Chemotaxis on networks: Analysis and numerical approximation

Abstract

We consider the Keller-Segel model of chemotaxis on one-dimensional networks. Using a variational characterization of solutions, positivity preservation, conservation of mass, and energy estimates, we establish global existence of weak solutions and uniform bounds. This extends related results of Osaki and Yagi to the network context. We then analyze the discretization of the system by finite elements and an implicit time-stepping scheme. Mass lumping and upwinding are used to guarantee the positivity of the solutions on the discrete level. This allows us to deduce uniform bounds for the numerical approximations and to establish order optimal convergence of the discrete approximations to the continuous solution without artificial smoothness requirements. In addition, we prove convergence rates under reasonable assumptions. Some numerical tests are presented to illustrate the theoretical results.

A comparison of PINN approaches for drift-diffusion equations on metric graphs

“…Having introduced the complete continuous model, we state the following existence and uniqueness result, which can be obtained by combining the proofs of [39,35].…”

“…Classical approaches are Galerkin finite element methods [34,35] to discretize of the governing equations. In particular, discontinuous Galerkin finite element method (DG) [36] or finite volume methods [37] are often the method of choice given they are guaranteeing local conservation of the flows.…”

Discretization of the wave equation on a metric graph

“…DENs, classically, arise in the study of stability, health, and oscillations of flexible structures that are made of strings, beams, cables, and struts [1][2][3][4] -Water, Electricity, Gas, and Traffic Networks. An important example of DENs is the Saint-Venant system of equations, which model hydraulic networks for water supply and irrigation [5] and first-order hyperbolic equations [6][7][8][9][10][11] and the isothermal Euler equations for describing the gas flow through pipelines [12,13].Other important examples of DENs include the telegrapher equation for modeling electric networks [14], the diffusion equations in power networks [15], and Aw-Rascle equations for describing road traffic dynamics [16], see also [17] for traffic flow on networks and [18][19][20] for modeling groundwater flow.…”