Bradji, Abdallah; Fuhrmann, Jürgen Some new error estimates for finite element methods for second order hyperbolic equations using the Newmark method. (English) Zbl 1340.65217 Math. Bohem. 139, No. 2, 125-136 (2014). Summary: We consider a family of conforming finite element schemes with piecewise polynomial space of degree \(k\) in space for solving the wave equation, as a model for second-order hyperbolic equations. The discretization in time is performed using the Newmark method. A new a priori estimate is proved. Thanks to this new a priori estimate, it is proved that the convergence order of the error is \(h^{k}+\tau ^{2}\) in the discrete norms of \(\mathcal {L}^{\infty}(0,T;\mathcal {H}^1(\Omega))\) and \(\mathcal {W}^{1,\infty}(0,T;\mathcal {L}^2(\Omega))\), where \(h\) and \(\tau \) are the mesh size of the spatial and temporal discretization, respectively.These error estimates are useful since they allow us to get second order time accurate approximations for not only the exact solution of the wave equation but also for its first derivatives (both spatial and temporal).Even though the proof presented in this note is in some sense standard, the stated error estimates seem not to be present in the existing literature on the finite element methods which use the Newmark method for the wave equation (or general second order hyperbolic equations). Cited in 1 Document MSC: 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 35L15 Initial value problems for second-order hyperbolic equations 35L05 Wave equation Keywords:acoustic wave equation; finite element method; Newmark method; new error estimate; second-order hyperbolic equations; convergence × Cite Format Result Cite Review PDF Full Text: DOI Link