Gehring theory for time-discrete hyperbolic differential equations

Keisuke Hoshino; Norio Kikuchi

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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 $h^{k} + τ^{2}$ in the discrete norms of $ℒ^{\infty} (0, T; ℋ^{1} (Ω))$ and $𝒲^{1, \infty} (0, T; ℒ^{2} (Ω))$ , where $h$ and $τ$ are the mesh size of the spatial and temporal discretization,...