Crank-Nicolson finite element discretizations for a two-dimensional linear Schrödinger-type equation posed in a noncylindrical domain

Antonopoulou, D.; Karali, G.; Plexousakis, M.; Zouraris, G.

doi:10.1090/S0025-5718-2014-02900-1

Crank-Nicolson finite element discretizations for a two-dimensional linear Schrödinger-type equation posed in a noncylindrical domain
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by D. C. Antonopoulou, G. D. Karali, M. Plexousakis and G. E. Zouraris PDF

Math. Comp. 84 (2015), 1571-1598 Request permission

Abstract:

Motivated by the paraxial narrow–angle approximation of the Helmholtz equation in domains of variable topography, we consider an initial- and boundary-value problem for a general Schrödinger-type equation posed on a two space-dimensional noncylindrical domain with mixed boundary conditions. The problem is transformed into an equivalent one posed on a rectangular domain, and we approximate its solution by a Crank–Nicolson finite element method. For the proposed numerical method, we derive an optimal order error estimate in the $L^2$ norm, and to support the error analysis we prove a global elliptic regularity theorem for complex elliptic boundary value problems with mixed boundary conditions. Results from numerical experiments are presented which verify the optimal order of convergence of the method.

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