Year: 2019
Author: Kai Liu, Qingsong Zou
International Journal of Numerical Analysis and Modeling, Vol. 16 (2019), Iss. 6 : pp. 964–984
Abstract
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 $H^1$ space and the order $\mathcal{O}$($h^{3/2}$) in the $L^2$ space. Numerically, we observe that our method converges with the optimal convergence rate $\mathcal{O}$($h$) under the $H^1$ norm and with the the optimal convergence rate $\mathcal{O}$($h^2$) under the $L^2$ 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 $H^1$ norm error and diminished rate 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].
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2019-IJNAM-13262
International Journal of Numerical Analysis and Modeling, Vol. 16 (2019), Iss. 6 : pp. 964–984
Published online: 2019-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 21
Keywords: Immersed finite volume method stability optimal convergence rates immersed finite element method.