Conforming Discretizations of Mixed-Dimensional Partial Differential Equations
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Mixed-dimensional partial differential equations (PDEs) are coupled equations defined on connected manifolds of different dimensionalities. Two main examples of mixed-dimensional PDEs are considered in this dissertation, namely flow in fractured porous media and mechanics of composite materials. We focus on the discretization of these examples using hierarchical finite elements defined on coupled manifolds of codimension one, successively. By uncovering their underlying structure, we use the corresponding tools to define, analyze and discretize mixed-dimensional partial differential equations.
Our first example concerning mixed-dimensional PDEs arises in the context of fracture flow. Here, the planar fractures, intersection lines, as well as intersection points are represented as lower-dimensional manifolds. In turn, the entire embedded fracture network forms a mixed-dimensional geometry. We continue by defining the conservation and constitutive laws on the mixed-dimensional geometry, leading to a hierarchically coupled system of partial differential equations. Next, we extend these concepts from flow to derive the governing equations concerning mechanics of materials with thin inclusions in an analogous manner. Together, the embedded features and their surroundings form the mixed-dimensional geometry and the behavior of the system can be captured by prescribing significantly different material parameters. The analysis of these systems introduces several new concepts including mixed-dimensional function spaces and semi-discrete differential operators.
With the aim of discretization, we use finite element exterior calculus to construct mixed finite element schemes on the mixed-dimensional geometry. We focus on two families of mixed-dimensional finite elements, hierarchically ordered by dimensionality. We refer to these families as the first and second kind and show that both are of interest in the context of fracture flow, with different behavior in terms of convergence and computational cost. On the other hand, the mixed formulation of the mechanics equations requires the family of elements of the second kind.
For fracture flow, stability and optimal convergence of the discretization method are shown with the use of weighted, mixed-dimensional Sobolev spaces. A novel way of incorporating the fracture aperture leads to a scheme capable of handling arbitrarily small and spatially varying apertures. In case of fractures pinching out, the degeneration of the equations eliminates the possibility for flow resulting in a natural termination of fractures.
In a benchmark study concerning flow through fractured porous media, the proposed scheme is compared to various other numerical methods. Four two-dimensional test cases of varying complexity are considered, specifically designed to highlight the typical difficulties with modeling flow through fracture networks.With eight participating numerical schemes, a clear view is given of the performance and limitations of the state-of-the-art numerical schemes.
Finally, we consider the evolution of the water table and identify the water table itself as a lower-dimensional manifold. Its location is governed by a partial differential equation which is coupled to the underlying saturated region. To solve this problem, a numerical scheme is proposed which maps the problem to a stationary reference domain. We analyze the properties of this scheme and successfully apply it to a real world problem concerning ground flow patterns surrounding meandering streams.