Conforming and nonconforming conservative virtual element methods for nonlinear Schrödinger equation: A unified framework

Abstract We present, in a unified framework, conforming and nonconforming virtual element methods for nonlinear Schrodinger equation. The constructed schemes conserve not only the mass but also the energy in the discrete senses. Then, by using the Brouwer fixed point theorem, the Gagliardo–Nirenberg inequality and the classical Ritz projection, we prove the boundedness, unique solvability and optimal convergence of the conforming virtual element scheme. To obtain the boundedness, unique solvability and optimal convergence of the nonconforming virtual element scheme, we utilize a new defined Ritz projection and a new type of Gagliardo–Nirenberg inequality. The optimal rates of convergence in the discrete L 2 -norm are derived without any restrictions on the grid ratio for both types of virtual element methods. Finally, numerical examples on a set of polygonal meshes are given to support the theoretical analysis. As a contrast, we also supply another classical proof method of the optimal convergence which has to limit the time–space grid ratio τ = o ( h 1 ∕ 2 ) .