On polynomial Trefftz spaces for the linear time-dependent Schrödinger equation

Sergio Gómez; Andrea Moiola; Ilaria Perugia; Paul Stocker

doi:10.1016/j.aml.2023.108824

paper

On polynomial Trefftz spaces for the linear time-dependent Schrödinger equation

arXiv:2306.09571 · doi:10.1016/j.aml.2023.108824

Abstract

We study the approximation properties of complex-valued polynomial Trefftz spaces for the $(d + 1)$ -dimensional linear time-dependent Schrödinger equation. More precisely, we prove that for the space-time Trefftz discontinuous Galerkin variational formulation proposed by Gómez, Moiola (SIAM. J. Num. Anal. 60(2): 688-714, 2022), the same $h$ -convergence rates as for polynomials of degree $p$ in $(d + 1)$ variables can be obtained in a mesh-dependent norm by using a space of Trefftz polynomials of anisotropic degree. For such a space, the dimension is equal to that of the space of polynomials of degree $2 p$ in $d$ variables, and bases are easily constructed.