A spectral method for fractional porous medium equations

Abstract

This master's thesis considers the fractional general porous medium equation; a nonlocal equation with nonlinear diffusivity. Properties of the nonlocal operator are derived. Existence of distributional solutions are proved, together with $L^1$-contraction and distance to the family of vanishing viscosity solutions. Then a Fourier Galerkin method with spectral vanishing viscosity (SVV) is proposed and shown to be convergent under suitable conditions to the distributional solution.

Lastly, numerical experiments for some important special cases of the problem are provided, together with convergence plots. This gives some information about when it is suitable to use SVV.