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Crank-Nicolson Galerkin approximations to nonlinear Schrödinger equations with rough potentials. (English) Zbl 1377.65128
Authors’ abstract: This paper analyzes the numerical solution of a class of nonlinear Schrödinger equations by Galerkin finite elements in space and a mass and energy conserving variant of the Crank-Nicolson method due to Sanz-Serna in time. The main aspects of the analysis are the incorporation of rough, discontinuous potentials in the context of weak and strong disorder, the consideration of some general class of nonlinearities, and the proof of convergence with rates in $$L^\infty(L^2)$$ under moderate regularity assumptions that are compatible with discontinuous potentials. For sufficiently smooth potentials, the rates are optimal without any coupling condition between the time step size and the spatial mesh width.

##### MSC:
 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 35Q55 NLS equations (nonlinear Schrödinger equations) 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
##### Software:
GPELab; GP-SCL; GP-SCL-HYB; imagtime3d-hyb; realtime3d-hyb
Full Text:
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