Third-order upwind finite element formulations for incompressible viscous flow problems

Abstract

A third-order upwind finite element scheme based on the Petrov-Galerkin formulation is presented for highly accurate solutions of convection dominated flows. A new modified weighting function which is expressed by the sum of a standard weighting function and its second and third derivatives is applied to the formulation. The discrete forms for artificial dissipation terms in the upwind scheme can be rewritten by expressions of fourth and fifth spatial derivatives of an unknown function by applying the Taylor-series expansion. The third-order upwinding technique is first applied to the one-dimensional advection-diffusion equation so that the structures of the upwind scheme are explained in detail. Next the upwind scheme is extended to the incompressible Navier-Stokes equations in multidimensions. Numerical results for a driven cavity flow in a two-dimensional square region are presented to demonstrate the effectiveness and applicability of the upwind finite element scheme proposed in this paper.