This paper discusses lowest-order nonstandard finite element methods for space discretization and explicit and implicit schemes for time discretization of the biharmonic wave equation with clamped boundary conditions. A modified Ritz projection operator defined on $H^2_0(\Omega)$ ensures error estimates under appropriate regularity assumptions on the solution. Stability results and error estimates of optimal order are established in suitable norms for the semidiscrete and explicit/implicit fully-discrete versions of the proposed schemes. Finally, we report on numerical experiments using explicit and implicit schemes for time discretization and Morley, discontinuous Galerkin, and {C$^0$ interior} penalty schemes for space discretization, that validate the theoretical error estimates.

翻译：本文讨论了具有固定边界条件的双调和波方程的空间离散化最低阶非标准有限元方法，以及时间离散化的显式和隐式格式。定义于 $H^2_0(\Omega)$ 上的修正 Ritz 投影算子在适当解的正则性假设下保证了误差估计。针对所提方案的半离散形式及显式/隐式全离散形式，在合适的范数下建立了稳定性结果与最优阶误差估计。最后，我们报告了采用显式与隐式时间离散格式，并结合 Morley 元、间断 Galerkin 方法及 {C$^0$ 内部} 罚方法进行空间离散的数值实验，验证了理论误差估计。