Conservative Conforming and Nonconforming VEMs for Fourth Order Nonlinear Schrödinger Equations with Trapped Term (Journal of Computational Mathematics, 2024)
This paper aims to construct and analyze the conforming and nonconforming virtualelement methods for a class of fourth order nonlinear Schrödinger equations with trappedterm. We mainly consider three types of virtual elements, including $H^2$ conforming virtual element, $C^0$ nonconforming virtual element and Morley-type nonconforming virtualelement. The fully discrete schemes are constructed by virtue of virtual element methodsin space and modified Crank-Nicolson method in time. We prove the mass and energyconservation, the boundedness and the unique solvability of the fully discrete schemes.After introducing a new type of the Ritz projection, the optimal and unconditional errorestimates for the fully discrete schemes are presented and proved. Finally, two numericalexamples are investigated to confirm our theoretical analysis.
Keywords: $H^2$ conforming virtual element; $C^0$ nonconforming virtual element; Morley-type nonconforming virtual element; Nonlinear Schrödinger equation; Conservation; Convergence.