  This spreading of prion proteins is at the basis of brain neurodegeneration.
This paper deals with the numerical modelling of the misfolding process of
$\alpha$-synuclein in Parkinson's disease. We introduce and analyze a
discontinuous Galerkin method for the semi-discrete approximation of the
Fisher-Kolmogorov (FK) equation that can be employed to model the process. We
employ a discontinuous Galerkin method on polygonal and polyhedral grids
(PolyDG) for space discretization, to accurately simulate the wavefronts
typically observed in the prionic spreading and we prove stability and a priori
error estimates. Next, we use a Crank-Nicolson scheme to advance in time. For
the numerical verification of our numerical model, we first consider a
manufactured solution, and then we consider a case with wavefront propagation
in two-dimensional polygonal grids. Next, we carry out a simulation of
$\alpha$-synuclein spreading in a two-dimensional brain slice in the sagittal
plane with a polygonal agglomerated grid that takes full advantage of the
flexibility of PolyDG approximation. Finally, we present a simulation in a
three-dimensional geometry reconstructed from magnetic resonance images of a
patient's brain.

Numerical Analysis;Numerical Analysis