Energy-conservative finite element methods for nonlinear Schrödinger equations
Abstract: This thesis is devoted to numerical methods for nonlinear Schrödingerequations (NLSEs). These equations have applications in a variety offields such as optics, fluid dynamics, solid-state physics and, of course,quantum mechanics. Notably, the cubic NLSE describes the dynamics of Bose-Einstein condensates. The Physics community has a livleyinterest in this phenomenon as it offers a way of studying quantumphysics on, in some sense, macroscopic scales. This particular application is a major motivation behind this thesis. A central focus when numerically solving NLSEs has been to preserve the time invariants ofthese equations in the discrete setting. This aspect is studied in this thesis by means of both analysis and numerical examples. A novel approach for solving the cubic NLSE, that is both efficient and robust, issuggested. The method combines a spatial discretization based on themethod of Localized Orthogonal Decomposition with a conservativetime integrator. The thesis consists of 3 papers and an introduction.In Paper A is presented a numerical comparison of various mass conservative discretizations for the time-dependent cubic NLSE. Themain observation of this paper is that mass conservation alone is insufficient in cases of reduced regularity and additional potential terms.In paper B we prove optimal L∞(H1)-error estimates of the Crank Nicolson discretization, both in the semi-discrete Hilbert space setting,as well as in fully-discrete finite element settings. We also suggest afixed-point iteration to solve the arising nonlinear system of equationsthat makes the method easy to implement and efficient. This is illustrated by numerical experiments.In Paper C we present a novel method for solving the cubic NLSE.We show that using a spatial discretization based on the method of Localized Orthogonal Decomposition, the time invariants of the equation are initially approximated to O(H6) with respect to the chosenmesh size H, while only requiring H4-regularity. Furthermore, the low errors with respect to the time invariants are preserved in time by a highly efficient modified Crank-Nicolson time integrator tailored forthe LOD-space. Finally, we demonstrate the dramatic effect inaccurate representation of the time invariants can have on the numerical solution.
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