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arXiv · 2106.04724

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

Sergio Gómez·Andrea Moiola
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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.

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Sergio Gómez, Andrea Moiola. 2021-06-08. A space-time Trefftz discontinuous Galerkin method for the linear Schr\"odinger equation. https://doi.org/10.1137/21m1426079

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