New Theory of Hybrid-Upwind Technique and Discrete Del Operator for Finite-Element Method

In this paper, we present a new hybrid-streamline-upwind finite-element method, which is based on the finite analytic method developed in the field of the finite difference method. In order to obtain an optimal shape function, we introduce an adjoint differential operator to the differential operator for a steady advection-diffusion equation. The shape functions which satisfy these differential operators are mutually dual. One of them interpolates accurately the functions appearing in convection-dominated flows, the other becomes a hybrid-streamline-upwind weighting function. Furthermore, we define a discrete del operator for the reduction of memory storage in the computer. As a result, we achieve simplicity of formulation and high-speed calculation of the finite-element method.