Optimal preconditioning techniques for finite volume approximation of three-dimensional conservative space-fractional diffusion equations

TL;DR

The paper addresses the computational challenge of solving large, dense, Toeplitz-like linear systems arising from a Crank–Nicolson finite volume discretization of three-dimensional conservative space-fractional diffusion equations. It develops sine transform–based preconditioners that diagonalize the Kronecker-structured coefficient, enabling fast application and provable mesh-size–robust convergence: the symmetric problem yields spectra of $P^{-1/2}AP^{-1/2}$ contained in $(1/2,3/2)$, while the non-symmetric case achieves linear, mesh-size–independent convergence under GMRES with a two-sided preconditioning framework. Theoretical results are complemented by extensive 2D and 3D numerical experiments showing near-constant iteration counts and reduced CPU time relative to circulant preconditioners. The work broadens the applicability of sine-transform preconditioning to finite-volume discretizations of fractional diffusion and promises scalable performance for large-scale simulations, with future extensions to high-dimensional variable-coefficient problems.

Abstract

A Crank-Nicolson finite volume approximation for three-dimensional conservative space-fractional diffusion equation results in large and dense three-level Toeplitz discrete linear systems. Preconditioned Krylov subspace methods with sine transform-based preconditioners are developed to solve these systems, including the preconditioned conjugate gradient (PCG) method for the symmetric case and the preconditioned generalized minimal residual (PGMRES) method for the non-symmetric case. Moreover, we provide detailed analysis of the convergence of these Krylov subspace methods. Specifically, for the symmetric case, we prove the spectra of the preconditioned matrices are uniformly bounded in the open interval (1/2, 3/2), which results in a linear convergence rate of the PCG method. For the non-symmetric case, we demonstrate that the PGMRES method also achieves a linear convergence rate independent of discretization stepsizes from the residual point of view. These results imply that the iteration counts of the PCG and PGMRES methods are uniformly bounded and independent of the matrix sizes. Numerical experiments in both symmetric and non-symmetric cases in two- and three-dimensions are conducted to confirm the optimal performance of the proposed preconditioned Krylov subspace methods.

Theorems & Definitions (29)

• Remark 2.1
• Remark 2.2
• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3