OPTIMIZED AND QUASI-OPTIMAL SCHWARZ WAVEFORM RELAXATION FOR THE ONE-DIMENSIONAL SCHRÖDINGER EQUATION

Abstract

Schwarz waveform relaxation algorithms are designed for the linear Schrödinger equation with potential. Two classes of algorithms are introduced: the quasi-optimal algorithm, based on the transparent continuous or discrete boundary condition, and the optimized complex Robin algorithm. We analyze their properties in one dimension. First, well-posedness and convergence are studied, in the overlapping and the non-overlapping case, for constant or non-constant potentials. Then discrete algorithms are established, for which convergence is proved through discrete energies or Fourier transforms, as in the continuous case. Numerical results illustrate the efficiency of the methods, for various types of potentials and any number of subdomains.