Analysis of Two-Grid Characteristic Finite Element Methods for Convection-Diffusion Equations

Abstract

In this paper, two efficient two-grid algorithms for the convection-diffusion problem with a modified characteristic finite element method are studied. We present an optimal error estimate in $L^p$ -norm for the characteristic finite element method unconditionally, while all previous works require certain time-step restrictions. To linearize the characteristic method equations, two-grid algorithms based on the Newton iteration approach and the correction method are applied. The error estimate and the convergence result of the two-grid method are derived in detail. It is shown that the coarse space can be extremely coarse and achieve asymptotically optimal approximations as long as the mesh sizes $H=\mathcal{O}\left(h^{1/3}\right)$ in the first algorithm and $H=\mathcal{O}\left(h^{1/4}\right)$ in the second algorithm, respectively. Finally, two numerical examples are presented to demonstrate the theoretical analysis.