A Space-Time Continuous Galerkin Finite Element Method for Linear Schrödinger Equations
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913787284553728 |
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| author | Zank, Marco |
| author_facet | Zank, Marco |
| contents | We introduce a space-time finite element method for the linear time-dependent Schrödinger equation with Dirichlet conditions in a bounded Lipschitz domain. The proposed discretization scheme is based on a space-time variational formulation of the time-dependent Schrödinger equation. In particular, the space-time method is conforming and is of Galerkin-type, i.e., trial and test spaces are equal. We consider a tensor-product approach with respect to time and space, using piecewise polynomial, continuous trial and test functions. In this case, we state the global linear system and efficient direct space-time solvers based on exploiting the Kronecker structure of the global system matrix. This leads to the Bartels-Stewart method and the fast diagonalization method. Both methods result in solving a sequence of spatial subproblems. In particular, the fast diagonalization method allows for solving the spatial subproblems in parallel, i.e., a time parallelization is possible. Numerical examples for a two-dimensional spatial domain illustrate convergence in space-time norms and show the potential of the proposed solvers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_07269 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Space-Time Continuous Galerkin Finite Element Method for Linear Schrödinger Equations Zank, Marco Numerical Analysis 65F05, 65M60 We introduce a space-time finite element method for the linear time-dependent Schrödinger equation with Dirichlet conditions in a bounded Lipschitz domain. The proposed discretization scheme is based on a space-time variational formulation of the time-dependent Schrödinger equation. In particular, the space-time method is conforming and is of Galerkin-type, i.e., trial and test spaces are equal. We consider a tensor-product approach with respect to time and space, using piecewise polynomial, continuous trial and test functions. In this case, we state the global linear system and efficient direct space-time solvers based on exploiting the Kronecker structure of the global system matrix. This leads to the Bartels-Stewart method and the fast diagonalization method. Both methods result in solving a sequence of spatial subproblems. In particular, the fast diagonalization method allows for solving the spatial subproblems in parallel, i.e., a time parallelization is possible. Numerical examples for a two-dimensional spatial domain illustrate convergence in space-time norms and show the potential of the proposed solvers. |
| title | A Space-Time Continuous Galerkin Finite Element Method for Linear Schrödinger Equations |
| topic | Numerical Analysis 65F05, 65M60 |
| url | https://arxiv.org/abs/2504.07269 |