A mesh- and horizon-robust preconditioner for finite element discretizations of volume-constrained nonlocal diffusion problems
Abstract
In this paper, we develop a preconditioner that is robust with respect to both the mesh size and the interaction horizon for finite element discretizations of volume-constrained nonlocal diffusion problems. Motivated by the Fourier multiplier of the nonlocal operator, we construct the preconditioner Bh=Gh+βδ2Dh−1, where Gh is a symmetric local Poisson solver, Dh is the lumped mass matrix, and β is a kernel-dependent constant. We rigorously prove that the spectrum of the Bh-preconditioned system is bounded above and away from zero uniformly with respect to the mesh size and horizon. Moreover, we develop a pair-cell assembly for the finite element matrix that preserves the symmetry and positive definiteness required by the preconditioner and PCG, and establish an L2-error estimate for the finite element discretization. Finally, we use a single symmetric geometric multigrid V-cycle for Gh in the numerical experiments, and the corresponding numerical results confirm the predicted convergence rate and show bounded condition numbers and PCG iteration counts over the tested meshes, horizons, and kernels.
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BibTeX
@misc{lu2026mesh,
title = {{A mesh- and horizon-robust preconditioner for finite element discretizations of volume-constrained nonlocal diffusion problems}},
author = {Jiashu Lu and Yufeng Nie and Haolun Zhang and Pingrui Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24143},
eprint = {2609.24143},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24143},
}
RIS
TY - UNPB
TI - A mesh- and horizon-robust preconditioner for finite element discretizations of volume-constrained nonlocal diffusion problems
AU - Jiashu Lu
AU - Yufeng Nie
AU - Haolun Zhang
AU - Pingrui Zhang
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24143
UR - https://arxiv.org/abs/2609.24143
AB - In this paper, we develop a preconditioner that is robust with respect to both the mesh size and the interaction horizon for finite element discretizations of volume-constrained nonlocal diffusion problems. Motivated by the Fourier multiplier of the nonlocal operator, we construct the preconditioner B_h=G_h+βδ^2D_h^−1, where G_h is a symmetric local Poisson solver, D_h is the lumped mass matrix, and β is a kernel-dependent constant. We rigorously prove that the spectrum of the B_h-preconditioned system is bounded above and away from zero uniformly with respect to the mesh size and horizon. Moreover, we develop a pair-cell assembly for the finite element matrix that preserves the symmetry and positive definiteness required by the preconditioner and PCG, and establish an L^2-error estimate for the finite element discretization. Finally, we use a single symmetric geometric multigrid V-cycle for G_h in the numerical experiments, and the corresponding numerical results confirm the predicted convergence rate and show bounded condition numbers and PCG iteration counts over the tested meshes, horizons, and kernels.
ER -