In this work, following the Discrete de Rham (DDR) paradigm, we develop an arbitrary-order discrete divdiv complex on general polyhedral meshes. The construction rests 1) on discrete spaces that are spanned by vectors of polynomials whose components are attached to mesh entities and 2) on discrete operators obtained mimicking integration by parts formulas. We provide an in-depth study of the algebraic properties of the local complex, showing that it is exact on mesh elements with trivial topology. The new DDR complex is used to design a numerical scheme for the approximation of biharmonic problems, for which we provide detailed stability and convergence analyses.

翻译：本文遵循离散de Rham（DDR）范式，在一般多面体网格上构建了任意阶离散divdiv复形。该构造基于：1）由附着于网格实体的多项式分量向量张成的离散空间；2）通过模仿分部积分公式得到的离散算子。我们深入研究了局部复形的代数性质，证明其在拓扑平凡的网格单元上是正合的。利用该新型DDR复形设计了双调和问题逼近的数值格式，并提供了详细的稳定性与收敛性分析。