An ultra-weak space-time variational formulation for the Schrödinger equation

IF 1.8 2区数学 Q1 MATHEMATICS Journal of Complexity Pub Date : 2024-05-31 DOI:10.1016/j.jco.2024.101868

Stefan Hain, Karsten Urban

引用次数: 0

Abstract

We present a well-posed ultra-weak space-time variational formulation for the time-dependent version of the linear Schrödinger equation with an instationary Hamiltonian. We prove optimal inf-sup stability and introduce a space-time Petrov-Galerkin discretization with optimal discrete inf-sup stability.

We show norm-preservation of the ultra-weak formulation. The inf-sup optimal Petrov-Galerkin discretization is shown to be asymptotically norm-preserving, where the deviation is shown to be in the order of the discretization. In addition, we introduce a Galerkin discretization, which has suboptimal inf-sup stability but exact norm-preservation.

Numerical experiments underline the performance of the ultra-weak space-time variational formulation, especially for non-smooth initial data.

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薛定谔方程的超弱时空变分公式

我们针对线性薛定谔方程的时间依赖版本，提出了一个具有固定哈密顿的、假设良好的超弱时空变分公式。我们证明了最优 inf-sup 稳定性，并引入了具有最优离散 inf-sup 稳定性的时空 Petrov-Galerkin 离散化。我们证明了超弱公式的规范保留性，并证明了 inf-sup 最佳 Petrov-Galerkin 离散化具有渐近的规范保留性，其偏差在离散化的阶次上。此外，我们还引入了一种 Galerkin 离散化方法，它具有次优 inf-sup 稳定性，但具有精确的规范保留性。数值实验强调了超弱时空变分公式的性能，特别是对于非光滑初始数据。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Complexity 工程技术-计算机：理论方法

CiteScore

3.10

自引率

17.60%

发文量

审稿时长

>12 weeks

期刊介绍： The multidisciplinary Journal of Complexity publishes original research papers that contain substantial mathematical results on complexity as broadly conceived. Outstanding review papers will also be published. In the area of computational complexity, the focus is on complexity over the reals, with the emphasis on lower bounds and optimal algorithms. The Journal of Complexity also publishes articles that provide major new algorithms or make important progress on upper bounds. Other models of computation, such as the Turing machine model, are also of interest. Computational complexity results in a wide variety of areas are solicited. Areas Include: • Approximation theory • Biomedical computing • Compressed computing and sensing • Computational finance • Computational number theory • Computational stochastics • Control theory • Cryptography • Design of experiments • Differential equations • Discrete problems • Distributed and parallel computation • High and infinite-dimensional problems • Information-based complexity • Inverse and ill-posed problems • Machine learning • Markov chain Monte Carlo • Monte Carlo and quasi-Monte Carlo • Multivariate integration and approximation • Noisy data • Nonlinear and algebraic equations • Numerical analysis • Operator equations • Optimization • Quantum computing • Scientific computation • Tractability of multivariate problems • Vision and image understanding.