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

Kai Liu and Qingsong Zou

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

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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 H1-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 O(h) in the Hspace and the order O(h3/2) in the Lspace. Numerically, we observe that our method converges with the optimal convergence rate O(h) under the H1 norm and with the the optimal convergence rate O(h2) under the L2 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 O(h0.82) for the H1 norm error and diminished rate up to O(h1.1) for L2-norm error), when h is getting small, as illustrated in Tables 4 and 5 of [35].

  • History

Published online: 2019-08

  • AMS Subject Headings

35R35, 49J40, 60G40

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