Volume 16, Issue 6
Analysis of a Special Immersed Finite Volume Method for Elliptic Interface Problems

Int. J. Numer. Anal. Mod., 16 (2019), pp. 964-984.

Published online: 2019-08

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• 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].

• Keywords

Immersed finite volume method, stability, optimal convergence rates, immersed finite element method.

35R35, 49J40, 60G40

liuk53@mail2.sysu.edu.cn (Kai Liu)

mcszqs@mail.sysu.edu.cn (Qingsong Zou)

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@Article{IJNAM-16-964, author = {Kai and Liu and liuk53@mail2.sysu.edu.cn and 13825 and Department of Hand Surgery, The First Hospital, Jilin University, Changchun 130027, China and Kai Liu and Qingsong and Zou and mcszqs@mail.sysu.edu.cn and 12386 and School of Data and Computer Science, Guangdong Province Key Laboratory of Computational Science, Sun Yat-sen University, Guangzhou, 510006, P.R. China and Qingsong Zou}, 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 = {

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].

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/13262.html} }
TY - JOUR T1 - Analysis of a Special Immersed Finite Volume Method for Elliptic Interface Problems AU - Liu , Kai AU - Zou , Qingsong JO - International Journal of Numerical Analysis and Modeling VL - 6 SP - 964 EP - 984 PY - 2019 DA - 2019/08 SN - 16 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/13262.html KW - Immersed finite volume method, stability, optimal convergence rates, immersed finite element method. AB -

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].

Kai Liu & Qingsong Zou. (2019). Analysis of a Special Immersed Finite Volume Method for Elliptic Interface Problems. International Journal of Numerical Analysis and Modeling. 16 (6). 964-984. doi:
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