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Conditioning of the finite volume element method for diffusion problems with general simplicial meshes


Authors: Xiang Wang, Weizhang Huang and Yonghai Li
Journal: Math. Comp. 88 (2019), 2665-2696
MSC (2010): Primary 65N08, 65F35
DOI: https://doi.org/10.1090/mcom/3423
Published electronically: March 26, 2019
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Abstract: The conditioning of the linear finite volume element discretization for general diffusion equations is studied on arbitrary simplicial meshes. The condition number is defined as the ratio of the maximal singular value of the stiffness matrix to the minimal eigenvalue of its symmetric part. This definition is motivated by the fact that the convergence rate of the generalized minimal residual method for the corresponding linear systems is determined by the ratio. An upper bound for the ratio is established by developing an upper bound for the maximal singular value and a lower bound for the minimal eigenvalue of the symmetric part. It is shown that the bound depends on three factors: the number of the elements in the mesh, the mesh nonuniformity measured in the Euclidean metric, and the mesh nonuniformity measured in the metric specified by the inverse diffusion matrix. It is also shown that the diagonal scaling can effectively eliminate the effects from the mesh nonuniformity measured in the Euclidean metric. Numerical results for a selection of examples in one, two, and three dimensions are presented.


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Additional Information

Xiang Wang
Affiliation: School of Mathematics, Jilin University, Changchun 130012, People’s Republic of China
Email: wxjldx@jlu.edu.cn

Weizhang Huang
Affiliation: Department of Mathematics, The University of Kansas, Lawrence, Kansas 66045
Email: whuang@ku.edu

Yonghai Li
Affiliation: School of Mathematics, Jilin University, Changchun 130012, People’s Republic of China
Email: yonghai@jlu.edu.cn

DOI: https://doi.org/10.1090/mcom/3423
Received by editor(s): February 4, 2018
Received by editor(s) in revised form: August 2, 2018, and January 11, 2019
Published electronically: March 26, 2019
Additional Notes: This work was supported in part by the National Natural Science Foundation of China through grants 11701211 and 11371170, the China Postdoctoral Science Foundation through grant 2017M620106, the Joint Fund of the National Natural Science Foundation of China and the China Academy of Engineering Physics (NASF) through grant U1630249, and the Science Challenge Program (China) through grant JCKY2016212A502.
The first-named author was supported by China Scholarship Council (CSC) under grant 201506170088 for his research visit to the University of Kansas from September of 2015 to September of 2016.
Article copyright: © Copyright 2019 American Mathematical Society