A generalized finite-difference method based on minimizing global residual |
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| Affiliation: | 1. Institute of Crystallography FSRC “Crystallography and Photonics”, Russian Academy of Sciences, Leninskii pr. 59, Moscow 119333, Russia;2. National University of Science and Technology “MISIS”, Leninsky pr. 4, Moscow 119049, Russia;3. IFW Dresden, SAWLab Saxony, P.O. 27 00 16, D-01171 Dresden, Germany;4. InnoXacs, Am Muehlfeld 34, D-01744 Dippoldiswalde, Germany;1. School of Mathematics and Quantitative Economics, Dongbei University of Finance and Economics, Dalian 116025, China;2. School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China;1. Department of Medicine/Cardiology, The Zena and Michael A. Weiner Cardiovascular Institute and the Marie-Josée and Henry R. Kravis Cardiovascular Health Center, Icahn School of Medicine at Mount Sinai, New York, New York;2. Department of Pathology and Laboratory Medicine, University of Kentucky Medical Center, Lexington, Kentucky;1. Moscow State University of Civil Engineering (National Research University), 26 Yaroslavskoye Shosse, Moscow, 129337, Russia |
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| Abstract: | An improved generalized finite-difference method is proposed in this paper, as an alternative meshless method to solve differential equations. The method establishes discrete equations by minimizing a global residual. A general frame for constructing difference schemes is first described. As one choice the moving least square method is used in this paper. Compared with other generalized finite-difference methods, the improved method yields a set of discrete equations having the favorable properties such as symmetric, positive definite and well conditioned. Compared with meshless methods based on a variational principle or a weak form, the method described in this paper does not need a numerical integration and thus provides an alternative way to avoid the difficulties in implementing a numerical integration. In the proposed method there is no such inconvenience in applying essential boundary conditions as commonly encountered in other meshless methods. Numerical examples show that the improved method has a high convergence rate and can produce accurate results even with a coarse mesh. |
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