On Error Estimates of the Crank-Nicolson-Polylinear Finite Element Method with the Discrete TBC for the Generalized Schrödinger Equation in an Unbounded Parallelepiped
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arXiv
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| Format: | Preprint |
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2015
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| _version_ | 1866917178669793280 |
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| author | Zlotnik, Alexander |
| author_facet | Zlotnik, Alexander |
| contents | We deal with an initial-boundary value problem for the generalized time-dependent Schrödinger equation with variable coefficients in an unbounded $n$--dimensional parallelepiped ($n\geq 1$). To solve it, the Crank-Nicolson in time and the polylinear finite element in space method with the discrete transpa\-rent boundary conditions is considered. We present its stability properties and derive new error estimates $O(τ^2+|h|^2)$ uniformly in time in $L^2$ space norm, for $n\geq 1$, and mesh $H^1$ space norm, for $1\leq n\leq 3$ (a superconvergence result), under the Sobolev-type assumptions on the initial function. Such estimates are proved for methods with the discrete TBCs for the first time. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1503_07811 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | On Error Estimates of the Crank-Nicolson-Polylinear Finite Element Method with the Discrete TBC for the Generalized Schrödinger Equation in an Unbounded Parallelepiped Zlotnik, Alexander Numerical Analysis 65M60, 65M12, 35Q40 We deal with an initial-boundary value problem for the generalized time-dependent Schrödinger equation with variable coefficients in an unbounded $n$--dimensional parallelepiped ($n\geq 1$). To solve it, the Crank-Nicolson in time and the polylinear finite element in space method with the discrete transpa\-rent boundary conditions is considered. We present its stability properties and derive new error estimates $O(τ^2+|h|^2)$ uniformly in time in $L^2$ space norm, for $n\geq 1$, and mesh $H^1$ space norm, for $1\leq n\leq 3$ (a superconvergence result), under the Sobolev-type assumptions on the initial function. Such estimates are proved for methods with the discrete TBCs for the first time. |
| title | On Error Estimates of the Crank-Nicolson-Polylinear Finite Element Method with the Discrete TBC for the Generalized Schrödinger Equation in an Unbounded Parallelepiped |
| topic | Numerical Analysis 65M60, 65M12, 35Q40 |
| url | https://arxiv.org/abs/1503.07811 |