Volume 11, Issue 4
Quadratic Finite Volume Method for a Nonlinear Elliptic Problem

Yanwei Du, Yonghai Li & Zhiqiang Sheng

Adv. Appl. Math. Mech., 11 (2019), pp. 838-869.

Published online: 2019-06

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  • 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.

  • Keywords

Nonlinear elliptic problem, quadratic finite volume method, optimal error estimates, orthogonal conditions.

  • AMS Subject Headings

65N08, 65N15

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-11-838, author = {}, title = {Quadratic Finite Volume Method for a Nonlinear Elliptic Problem}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2019}, volume = {11}, number = {4}, pages = {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.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2017-0231}, url = {http://global-sci.org/intro/article_detail/aamm/13191.html} }
TY - JOUR T1 - Quadratic Finite Volume Method for a Nonlinear Elliptic Problem JO - Advances in Applied Mathematics and Mechanics VL - 4 SP - 838 EP - 869 PY - 2019 DA - 2019/06 SN - 11 DO - http://dor.org/10.4208/aamm.OA-2017-0231 UR - https://global-sci.org/intro/article_detail/aamm/13191.html KW - Nonlinear elliptic problem, quadratic finite volume method, optimal error estimates, orthogonal conditions. AB -

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.

Yanwei Du, Yonghai Li & Zhiqiang Sheng. (2019). Quadratic Finite Volume Method for a Nonlinear Elliptic Problem. Advances in Applied Mathematics and Mechanics. 11 (4). 838-869. doi:10.4208/aamm.OA-2017-0231
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