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Journal of Scientific Computing

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08-08-2017

Superconvergence of Immersed Finite Volume Methods for One-Dimensional Interface Problems

Authors: Waixiang Cao, Xu Zhang, Zhimin Zhang, Qingsong Zou

Published in: Journal of Scientific Computing | Issue 2-3/2017

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Abstract

In this paper, we introduce a class of high order immersed finite volume methods (IFVM) for one-dimensional interface problems. We show the optimal convergence of IFVM in \(H^1\)- and \(L^2\)-norms. We also prove some superconvergence results of IFVM. To be more precise, the IFVM solution is superconvergent of order \(p+2\) at the roots of generalized Lobatto polynomials, and the flux is superconvergent of order \(p+1\) at generalized Gauss points on each element including the interface element. Furthermore, for diffusion interface problems, the convergence rates for IFVM solution at the mesh points and the flux at generalized Gauss points can both be raised to 2p. These superconvergence results are consistent with those for the standard finite volume methods. Numerical examples are provided to confirm our theoretical analysis.

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Title: Superconvergence of Immersed Finite Volume Methods for One-Dimensional Interface Problems
Authors: Waixiang Cao
Xu Zhang
Zhimin Zhang
Qingsong Zou
Publication date: 08-08-2017
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 2-3/2017
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-017-0532-6

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