Hierarchical network structures occur in many biological systems, since they are responsible for the transport of fluids, nutrients or oxygen. Such a network structure is, for example, the blood vessel network supplying organs with oxygenated blood or removing metabolic waste from the tissue. A further example is the root network of a plant, ensuring the plant‘s water supply.
One way to obtain a realistic mathematical model for such processes is based on a decomposition approach. Thereby, the network structure is separated from the surrounding medium and different models are assigned to both domains. Often, the surrounding medium (e.g. tissue or soil) is considered a three-dimensional (3D) porous medium. To decrease computational costs while maintaining a certain degree of accuracy, flow and transport processes within the networks are modeled by one-dimensional (1D) PDE-systems based on cross-section averaged quantities. A coupling of the network and the porous medium model is achieved by first averaging the 3D quantities and projecting them onto the 1D network structure. As a next step, a transfer function based on the difference of the averaged 3D and 1D quantities is computed and incorporated into the source/sink terms of the corresponding models. The source term of the 3D problem exhibits a Dirac measure concentrated on the 1D network.
In this talk we present several application areas for this kind of coupling concepts. Furthermore, we are concerned with the numerical analysis of PDE systems arising in the context of this model concept. In particular, it is investigated how the Dirac source terms and averaging operators affect the convergence behavior of standard finite element methods. Therefore, elliptic model problems with Dirac source terms and averaging operators are investigated. Our theoretical results are confirmed by numerical tests.
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