We are interested in differential forms on mixed-dimensional geometries, in the sense of a domain containing sets of d-dimensional manifolds, structured hierarchically so that each d-dimensional manifold is contained in the boundary of one or more \(d + 1\)-dimensional manifolds. On any given d-dimensional manifold, we then consider differential operators tangent to the manifold as well as discrete differential operators (jumps) normal to the manifold. The combined action of these operators leads to the notion of a semi-discrete differential operator coupling manifolds of different dimensions. We refer to the resulting systems of equations as mixed-dimensional, which have become a popular modeling technique for physical applications including fractured and composite materials. We establish analytical tools in the mixed-dimensional setting, including suitable inner products, differential and codifferential operators, Poincaré lemma, and Poincaré–Friedrichs inequality. The manuscript is concluded by defining the mixed-dimensional minimization problem corresponding to the Hodge Laplacian, and we show that this minimization problem is well-posed.

我们对混合维几何上的微分形式感兴趣，就一个域而言，它包含一组d维流形，这些维是层次结构的，因此每个d维流形都包含在一个或多个\（d + 1 \ ）尺寸歧管。在任何给定的d维流形，然后考虑与流形相切的微分算子以及垂直于流形的离散微分算子（跳跃）。这些操作员的共同作用导致了不同尺寸的半离散差动操作员联接歧管的概念。我们将方程的结果系统称为混合维，这已成为包括断裂和复合材料在内的物理应用的流行建模技术。我们在混合维度环境中建立了分析工具，包括合适的内积，微分和协微算子，庞加莱引理和庞加莱-弗里德里希斯不等式。通过定义与Hodge Laplacian相对应的混合维最小化问题来得出手稿，