A space-time Trefftz discontinuous Galerkin method for the linear Schr\"odinger equation

TL;DR

This paper introduces a novel space-time Trefftz discontinuous Galerkin method for solving the Schrödinger equation, achieving reduced degrees of freedom and high-order convergence, with proven stability and numerical validation.

Contribution

The paper develops a Trefftz DG method using wave functions that satisfy the Schrödinger equation locally, reducing degrees of freedom and providing optimal error estimates.

Findings

Proven well-posedness and stability of the method.

Achieved optimal high-order convergence in 1D and 2D cases.

Numerical experiments confirm theoretical error estimates.

Abstract

A space-time Trefftz discontinuous Galerkin method for the Schr\"odinger equation with piecewise-constant potential is proposed and analyzed. Following the spirit of Trefftz methods, trial and test spaces are spanned by non-polynomial complex wave functions that satisfy the Schro\"odinger equation locally on each element of the space-time mesh. This allows for a significant reduction in the number of degrees of freedom in comparison with full polynomial spaces. We prove well-posedness and stability of the method, and, for the one- and two- dimensional cases, optimal, high-order, h-convergence error estimates in a skeleton norm. Some numerical experiments validate the theoretical results presented.