Numerical simulation and error estimation of the Davey-Stewartson equations with virtual element method

Abstract

This paper aims to present the virtual element method (VEM) for solving the Davey–Stewartson equations with application in fluid mechanics. The VEM is a recent technology that can be regarded as a generalization of the standard finite element method (FEM) to general meshes without the need to integrate complex nonpolynomial functions on the elements. This method only utilizes degrees of freedom associated with the boundary, hence reducing computational complexity compared to the standard FEM. To obtain a full- discrete scheme we combine a semi-implicit scheme with the VEM for time and space variable discretizations, respectively. Furthermore, we obtain an error bound for the full-discrete scheme. The theoretical analysis demonstrates that the convergence rate in the $L^2$ norm is $O(h^2+\tau )$. Numerical examples confirm efficiency and applicability of the presented method and validate the theoretical outcomes.