A spacetime DPG method for the Schrodinger equation

TL;DR

This paper introduces a novel spacetime Discontinuous Petrov Galerkin method for the linear Schrödinger equation, effectively handling irregular solutions and providing well-posed formulations with convergence analysis and numerical validation.

Contribution

The paper develops a second order spacetime DPG method for the Schrödinger equation, including well-posed strong and ultraweak formulations, and analyzes convergence and applicability to pulse propagation.

Findings

Well-posed strong and ultraweak formulations established.

Convergence of the ultraweak DPG method demonstrated.

Numerical experiments validate the method for pulse propagation.

Abstract

A spacetime Discontinuous Petrov Galerkin (DPG) method for the linear time-dependent Schrodinger equation is proposed. The spacetime approach is particularly attractive for capturing irregular solutions. Motivated by the fact that some irregular Schrodinger solutions cannot be solutions of certain first order reformulations, the proposed spacetime method uses the second order Schrodinger operator. Two variational formulations are proved to be well-posed: a strong formulation (with no relaxation of the original equation) and a weak formulation (also called the ultraweak formulation, that transfers all derivatives onto test functions). The convergence of the DPG method based on the ultraweak formulation is investigated using an interpolation operator. A standalone appendix analyzes the ultraweak formulation for general differential operators. Reports of numerical experiments motivated by pulse propagation in dispersive optical fibers are also included.