An ultra-weak space-time variational formulation for the Schrödinger equation. (English) Zbl 07922209
Summary: 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.
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.
MSC:
| 35Q41 | Time-dependent Schrödinger equations and Dirac equations |
| 35L15 | Initial value problems for second-order hyperbolic equations |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
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