We analyze the wave equation in mixed form, with periodic and/or Dirichlet homogeneous boundary conditions, and nonconstant coefficients that depend on the spatial variable. For the discretization, the weak form of the second equation is replaced by a strong form, written in terms of a projection operator. The system of equations is discretized with B-splines forming a De Rham complex along with suitable commutative projectors for the approximation of the second equation. The discrete scheme is energy conservative when discretized in time with a conservative method such as Crank-Nicolson. We propose a convergence analysis of the method to study the dependence with respect to the mesh size $h$, with focus on the consistency error. Numerical results show optimal convergence of the error in energy norm, and a relative error in energy conservation for long-time simulations of the order of machine precision.

翻译：我们分析了混合形式的波动方程，包含周期性边界条件和/或齐次狄利克雷边界条件，以及依赖于空间变量的非常系数。在离散化过程中，第二个方程的弱形式被替换为以投影算子表示的强形式。该系统使用形成De Rham复形的B样条，并结合适用于第二方程逼近的交换投影子进行离散。当采用Crank-Nicolson等守恒方法进行时间离散时，该离散格式具有能量守恒性质。我们提出了该方法的收敛性分析，重点研究依赖于网格尺寸$h$的一致误差。数值结果表明，能量范数下的误差达到最优收敛阶，且长时间模拟中能量守恒的相对误差达到机器精度量级。