@Article{NMTMA-14-1,
author = {Xiang, Wang and Zhang, Yuqing},
title = {On the Construction and Analysis of Finite Volume Element Schemes with Optimal $L^2$ Convergence Rate},
journal = {Numerical Mathematics: Theory, Methods and Applications},
year = {2021},
volume = {14},
number = {1},
pages = {47--70},
abstract = {<p style="text-align: justify;">We provide a general construction method for a finite volume element
(FVE) scheme with the optimal $L^2$ convergence rate. The $k$-($k$-1)-order orthogonal
condition (generalized) is proved to be a sufficient and necessary condition for a $k$-order FVE scheme to have the optimal $L^2$ convergence rate in 1D, in which the
independent dual parameters constitute a ($k$-1)-dimension surface in the reasonable
domain in $k$-dimension.<br/>In the analysis, the dual strategies in different primary elements are not necessarily to be the same, and they are allowed to be asymmetric in each primary element,
which open up more possibilities of the FVE schemes to be applied to some complex
problems, such as the convection-dominated problems. It worth mentioning that,
the construction can be extended to the quadrilateral meshes in 2D. The stability
and $H^1$ estimate are proved for completeness. All the above results are demonstrated by numerical experiments.</p>},
issn = {2079-7338},
doi = {https://doi.org/10.4208/nmtma.OA-2020-0027},
url = {https://global-sci.com/article/90291/on-the-construction-and-analysis-of-finite-volume-element-schemes-with-optimal-l2-convergence-rate}
}