Error Estimates for an Immersed Finite Element Method for Second Order Hyperbolic Equations in Inhomogeneous Media
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Error Estimates for an Immersed Finite Element Method for Second Order Hyperbolic Equations in Inhomogeneous Media Slimane Adjerid1 · Tao Lin1 · Qiao Zhuang1 Received: 20 May 2019 / Revised: 18 February 2020 / Accepted: 15 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
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. Keywords Error estimates · Immersed finite element methods · Hyperbolic equations · Inhomogeneous media
1 Introduction Wave propagation in inhomogeneous media appears often in science and engineering such as in acoustics, elasticity, and electromagnetism. For instance, an acoustic wave propagating at different speeds in different media is modeled by the second-order wave equation with discontinuous coefficients. In addition to the usual initial and boundary conditions, jump conditions across the material interface are also required in order for the pertinent partial differential equation to have a unique solution. These considerations lead us to consider the
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Qiao Zhuang [email protected] Slimane Adjerid [email protected] Tao Lin [email protected]
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Department of Mathematics, Virginia Tech, Blacksburg, VA 24061, USA 0123456789().: V,-vol
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Journal of Scientific Computing
(2020) 84:35
following second-order hyperbolic interface problem: u tt − ∇ · (c2 ∇u) = f , u|∂ = g(X , t), u(X , 0) = w0 (X ), u t (X , 0) = w1 (X ),
in − ∪ + , t ∈ [0, T ],
(1.1a)
t ∈ [0, T ],
(1.1b)
X ∈ ,
(1.1c)
where we have assumed that the domain ⊆ R2 is divided by an interface curve into two subdomains − and + such that = − ∪ + ∪ , the speed c is a piecewise positive constant function such that − c for X ∈ − , (1.1d) c(X ) = c+ for X ∈ + . To close the problem we assume that the exact solution u satisfies the following jump conditions across the interface [u] :=u − | − u + | = 0, 2 c ∇u · n :=(c− )2 ∇u − · n| − (c+ )2 ∇u + · n| = 0,
(1.1e) (1.1f)
where n is the unit normal vector to the interface . Conventional finite element methods on body-fitting meshes [10,16,17,36] have been developed for solving interface problems, but they are usually su
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