Valentina Pederzoli | Searcharxiv

arXiv2026

Polytopal Discontinuous Galerkin Discretizations of Coupled Non-Newtonian Stokes-Darcy Systems

We propose and analyze a polytopal discontinuous Galerkin method for the numerical approximation of a coupled non-Newtonian Stokes-Darcy system modeling the interaction between a non-Newtonian free-flow fluid and a non-Newtonian flow through a porous medium. Due to its geometric flexibility and arbitrary-order accuracy, the proposed discretization scheme is well-suited to configurations with complex geometries. We provide a complete a-priori analysis that considers shear-dependent and velocity-dependent non-Newtonian viscosity models for the free-flow and porous media regions, respectively. The well-posedness, stability, and error bounds of the method are established in the framework of generalized inf-sup theory. Error estimates are confirmed by numerical results.

math.NA↗

arXiv2024

A coupled mathematical and numerical model for protein spreading and tissue atrophy, applied to Alzheimer's disease

The aim of this paper is to introduce, analyse and test in practice a new mathematical model describing the interplay between biological tissue atrophy driven by pathogen diffusion, with applications to neurodegenerative disorders. This study introduces a novel mathematical and computational model comprising a Fisher-Kolmogorov equation for species diffusion coupled with an elasticity equation governing mass loss. These equations intertwine through a logistic law dictating the reduction of the medium's mass. One potential application of this model lies in understanding the onset and development of Alzheimer's disease. Here, the equations can describe the propagation of misfolded tau-proteins and the ensuing brain atrophy characteristic of the disease. To address numerically the inherited complexities, we propose a Polygonal Discontinuous Galerkin method on polygonal/polyhedral grids for spatial discretization, while time integration relies on the theta-method. We present the mathematical model, delving into its characteristics and propose discretization applied. Furthermore, convergence results are presented to validate the model, accompanied by simulations illustrating the application scenario of the onset of Alzheimer's disease.

math.NA↗