Finite Element Methods For Schr(?)dinger Equation With Complicated Nonlinear Terms

The Schrodinger equation with logarithmic nonlinear term has very complicated properties because logarithmic term does not have the condition of Lipschitz continuity.We can make the logarithm term satisfy the Lipschitz condition by the regularized technique.In this thesis,the Crank-Nicolson scheme is adopted to establish the finite element method,which satisfy the mass conservation and the energy conservation and its convergence is proved.Next,This idea is introduced into the nonlocal Maxwell-Schr(?)dinger coupled equations.The Maxwell-Schr(?)dinger equations with nonlocal effects are derived from the fluid dynamics model by assuming the separation of free and bound electrons in the metal response.The nonlinear term of the Schrodinger equation in the system is formed by the coupling of the vector potential function in the electromagnetic field and the wave function in the Schrodinger equation,which satisfies the Hamilton system and thus keeps energy conservation.The Maxwell-Schr(?)equations with nonlocal response are discretized by the frog-leaping finite element scheme and the mixed energy stability analysis and error estimation of the discretized scheme are obtained.Finally,the correctness of the theoretical analysis is verified by the numerical examples.