The mass- and energy-conserving relaxation virtual element method for the nonlinear Schrödinger equation

Abstract

This paper develops a conservative relaxation virtual element method for the nonlinear Schrödinger equation on polygonal meshes. The advantage of this method is to build the virtual element space where the basis functions do not need to be explicitly defined for each local element, and the bilinear forms and nonlinear terms can be computed by using elementwise polynomial projections and pre-defined degrees of freedom. Furthermore, the constructed schemes ensure the conservation of both mass and energy in discrete senses. By using the Brouwer fixed point theorem, we prove the unique solvability of the fully discrete scheme. Finally, some numerical experiments are implemented to verify the theoretical results.