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A scalable solver for a class of variable-order fractional diffusion problems.

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Title: A scalable solver for a class of variable-order fractional diffusion problems.
Authors: Lei, Wenyu1 (AUTHOR) wenyu.lei@uestc.edu.cn, Turkiyyah, George2 (AUTHOR), Knio, Omar2 (AUTHOR)
Source: Computers & Mathematics with Applications. Oct2026, Vol. 219, p317-332. 16p.
Subjects: Sparse matrices, Integral operators, Numerical analysis
Abstract: We present a scalable method to solve multidimensional fractional diffusion problems with spatially varying fractional order and diffusivity. The nonlocal nature of the operator yields dense discrete systems that are expensive to solve at scale, making the development of scalable solvers for the variable-order case needed in many practical applications an important challenge. In this work, we propose a V-cycle multilevel preconditioner and demonstrate its scalability in general geometries and in the presence of discontinuities in the fractional order. At every level of the spatial hierarchy, the preconditioner represents the discrete system matrix as the sum of a sparse matrix related to the singular near-field interactions, a hierarchical low-rank approximation of the dense matrix representing the far-field effects of the nonlocal setting, and a sparse weighted mass matrix representing nonlocal boundary (volume) conditions. The overall storage requirements and construction costs of the preconditioner are linear in both storage and construction cost. We prove that the condition number of the preconditioned system is bounded independently of the mesh size, and numerical experiments on two-dimensional test problems confirm this finding, demonstrating that end-to-end linear-cost solution of variable-order fractional diffusion problems on general geometries is achievable. [ABSTRACT FROM AUTHOR]
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Database: Engineering Source
Description
Abstract:We present a scalable method to solve multidimensional fractional diffusion problems with spatially varying fractional order and diffusivity. The nonlocal nature of the operator yields dense discrete systems that are expensive to solve at scale, making the development of scalable solvers for the variable-order case needed in many practical applications an important challenge. In this work, we propose a V-cycle multilevel preconditioner and demonstrate its scalability in general geometries and in the presence of discontinuities in the fractional order. At every level of the spatial hierarchy, the preconditioner represents the discrete system matrix as the sum of a sparse matrix related to the singular near-field interactions, a hierarchical low-rank approximation of the dense matrix representing the far-field effects of the nonlocal setting, and a sparse weighted mass matrix representing nonlocal boundary (volume) conditions. The overall storage requirements and construction costs of the preconditioner are linear in both storage and construction cost. We prove that the condition number of the preconditioned system is bounded independently of the mesh size, and numerical experiments on two-dimensional test problems confirm this finding, demonstrating that end-to-end linear-cost solution of variable-order fractional diffusion problems on general geometries is achievable. [ABSTRACT FROM AUTHOR]
ISSN:08981221
DOI:10.1016/j.camwa.2026.07.021