Convergence Analysis of a Finite-Volume Scheme for a Microglia--Amyloid Chemotaxis Model with Measure-Valued Vascular Boundary Sources

Abstract

We study a parabolic-parabolic chemotaxis system motivated by microglial recruitment toward an amyloid-$β$-associated signal in Alzheimer's disease. The signal is subject to a nonnegative Radon measure-valued Neumann influx on a vascular portion of the boundary, while microglial cells respond to a nonlocal spatial average of the signal. For a fixed sensing length $σ>0$, the chemotactic velocity is $b_σ[v]=\nabla K_σ[v]$. For every fixed $σ>0$, the nonlocal operator maps finite signal mass into a bounded spatially Lipschitz velocity field. We introduce a weak-solution concept adapted to the low regularity induced by the boundary measure and construct a fully implicit upwind finite-volume approximation in which the boundary source is discretized through its exact mass on each boundary face-time cell. We establish existence and positivity of discrete solutions, together with uniform mass, energy, discrete-gradient, and compactness estimates. Finally, we prove subsequential convergence of the discrete solutions toward a nonnegative weak solution of the continuous problem.