Numerische Mathematik Numerical analysis of a multiscale finite element scheme for the resolution of the stationary Schrödinger equation
Numerische Mathematik Numerical analysis of a multiscale finite element scheme for the resolution of the stationary Schrödinger equation
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数值数学 用于解决稳态薛定谔方程的多尺度有限元方案的数值分析
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发表时间：
2008
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通讯作者：
C. Negulescu
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作者：
C. Negulescu
N. Ben Abdallah和O. Pinaud在《开放量子系统中输运的多尺度模拟：共振和WKB插值》中提出了一种求解一维开放边界条件下Schrödinger方程的数值方法。[j] .物理学报，2013(1),387 - 398（2006）。该方法的主要特点是显著降低了所需精度的计算成本。它特别基于WKB基函数的推导，比标准多项式插值函数更适合于高振荡波函数的近似。本文对该方法进行了数值分析。给出了一致性和稳定性结果。建立了与波长λ无关的网格尺寸误差估计。这种特性说明了该方法的重要性，因为可以选择多波长网格来获得准确的结果，从而减少了仿真时间。数学学科分类（2000）35Q40·35J10·65L20·65L70·65L10
A numerical method for the resolution of the one-dimensional Schrödinger equation with open boundary conditions was presented in N. Ben Abdallah and O. Pinaud (Multiscale simulation of transport in an open quantum system: resonances and WKB interpolation. J. Comp. Phys. 213(1), 288–310 (2006)). The main attribute of this method is a significant reduction of the computational cost for a desired accuracy. It is based particularly on the derivation of WKB basis functions, better suited for the approximation of highly oscillating wave functions than the standard polynomial interpolation functions. The present paper is concerned with the numerical analysis of this method. Consistency and stability results are presented. An error estimate in terms of the mesh size and independent on the wavelength λ is established. This property illustrates the importance of this method, as multiwavelength grids can be chosen to get accurate results, reducing by this manner the simulation time. Mathematics Subject Classification (2000) 35Q40 · 35J10 · 65L20 · 65L70 · 65L10