{
  "id": "2212.14398",
  "version": "v1",
  "published": "2022-12-29T18:12:43.000Z",
  "updated": "2022-12-29T18:12:43.000Z",
  "title": "An Ultra-Weak Space-Time Variational Formulation for the Schrödinger Equation",
  "authors": [
    "Stefan Hain",
    "Karsten Urban"
  ],
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "We present a well-posed ultra-weak space-time variational formulation for the time-dependent version of the linear Schr\\\"odinger 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-29T18:12:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35L15",
      "65M15",
      "65M60"
    ],
    "keywords": [
      "schrödinger equation",
      "well-posed ultra-weak space-time variational formulation",
      "optimal discrete inf-sup stability",
      "inf-sup optimal petrov-galerkin discretization",
      "space-time petrov-galerkin discretization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}