Biharmonic wave equations are of importance to various applications including thin plate analyses. The innovation of this work comes through the numerical approximation of their solutions by a $C^1$-conforming in space and time finite element approach. Therein, the smoothness properties of solutions to the continuous evolution problem are embodied. Time discretization is based on a combined Galerkin and collocation technique. For space discretization the Bogner–Fox–Schmit element is applied. Optimal order error estimates are proven. The convergence and performance properties are illustrated by numerical experiments with complex wave profiles in homogeneous and heterogeneous media, illustrating that the approach offers high potential also for sophisticated multi-physics and/or multi-scale systems.

双谐波波动方程对包括薄板分析在内的各种应用都很重要。这项工作的创新来自于他们的解决方案的数值近似$C^1$-符合空间和时间有限元方法。其中体现了连续演化问题解的平滑性。时间离散化基于 Galerkin 和搭配技术的结合。对于空间离散化，应用 Bogner-Fox-Schmit 元素。证明了最优阶误差估计。通过在均匀和非均匀介质中具有复杂波廓的数值实验说明了收敛性和性能特性，说明该方法也为复杂的多物理场和/或多尺度系统提供了巨大的潜力。