TY - JOUR
T1 - Convergence of a finite element scheme for the two-dimensional time-dependent Schrödinger equation in a long strip
AU - Jin, Jicheng
AU - Wu, Xiaonan
N1 - Funding Information:
The research of the first author was supported by NSFC (10971059) and Hunan Provincial Natural Science Foundation of China (09JJ3007) and the research of the second author was supported by RGC of Hong Kong and FRG of Hong Kong Baptist University.
PY - 2010/6/1
Y1 - 2010/6/1
N2 - This paper addresses the finite element method for the two-dimensional time-dependent Schrödinger equation on an infinite strip by using artificial boundary conditions. We first reduce the original problem into an initial-boundary value problem in a bounded domain by introducing a transparent boundary condition, then fully discretize this reduced problem by applying the Crank-Nicolson scheme in time and a bilinear or quadratic finite element approximation in space. This scheme, by a rigorous analysis, has been proved to be unconditionally stable and convergent, and its convergence order has also been obtained. Finally, two numerical examples are given to verify the accuracy of the scheme.
AB - This paper addresses the finite element method for the two-dimensional time-dependent Schrödinger equation on an infinite strip by using artificial boundary conditions. We first reduce the original problem into an initial-boundary value problem in a bounded domain by introducing a transparent boundary condition, then fully discretize this reduced problem by applying the Crank-Nicolson scheme in time and a bilinear or quadratic finite element approximation in space. This scheme, by a rigorous analysis, has been proved to be unconditionally stable and convergent, and its convergence order has also been obtained. Finally, two numerical examples are given to verify the accuracy of the scheme.
KW - Artificial boundary condition
KW - Finite element method
KW - Schrödinger equation
UR - http://www.scopus.com/inward/record.url?scp=77949491345&partnerID=8YFLogxK
U2 - 10.1016/j.cam.2010.01.042
DO - 10.1016/j.cam.2010.01.042
M3 - Journal article
AN - SCOPUS:77949491345
SN - 0377-0427
VL - 234
SP - 777
EP - 793
JO - Journal of Computational and Applied Mathematics
JF - Journal of Computational and Applied Mathematics
IS - 3
ER -