Computation of incompressible viscous flows by the third‐order upwind finite element method

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Abstract
A third‐order upwind finite element scheme is presented for numerical solutions of incompressible viscous flow problems. In order to achieve the third‐order upwind approximation for only the convection term in the Navier‐Stokes equations, a simplified Petrov‐Galerkin formulation in which a modified weighting function is expressed by the sum of a standard weighting function and its second and third spatial derivatives is employed. The mixed method is also employed in the formulation so that a discretization with high‐order accuracy in space is carried out by the use of linear elements. Because a truncation error caused by the third‐order upwind approximation is smaller than that of a first‐order upwind scheme, it is expected that the third‐order upwind scheme will greatly improve the numerical solutions of the Navier‐Stokes equations. Numerical results in one and two dimensions are presented to illustrate the effectiveness of the proposed scheme.