Volume 13, Issue 2
Linear and Quadratic Finite Volume Methods on Triangular Meshes for Elliptic Equations with Singular Solutions

G.-H. Jin, H.-G. Li, Q.-H. Zhang & Q.-S. Zou

Int. J. Numer. Anal. Mod., 13 (2016), pp. 244-264

Published online: 2016-03

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  • Abstract

This paper is devoted to the presentation and analysis of some linear and quadratic finite volume (FV) schemes for elliptic problems with singular solutions due to the non-smoothness of the domain. Our FV schemes are constructed over specially-designed graded triangular meshes. We provide sharp parameter selection criteria for the graded mesh, such that both the linear and quadratic FV schemes achieve the optimal convergence rate approximating singular solutions in H¹. In addition, we show that on the same mesh, a linear FV scheme obtains the optimal rate of convergence in L². Numerical tests are provided to verify the analysis.

  • Keywords

Finite volume method singular solution optimal convergence rate

  • AMS Subject Headings

65N08 65N15 65N50 35J15

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COPYRIGHT: © Global Science Press

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@Article{IJNAM-13-244, author = {G.-H. Jin, H.-G. Li, Q.-H. Zhang and Q.-S. Zou}, title = {Linear and Quadratic Finite Volume Methods on Triangular Meshes for Elliptic Equations with Singular Solutions}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2016}, volume = {13}, number = {2}, pages = {244--264}, abstract = {This paper is devoted to the presentation and analysis of some linear and quadratic finite volume (FV) schemes for elliptic problems with singular solutions due to the non-smoothness of the domain. Our FV schemes are constructed over specially-designed graded triangular meshes. We provide sharp parameter selection criteria for the graded mesh, such that both the linear and quadratic FV schemes achieve the optimal convergence rate approximating singular solutions in H¹. In addition, we show that on the same mesh, a linear FV scheme obtains the optimal rate of convergence in L². Numerical tests are provided to verify the analysis.}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/437.html} }
TY - JOUR T1 - Linear and Quadratic Finite Volume Methods on Triangular Meshes for Elliptic Equations with Singular Solutions AU - G.-H. Jin, H.-G. Li, Q.-H. Zhang & Q.-S. Zou JO - International Journal of Numerical Analysis and Modeling VL - 2 SP - 244 EP - 264 PY - 2016 DA - 2016/03 SN - 13 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/437.html KW - Finite volume method KW - singular solution KW - optimal convergence rate AB - This paper is devoted to the presentation and analysis of some linear and quadratic finite volume (FV) schemes for elliptic problems with singular solutions due to the non-smoothness of the domain. Our FV schemes are constructed over specially-designed graded triangular meshes. We provide sharp parameter selection criteria for the graded mesh, such that both the linear and quadratic FV schemes achieve the optimal convergence rate approximating singular solutions in H¹. In addition, we show that on the same mesh, a linear FV scheme obtains the optimal rate of convergence in L². Numerical tests are provided to verify the analysis.
G.-H. Jin, H.-G. Li, Q.-H. Zhang & Q.-S. Zou. (1970). Linear and Quadratic Finite Volume Methods on Triangular Meshes for Elliptic Equations with Singular Solutions. International Journal of Numerical Analysis and Modeling. 13 (2). 244-264. doi:
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