Year: 2019
Author: Yanwei Du, Yonghai Li, Zhiqiang Sheng
Advances in Applied Mathematics and Mechanics, Vol. 11 (2019), Iss. 4 : pp. 838–869
Abstract
In this article, a quadratic finite volume method is applied to solve the nonlinear elliptic equation. Firstly, we construct a finite volume scheme for this nonlinear equation. Then, under certain assumptions, the boundedness and ellipticity of the corresponding bilinear form are obtained. Moreover, we get the optimal error estimates not only in $H^{1}$-norm but also in $L^{2}$-norm where the optimal error estimate in $L^{2}$-norm depends on the optimal dual partition. In addition, the effect of numerical integration is analyzed. To confirm the theoretical analysis, we solve the nonlinear equation by the Newton iteration method and prove the quadratic rate of convergence. The numerical results show the effectiveness of our method.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/aamm.OA-2017-0231
Advances in Applied Mathematics and Mechanics, Vol. 11 (2019), Iss. 4 : pp. 838–869
Published online: 2019-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 32
Keywords: Nonlinear elliptic problem quadratic finite volume method optimal error estimates orthogonal conditions.