Strong Convergence of Discrete DG Solutions: A Unified Analysis

In this article, we delve into the realm of modeling physiological processes that occur within complex networks of vessels. By leveraging mathematical techniques, we can better understand how fluids and solutes move through these vessels, which is essential for studying various biological phenomena, such as drug transport in vascularized tissue or solute clearance through the lymphatic system of the brain.
To tackle these complex network systems, researchers often employ topological model reduction techniques that simplify the equations while preserving their essence. By collapsing the network into a set of centerlines, we can transform the 3D problem into a series of 1D equations that are more manageable to solve. These simplified models still capture the essential physics of the original problem, making them useful for understanding complex physiological processes.
One key challenge in modeling these systems is dealing with the small diameter of the vessels compared to the surrounding domain. To address this issue, researchers have developed numerical methods that can efficiently solve the reduced 1D equations while accounting for the effects of the larger domain.
Throughout the article, we explore various mathematical techniques used to model physiological processes, including discontinuous Galerkin methods and singularity removal methods. These approaches enable researchers to accurately capture the complex behavior of fluids and solutes within vessels, which is crucial for understanding how our bodies function at a molecular level.
In summary, this article provides an in-depth look at the mathematical techniques used to model physiological processes involving complex networks of vessels. By leveraging these methods, researchers can gain valuable insights into how fluids and solutes move through these systems, which is essential for improving our understanding of various biological phenomena.