular Value Decomposition is presented.

After a synthesis of existing numerical approaches, a family of single-step time-marching upwind schemes is proposed for the convective-diffusive balance of a scalar in multidimensional incompressible laminar flows. A functional description of the new approach is given too. The schemes are adequate to the simulation of purely or highly convective transport phenomena. The proposed schemes were applied to a scalar test problem (rotating hill) and the results evaluated after an analysis of the intrinsic limitations of test itself, A third-order, fully upwind-biased scheme of the family is applied to the lid-driven and thermally driven square cavity problems. In the lid-driven case a steady-state solution is not achieved at Re = 104.