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Error Estimates for an Immersed Finite Element Method for Second Order Hyperbolic Equations in Inhomogeneous Media
Journal of Scientific Computing ( IF 2.5 ) Pub Date : 2020-08-01 , DOI: 10.1007/s10915-020-01283-0
Slimane Adjerid , Tao Lin , Qiao Zhuang

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.



中文翻译:

非均匀介质中二阶双曲方程浸入式有限元方法的误差估计

惩罚一组部分地浸入有限元(PPIFE),用于在阳(NUMER数学理论值方法申请11:272-298,2018)进行了讨论二阶双曲接口问题的方法,其中作者证明了它们的最佳Ôħ)会聚在次最佳分段(H ^ 3 \)正则性假设下的能量范数。在本文中,我们重新分析了Yang(2018)中提出的完全离散的PPIFE方法。利用郭等人最近给出的误差范围。(Int J Numer Anal Model 16(4):575–589,2019)对于椭圆界面问题,我们不仅可以在能量范数中而且在\(L ^ 2 \)在标准分段\(H ^ 2 \)下的范数精确解的空间变量中的正则性假设,而不是过多的分段(\ H ^ 3 \)正则性。给出了驻波和行波的数值模拟,证实了所报告的误差分析。

更新日期:2020-08-01
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