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Volume 14, Issue 4-5
A Finite Element Method for the One-Dimensional Prescribed Curvature Problem

Susanne C. Brenner, Li-Yeng Sung, Zhuo Wang & Yuesheng Xu

Int. J. Numer. Anal. Mod., 14 (2017), pp. 646-669.

Published online: 2017-08

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  • Abstract

We develop a finite element method for solving the Dirichlet problem of the one- dimensional prescribed curvature equation due to its irreplaceable role in applications. Specifically, we first analyze the existence and uniqueness of the solution of the problem and then develop a finite element method to solve it. The well-posedness of the finite element method is shown by employing the Banach fixed-point theorem. The optimal error estimates of the proposed method in both the $H^1$ norm and the $L^2$ norm are established. We also design a Newton type iteration scheme to solve the resulting discrete nonlinear system. Numerical experiments are presented to confirm the order of convergence of the proposed method.

  • Keywords

Prescribed curvature equation, finite element method, Newton iteration, Banach fixed-point theorem.

  • AMS Subject Headings

65N06, 65N15

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-14-646, author = {Susanne C. Brenner, Li-Yeng Sung, Zhuo Wang and Yuesheng Xu}, title = {A Finite Element Method for the One-Dimensional Prescribed Curvature Problem}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2017}, volume = {14}, number = {4-5}, pages = {646--669}, abstract = {

We develop a finite element method for solving the Dirichlet problem of the one- dimensional prescribed curvature equation due to its irreplaceable role in applications. Specifically, we first analyze the existence and uniqueness of the solution of the problem and then develop a finite element method to solve it. The well-posedness of the finite element method is shown by employing the Banach fixed-point theorem. The optimal error estimates of the proposed method in both the $H^1$ norm and the $L^2$ norm are established. We also design a Newton type iteration scheme to solve the resulting discrete nonlinear system. Numerical experiments are presented to confirm the order of convergence of the proposed method.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/10054.html} }
TY - JOUR T1 - A Finite Element Method for the One-Dimensional Prescribed Curvature Problem AU - Susanne C. Brenner, Li-Yeng Sung, Zhuo Wang & Yuesheng Xu JO - International Journal of Numerical Analysis and Modeling VL - 4-5 SP - 646 EP - 669 PY - 2017 DA - 2017/08 SN - 14 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/10054.html KW - Prescribed curvature equation, finite element method, Newton iteration, Banach fixed-point theorem. AB -

We develop a finite element method for solving the Dirichlet problem of the one- dimensional prescribed curvature equation due to its irreplaceable role in applications. Specifically, we first analyze the existence and uniqueness of the solution of the problem and then develop a finite element method to solve it. The well-posedness of the finite element method is shown by employing the Banach fixed-point theorem. The optimal error estimates of the proposed method in both the $H^1$ norm and the $L^2$ norm are established. We also design a Newton type iteration scheme to solve the resulting discrete nonlinear system. Numerical experiments are presented to confirm the order of convergence of the proposed method.

Susanne C. Brenner, Li-Yeng Sung, Zhuo Wang and Yuesheng Xu. (2017). A Finite Element Method for the One-Dimensional Prescribed Curvature Problem. International Journal of Numerical Analysis and Modeling. 14 (4-5). 646-669. doi:
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