A stability-preserving polytopal discontinuous Galerkin method for the Fisher-Kolmogorov model with applications to neurodegenerative diseases

The Fisher-Kolmogorov model is one of the most widely used models in the study of neurodegenerative diseases, owing to its simple structure as a nonlinear reaction-diffusion equation.

In particular, it is commonly employed to describe proteinopathies such as Alzheimer's and Parkinson's diseases.

Under suitable assumptions, non-negativity of the solution is guaranteed at the continuous level, which is physically relevant since the solution represents a relative concentration.

However, this property is not generally preserved at the discrete level, potentially leading to unphysical and unstable numerical approximations.

In this work, we analyze a modified version of the Fisher-Kolmogorov model that stabilizes the dynamics around the unstable equilibrium $c=0$.

For the spatial discretization, we adopt a discontinuous Galerkin method on polygonal and polyhedral meshes, coupled with the Crank-Nicolson scheme for time integration.

We derive stability and $a$-priori error estimates for the semi-discrete problem.

The theoretical findings are supported by numerical experiments, including convergence studies in both two and three dimensions.

Finally, we validate the model through simulations of $\alpha$-synuclein diffusion in a two-dimensional agglomerated brain section, demonstrating the high-order accuracy and robustness of the proposed method.