Algebra preconditionings for 2D Riesz distributed-order space-fractional diffusion equations on convex domains

When dealing with the discretization of differential equations on non-rectangular domains, a careful treatment of the boundary is mandatory and may result in implementation difficulties and in coefficient matrices without a prescribed structure. Here we examine the numerical solution of a two-dimensional constant coefficient distributed-order space-fractional diffusion equation with a nonlinear term on a convex domain. To avoid the aforementioned inconvenience, we resort to the volume-penalization method, which consists of embedding the domain into a rectangle and in adding a reaction penalization term to the original equation that dominates in the region outside the original domain and annihilates the solution correspondingly. Thanks to the volume-penalization, methods designed for problems in rectangular domains are available for those in convex domains and by applying an implicit finite difference scheme we obtain coefficient matrices with a 2-level Toeplitz structure plus a diagonal matrix which arises from the penalty term. As a consequence of the latter, we can describe the asymptotic eigenvalue distribution as the matrix size diverges as well as estimate the intrinsic asymptotic ill-conditioning of the involved matrices. On these bases, we discuss the performances of the conjugate gradient with circulant and tau-preconditioners and of the generalized minimal residual with split circulant and tau-preconditioners and conduct related numerical experiments.