A Posteriori Error Control and Adaptivity for the Linear Schrondinger Equation

On behalf of the Department of Mathematics and Statistics in the College of Arts and Sciences, you are cordially invited to a seminar by Dr. Theodoros Katsaounis from KAUST University, Saudi Arabia

Abstract:

We derive optimal order aposteriori error estimates for fully discrete approximations of linearSchrodinger-type equations. For the discretization in time we use theCrank-Nicolson method, while for the space discretization we use finite elementspaces that are allowed to change in time. The derivation of the estimators isbased on a novel elliptic reconstruction that leads to estimates which reflectthe physical properties of Schrodinger equations. The final estimates areobtained using energy techniques and residual-type estimators. Variousnumerical experiments for the one-dimensional linear Schrodinger equation inthe semi classical regime, verify and complement our theoretical results. Thenumerical implementations are performed with both uniform partitions andadaptivity in time and space. For adaptivity, we further develop and analyze anexisting time-space adaptive algorithm to the cases of Schrodinger equations.The adaptive algorithm reduces the computational cost substantially andprovides efficient error control for the solution and the observables of theproblem, especially for small values of the Planck constant.

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