Numerical analysis of a multiscale finite element scheme for the resolution of the stationary Schrödinger equation

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.

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