@Article{IJNAM-16-6,
author = {Liu, Kai and Zou, Qingsong},
title = {Analysis of a Special Immersed Finite Volume Method for Elliptic Interface Problems},
journal = {International Journal of Numerical Analysis and Modeling},
year = {2019},
volume = {16},
number = {6},
pages = {964--984},
abstract = {<p style="text-align: justify;">In this paper, we analyze a special immersed finite volume method that is different
from the classic immersed finite volume method by choosing special control volumes near the
interface. Using the elementwise stiffness matrix analysis technique and the $H^1$-norm-equivalence
between the immersed finite element space and the standard finite element space, we prove that
the special finite volume method is uniformly stable independent of the location of the interface.
Based on the stability, we show that our scheme converges with the optimal order $\mathcal{O}$($h$) in the $H^1$&nbsp;space and the order $\mathcal{O}$($h^{3/2}$) in the $L^2$&nbsp;space. Numerically, we observe that our method
converges with the optimal convergence rate $\mathcal{O}$($h$) under the $H^1$&nbsp;norm and with the the optimal
convergence rate $\mathcal{O}$($h^2$) under the $L^2$&nbsp;norm all the way even with very small mesh size $h$, while
the classic immersed finite element method is not able to maintain the optimal convergence rates (with diminished rate up to $\mathcal{O}$($h^{0.82}$) for the $H^1$&nbsp;norm&nbsp; error&nbsp; and&nbsp; diminished&nbsp; rate&nbsp; up to $\mathcal{O}$($h^{1.1}$)
for $L^2$-norm error), when $h$ is getting small, as illustrated in Tables 4 and 5 of [35].</p>},
issn = {2617-8710},
doi = {https://doi.org/2019-IJNAM-13262},
url = {https://global-sci.com/article/83086/analysis-of-a-special-immersed-finite-volume-method-for-elliptic-interface-problems}
}