We establish fully-discrete a priori and semi-discrete in time a posteriori error estimates for a discontinuous-continuous Galerkin discretization of the wave equation in second order formulation; the resulting method is a Petrov-Galerkin scheme based on piecewise and piecewise continuous polynomial in time test and trial spaces, respectively. Crucial tools in the a priori analysis for the fully-discrete formulation are the design of suitable projection and interpolation operators extending those used in the parabolic setting, and stability estimates based on a nonstandard choice of the test function; a priori estimates are shown, which are measured in $L^\infty$-type norms in time. For the semi-discrete in time formulation, we exhibit constant-free, reliable a posteriori error estimates for the error measured in the $L^\infty(L^2)$ norm; to this aim, we design a reconstruction operator into $\mathcal C^1$ piecewise polynomials over the time grid with optimal approximation properties in terms of the polynomial degree distribution and the time steps. Numerical examples illustrate the theoretical findings.

翻译：本文针对二阶形式波动方程的非连续-连续伽辽金离散，建立了全离散先验误差估计及时间半离散后验误差估计；该方法为基于分片多项式（试验空间）与分片连续多项式（试探空间）的彼得罗夫-伽辽金格式。全离散格式先验分析的关键工具包括：拓展抛物问题所用算子而设计的适当投影与插值算子，以及基于非标准试探函数选取的稳定性估计；所得先验误差估计以时间$L^\infty$型范数度量。针对时间半离散格式，我们给出了以$L^\infty(L^2)$范数度量误差的无常数可靠后验估计；为此构造了时间网格上到$\mathcal C^1$分片多项式空间的重构算子，该算子在多项式次数分布与时间步长方面具有最优逼近性质。数值算例验证了理论结果。