Applied PDE Group
Nonlinear Partial Differential Equations in Fluid and Solid Mechanics


An adaptive finite element method for the Linear Schrodinger equation

​Th. Katsaounis, I. Kyza, An adaptive finite element method for the linear Schrodinger equation, Numer. Math. 129(1):55-90, 2015​​​​​
Th. Katsaounis, I. Kyza
Aposteriori Error Control, Adaptivity, Schrodinger eq., Finite Element Method
We derive optimal order a posteriori error estimates for fully discrete approximations of linear Schrödinger-type equations, in the L^{\infty​}( L^2 )-norm. 
For the discretization in time we use the Crank–Nicolson method, while for the space discretization we use finite element spaces that are allowed to change in time. The derivation of the estimators is based on a novel elliptic reconstruction that leads to estimates which reflect the physical properties of Schrödinger equations. The final estimates are obtained using energy techniques and residual-type estimators. Various numerical experiments for the one-dimensional linear Schrödinger equation in the semiclassical regime, verify and complement our theoretical results. The numerical implementations are performed with both uniform partitions and adaptivity in time and space. For adaptivity, we further develop and analyze an existing time-space adaptive algorithm to the cases of Schrödinger equations. The adaptive algorithm reduces the computational cost substantially and provides efficient error control for the solution and the observables of the problem, especially for small values of the Planck constant.