A matrix-based approach to the stability of a space-time isogeometric method for the linear Schr\"odinger equation

TL;DR

This paper introduces a stable space-time isogeometric finite element method for the linear Schrödinger equation, using matrix analysis to prove unconditional stability and demonstrate optimal convergence.

Contribution

The paper develops a novel matrix-based stability analysis for a space-time isogeometric method, overcoming challenges posed by nonlocal spline bases.

Findings

The scheme is unconditionally stable due to well-conditioned Toeplitz-like matrices.

Numerical experiments confirm optimal convergence and stability.

The approach preserves mass and energy at the final time.

Abstract

We propose a space-time isogeometric finite element method for the linear Schr\"odinger equation, and establish its unconditional stability through a matrix-based analysis. Although maximal-regularity splines in time provide higher accuracy per degree of freedom compared to piecewise continuous polynomials, the nonlocal support of the spline bases precludes the use of standard variational arguments in the stability proofs. To overcome this, we show that the resulting scheme is governed by a family of nearly Toeplitz system matrices and, by studying the condition number of these matrices, we prove that the family is weakly well-conditioned, which guarantees the unconditional stability of the method. Furthermore, the discrete scheme preserves mass and energy at the final time. Numerical experiments confirm our theoretical findings and illustrate the optimal convergence behavior of the scheme. Finally, we exploit an algebraic connection between our formulation and a recent first-order-in-time space-time isogeometric method for the wave equation to derive a complete matrix-based stability analysis for the latter.