Optimal L^2 error estimates of mass- and energy-conserved FE schemes for a nonlinear Schrödinger-type system
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In this paper, we present an implicit Crank-Nicolson finite element (FE) scheme for solving a nonlinear Schrödinger-type system, which includes Schrödinger-Helmholz system and Schrödinger-Poisson system. In our numerical scheme, we employ an implicit Crank-Nicolson method for time discretization and a conforming FE method for spatial discretization. The proposed method is proved to be well-posedness and ensures mass and energy conservation at the discrete level. Furthermore, we prove optimal L^2 error estimates for the fully discrete solutions. Finally, some numerical examples are provided to verify the convergence rate and conservation properties.
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