We propose a new space-time variational formulation for wave equation initial-boundary value problems. The key property is that the formulation is coercive (sign-definite) and continuous in a norm stronger than $H^1(Q)$, $Q$ being the space-time cylinder. Coercivity holds for constant-coefficient impedance cavity problems posed in star-shaped domains, and for a class of impedance-Dirichlet problems. The formulation is defined using simple Morawetz multipliers and its coercivity is proved with elementary analytical tools, following earlier work on the Helmholtz equation. The formulation can be stably discretised with any $H^2(Q)$-conforming discrete space, leading to quasi-optimal space-time Galerkin schemes. Several numerical experiments show the excellent properties of the method.

翻译：本文提出了一种新的用于波动方程初边值问题的时空变分公式。该公式的关键性质在于：它在比$H^1(Q)$（$Q$为时空柱体）更强的范数下是强制（定号）且连续的。对于星形区域中常系数阻抗腔问题以及一类阻抗-狄利克雷问题，强制性质成立。该公式使用简单的Morawetz乘子定义，并遵循亥姆霍兹方程的早期工作，利用基本分析工具证明了其强制性质。该公式可通过任意$H^2(Q)$相容离散空间进行稳定离散，从而得到拟最优的时空伽辽金格式。多项数值实验表明该方法具有优异的性能。