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

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.