Application of C-Bézier and H-Bézier basis functions to numerical solution of convection-diffusion equations

Abstract

Convection-diffusion equation is widely used to describe many engineering and physical problems. The finite element method is one of the most common tools for computing numerical solution. In 2003, Wang et al. proposed C-Bézier and H-Bézier basis functions which are not only a generalization of classical Bernstein basis functions but also have a free shape parameter bringing a lot of flexibility to geometrical modeling. In this paper, we adopt C-Bézier and H-Bézier basis functions to construct test and trial function spaces of finite element method to get numerical solution of convection-diffusion equations. Compared with Lagrange basis functions, numerical accuracy is improved by $1-3$ 1 − 3 order-of magnitudes which implies a much better approximation in simulating convection-diffusion problems. Several examples are presented to verify the feasibility and effectiveness of our method.

Numerical Solutions of Second-Order Elliptic Equations with C-Bézier Basis

“…Spline basis functions have been used to produce better approximations of solutions, as was already discussed, and the C-Bézier basis is a particularly good spline function [27]. Sun et al [28] showed that the C-Bézier and H-Bézier basis functions have a much better approximation in simulating convection-diffusion problems. Sun et al [29] combine the Galerkin finite element method with C-Bézier basis functions to solve unsteady elastic equations, and the numerical results indicate that the method has much better precision in solving unsteady elastic equations.…”