APPLICATION OF A RESIDUAL-NORM-BASED ALGORITHM TO SOLVE ELLIPTIC BOUNDARY VALUE PROBLEMS

Authors

Yung-Wei Chen, Department of Marine Engineering, National Taiwan Ocean University, Keelung, Taiwan, R.O.CFollow
Chun-Ming Chang, Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung, Taiwan, R.O.C
Hui-Ming Fang, Department of Systems Engineering and Naval Architecture, National Taiwan Ocean University, Keelung, Taiwan, R.O.C.
Pai-Chen Guan, Department of Systems Engineering and Naval Architecture, National Taiwan Ocean University, Keelung, Taiwan, R.O.C.

Abstract

In this study, we use a residual-norm-based algorithm (RNBA) to solve nonlinear elliptic boundary value problems (BVPs) on an arbitrary planar domain. For complex geometries, BVPs are very difficult and time-consuming to solve using conventional finite difference methods (FDMs). To overcome these problems, we apply a novel finite difference method (NFDM). By adding a fictitious rectangular domain and a bilinear function, we can easily treat geometrically complex boundary conditions. Then, through the use of the internal residual and the boundary residual, we can easily obtain the solution without the necessity of computing a matrix inverse. The RNBA avoids the oscillations that can occur in the manifold-based exponentially convergent algorithm (MBECA) by maintaining the manifold properties while guaranteeing convergence order greater than one. The accuracy and the convergence behaviour of this new method are demonstrated with several examples.

Recommended Citation

Chen, Yung-Wei; Chang, Chun-Ming; Fang, Hui-Ming; and Guan, Pai-Chen (2016) "APPLICATION OF A RESIDUAL-NORM-BASED ALGORITHM TO SOLVE ELLIPTIC BOUNDARY VALUE PROBLEMS," Journal of Marine Science and Technology: Vol. 24: Iss. 4, Article 3.
DOI: 10.6119/JMST-016-0115-1
Available at: https://jmstt.ntou.edu.tw/journal/vol24/iss4/3