体积约束非局部扩散问题有限元离散的网格与水平鲁棒预条件子
A mesh- and horizon-robust preconditioner for finite element discretizations of volume-constrained nonlocal diffusion problems
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中文总结 AI 辅助
本文提出一种对网格尺寸和相互作用水平均鲁棒的预条件子，用于体积约束非局部扩散问题的有限元离散，通过理论分析和数值实验验证了其谱性质与收敛性。
中文摘要 AI 辅助
本文针对体积约束非局部扩散问题的有限元离散，开发了一种对网格尺寸和相互作用水平均具有鲁棒性的预条件子。受非局部算子傅里叶乘子的启发，我们构造了预条件子$B_h=G_h+\beta\delta^2D_h^{-1}$，其中$G_h$是对称的局部泊松求解器，$D_h$是集中质量矩阵，$\beta$是依赖于核的常数。我们严格证明了$B_h$预条件系统的谱关于网格尺寸和水平一致有上界且远离零。此外，我们开发了有限元矩阵的单元对组装方法，该方法保持了预条件子和PCG所需的对称性和正定性，并建立了有限元离散的$L^2$误差估计。最后，在数值实验中，我们对$G_h$使用单个对称几何多重网格V循环，相应的数值结果证实了预测的收敛速度，并显示了在测试的网格、水平和核上条件数和PCG迭代次数有界。
英文摘要
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.
发表机构
- Xi’an University of Technology（西安工业大学）
- Northwestern Polytechnical University（西北工业大学）
- Zhejiang University（浙江大学）
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