Crank-Nicolson Galerkin approximations to nonlinear Schr "odinger\n equations with rough potentials

2016/08/07 by Patrick Henning, Daniel Peterseim, Henning, Patrick +1 · 4 citations
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Advanced Numerical Methods in Computational Mathematics #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1608.02267

openalex publication_date 2016/08/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

This paper analyses the numerical solution of a class of non-linear\nSchr "odinger equations by Galerkin finite elements in space and a mass- and\nenergy conserving variant of the Crank-Nicolson method due to Sanz-Serna in\ntime. The novel aspects of the analysis are the incorporation of rough,\ndiscontinuous potentials in the context of weak and strong disorder, the\nconsideration of some general class of non-linearities, and the proof of\nconvergence with rates in L^\∞(L²) under moderate regularity\nassumptions that are compatible with discontinuous potentials. For sufficiently\nsmooth potentials, the rates are optimal without any coupling condition between\nthe time step size and the spatial mesh width.\n

Citations

• Structure of a quantized vortex in boson systems

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