Convergence of a full discrete finite element method for the Korteweg–de Vries equation

Abstract

In this paper, we put emphasis on discussing a full discrete finite element scheme for the Korteweg–de Vries equation, where nonlinear term is dealt with a semi-implicit scheme and temporal term is discreted by the Euler scheme. Theoretical analysis is based on error splitting technique, i.e., error function is split as temporal error function plus spatial error function, and then unconditionally optimal error estimates of the considered full discrete scheme are obtained. Numerical results are provided to confirm our theoretical analysis, which show that no time-step condition is needed.