mixeddimensional

Mixed-dimensional refers to mathematical and modeling frameworks that combine objects of different spatial dimensions within a single system. It is used to describe problems where a bulk domain coexists with lower-dimensional features, such as surfaces or lines embedded in the space, creating a single, interconnected model.

In a typical mixed-dimensional setting, one works with a hierarchy of domains of varying dimensions that are coupled together. Examples include a three-dimensional bulk region with two-dimensional interfaces (fractures, membranes) and one-dimensional networks (wires, vascular or neuronal networks) embedded within it. Each component carries its own governing equations, while coupling conditions enforce conservation, continuity, or exchange of quantities across dimensions. The analysis often relies on measures on lower-dimensional sets, trace results, and specialized function spaces that capture behavior across dimensions.

Governing equations in mixed-dimensional problems usually consist of PDEs defined on each dimension, together with interface or coupling conditions. For instance, diffusion or flow may occur in the bulk and along lower-dimensional features, with flux continuity and exchange terms linking the different dimensions. This leads to a system that is simultaneously solved on multiple scales and dimensions, often requiring dimensionally extended variational formulations.

Numerical approaches for mixed-dimensional problems include finite element and finite volume methods designed for hybrid domains, along with techniques such as mortar methods and non-mmatching grids to handle interfaces. Applications span modeling of flow in fractured porous media, composite materials with embedded inclusions, biological networks within tissues, and electrical or mechanical systems with interconnected dimensional components.

The term is used across mathematics, physics, and engineering to describe problems where dimensional heterogeneous structures interact in a single model. Related concepts include fractured media modeling, network PDEs, and dimensionally coupled simulations.