Multigrid for two-sided fractional differential equations discretized by finite volume elements on graded meshes
Additional information
Authors
Donatelli M.,
Krause R.,
Mazza M.,
Trotti K.
Type
Journal Article
Year
2024
Language
English
Abstract
It is known that the solution of a conservative steady-state two-sided fractional diffusion problem can exhibit singularities near the boundaries. As a consequence of this, and due to the conservative nature of the problem, we adopt a finite volume elements discretization approach over a generic non-uniform mesh. We focus on grids mapped by a smooth function which consists in a combination of a graded mesh near the singularity and a uniform mesh where the solution is smooth. Such a choice gives rise to Toeplitz-like discretization matrices and thus allows a low computational cost of the matrix–vector product and detailed spectral analysis. The obtained spectral information is used to develop an ad-hoc parameter-free multigrid preconditioner for GMRES, which is numerically shown to yield good convergence results in presence of graded meshes mapped by power functions that accumulate points near the singularity. The approximation order of the considered graded meshes is numerically compared with the one of a certain composite mesh given in literature that still leads to Toeplitz-like linear systems and is then still well-suited for our multigrid method. Several numerical tests confirm that power-graded meshes result in lower approximation errors than composite ones and that our solver has a wide range of applicability.
Keywords
Two-sided fractional problems, Finite volume elements methods, Toeplitz matrices, Spectral distribution, Multigrid methods
Journal
Journal of computational and applied mathematics
Volume
444
Number ( Month )
115787
Diffusion
License
CC BY
Visibility
Public
Status open access
Hybrid
