# Wietse Boon: Mixed-Dimensional PDEs: From Functional Analysis to Discretization Methods

**Time: **
Thu 2019-11-14 14.15 - 15.00

**Location: **
Room F11, Lindstedtsvägen 22, våningsplan 2, F-huset, KTH Campus.

**Participating: **
Wietse Boon, KTH

**Abstract:**

Mixed-dimensional models arise in cases where thin structures are represented as manifolds of lower dimension within the computational domain. As the leading example in this talk, we introduce the mixed-dimensional representation of a subsurface fracture network and consider a Darcy flow model. The governing equations are described using semi-discrete differential operators that map between manifolds of codimension one. The mixed-dimensional problem is then considered as a whole, and several fundamental tools are necessary to facilitate its analysis.

The focus of this work lies on the underlying structure of mixed-dimensional PDEs and we present extensions of well-established results from functional analysis to the mixed-dimensional setting, including appropriate Sobolev spaces and a mixed-dimensional de Rham complex. These results are then used to form stable discretization methods for mixed-dimensional PDEs, with a primary focus on the mixed finite element method.