We consider new degrees of freedom for higher order differential forms on cubical meshes. The approach is inspired by the idea of Rapetti and Bossavit to define higher order Whitney forms and their degrees of freedom using small simplices. We show that higher order differential forms on cubical meshes can be defined analogously using small cubes and prove that these small cubes yield unisolvent degrees of freedom. Significantly, this approach is compatible with discrete exterior calculus and expands the framework to cover higher order methods on cubical meshes, complementing the earlier strategy based on simplices.

翻译：本文研究了立方网格上高阶微分形式的新的自由度。该方法的灵感来源于Rapetti和Bossavit利用小单纯形定义高阶Whitney形式及其自由度的思想。我们证明，可以类似地利用小立方体来定义立方网格上的高阶微分形式，并证明这些小立方体能产生唯一可解的自由度。重要的是，该方法与离散外微分计算相兼容，并将该框架扩展至涵盖立方网格上的高阶方法，从而补充了此前基于单纯形的策略。