On polynomial Trefftz spaces for the linear time-dependent Schr\"odinger equation

TL;DR

This paper investigates the approximation capabilities of polynomial Trefftz spaces for the linear Schrödinger equation, demonstrating optimal convergence rates and explicit basis construction within a discontinuous Galerkin framework.

Contribution

It establishes that Trefftz polynomial spaces achieve optimal convergence rates similar to standard polynomials, with explicit construction and dimension analysis for the Schrödinger equation.

Findings

Achieves $h$-convergence rates comparable to polynomial degree $p$

Provides explicit basis construction for Trefftz spaces

Shows dimension matches polynomials of degree $2p$ in $d$ variables

Abstract

We study the approximation properties of complex-valued polynomial Trefftz spaces for the $(d+1)$-dimensional linear time-dependent Schr\"odinger equation. More precisely, we prove that for the space-time Trefftz discontinuous Galerkin variational formulation proposed by G\'omez, 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 $2p$ in $d$ variables, and bases are easily constructed.