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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 adiscontinuous Galerkin method for the semi-discrete approximation of theFisher-Kolmogorov (FK) equation that can be employed to model the process. Weemploy a discontinuous Galerkin method on polygonal and polyhedral grids(PolyDG) for space discretization, to accurately simulate the wavefrontstypically observed in the prionic spreading and we prove stability and a priorierror estimates. Next, we use a Crank-Nicolson scheme to advance in time. Forthe numerical verification of our numerical model, we first consider amanufactured solution, and then we consider a case with wavefront propagationin two-dimensional polygonal grids. Next, we carry out a simulation of$\alpha$-synuclein spreading in a two-dimensional brain slice in the sagittalplane with a polygonal agglomerated grid that takes full advantage of theflexibility of PolyDG approximation. Finally, we present a simulation in athree-dimensional geometry reconstructed from magnetic resonance images of apatient's brain.

Discontinuous Galerkin Methods for Fisher-Kolmogorov Equation with Application to $\alpha$-Synuclein Spreading in Parkinson's Disease