New Formulation using the Finite Element Method for the Two-Dimensional Convection-Diffusion Equation. Approach by the Error Analysis Technique.

Abstract

In our previous paper, the modified Galerkin method (MGM) was proposed for the one-dimensional convection-diffusion equation as one of the most efficient methods. In this paper, the MGM concept has been analytically extended to the two-dimensional convection-diffusion equation. The triangular linear and the rectangular linear elements were investigated by the error analysis technique combined with an analytical solution. Then, six correction coefficients were obtained for MGM and the Galerkin method (GL). At the same time, two finite difference methods were investigated using the error analysis technique. Then, all of these numerical methods were evaluated from the viewpoints of phase error and dissipation error. The MGM was proven to be the most efficient numerical method among these five methods. Through numerical experiments by MGM and GL, we confirmed the effectiveness of MGM and the general correspondence between the error analysis and the numerical simulation.