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Unconditional stability and convergence of Crank-Nicolson Galerkin FEMs for a nonlinear Schrödinger-Helmholtz system. (English) Zbl 1402.65119
Two linearized Crank-Nicolson Galerkin finite element methods (FEM) are presented for the complex nonlinear Schrödinger equations defined in a bounded and convex polygon or polyhedron in two and three dimensions, respectively. Homogeneous boundary conditions are considered and for simplification the Schrödinger-Helmholtz system is analyzed. Spatial and temporal error analyses are performed. Unconditionally optimal \(L^2\) error estimates are proven in both two and three space dimensions based on the uniform boundedness of the numerical solution in the \(L^\infty\) norm. A time-discrete system is introduced using a splitting technique.
Finally, three numerical examples are presented, which illustrate the theoretical convergence and stability results in 2d and 3d. The software FreeFem++ is used with uniform triangular grids on square and cubic domains.

MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
Software:
FreeFem++
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