Functional analysis and exterior calculus on mixed-dimensional geometries

Abstract

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.

PorePy: an open-source software for simulation of multiphysics processes in fractured porous media

“…From the dimension reduction, it follows that Γ j , Ω l , and ∂ j Ω h all coincide geometrically. For completeness, we note that the mathematical framework [36] on which our models are based considers the two sides of Ω l as Grids of all subdomains. Fracture intersections (1d) are represented by colored lines, the 0d grid by a red circle.…”

“…This allows for significant code reuse from the discretization of fixed-dimensional problems; thus, our design principles are also applicable to more general PDE software frameworks, such as FEniCS [33], Dune [34], and FireDrake [35]. Furthermore, for scalar and vector elliptic problems (flow and deformation), the models rest on a solid mathematical formulation [36][37][38].…”

Stable mixed finite elements for linear elasticity with thin inclusions

“…The structure of the derived equations fits well with the mixed-dimensional framework derived in [9,21]. We aim to preserve this structure and retain a local conservation of linear momentum after discretization with the use of conforming, mixed finite elements.…”

“…In this section, we introduce the mixed-dimensional geometry and establish notation. Here, we follow the conventions introduced in [9,10].…”

“…Proof The top row of Eq. 5.16 uses the differential operators from the mixed-dimensional De Rham complex [9]. The first equality then follows from the fact that exact forms are closed.…”