In this work, following the discrete de Rham (DDR) approach, we develop a discrete counterpart of a two-dimensional de Rham complex with enhanced regularity. The proposed construction supports general polygonal meshes and arbitrary approximation orders. We establish exactness on a contractible domain for both the versions of the complex with and without boundary conditions and, for the former, prove a complete set of Poincar\'e-type inequalities. The discrete complex is then used to derive a novel discretisation method for a quad-rot problem which, unlike other schemes in the literature, does not require the forcing term to be prepared. We carry out complete stability and convergence analyses for the proposed scheme and provide numerical validation of the results.

翻译：在这项工作中,我们采用离散的Rham(解甲返乡)方法,开发了一种分立的双维的Rham综合体,并增加了规律性。提议的构造支持一般多边网球和任意近似命令。我们为有边界条件和没有边界条件的复杂体的版本以及前者证明是一整套Poincar\'e型的不平等,确定了可承包域的精确性。离散综合体随后被用来为四分罗问题产生一种新颖的分解方法。与文献中的其他计划不同,这不需要编写强制术语。我们为拟议的计划进行全面的稳定性和趋同性分析,并对结果进行数字验证。