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Error Estimates for an Immersed Finite Element Method for Second Order Hyperbolic Equations in Inhomogeneous Media

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Abstract

A group of partially penalized immersed finite element (PPIFE) methods for second-order hyperbolic interface problems were discussed in Yang (Numer Math Theor Methods Appl 11:272–298, 2018) where the author proved their optimal O(h) convergence in an energy norm under a sub-optimal piecewise \(H^3\) regularity assumption. In this article, we reanalyze the fully discrete PPIFE method presented in Yang (2018). Utilizing the error bounds given recently in Guo et al. (Int J Numer Anal Model 16(4):575–589, 2019) for elliptic interface problems, we are able to derive optimal a-priori error bounds for this PPIFE method not only in the energy norm but also in \(L^2\) norm under the standard piecewise \(H^2\) regularity assumption in the space variable of the exact solution, rather than the excessive piecewise \(H^3\) regularity. Numerical simulations for standing and travelling waves are presented, which corroboratively confirm the reported error analysis.

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  1. Department of Mathematics, Virginia Tech, Blacksburg, VA, 24061, USA

    Slimane Adjerid, Tao Lin & Qiao Zhuang

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  1. Slimane Adjerid
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  2. Tao Lin
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  3. Qiao Zhuang
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Correspondence to Qiao Zhuang.

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Adjerid, S., Lin, T. & Zhuang, Q. Error Estimates for an Immersed Finite Element Method for Second Order Hyperbolic Equations in Inhomogeneous Media. J Sci Comput 84, 35 (2020). https://doi.org/10.1007/s10915-020-01283-0

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