度量图上分数阶扩散方程的数值逼近
Numerical approximation of fractional diffusion equations on metric graphs
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中文总结 AI 辅助
该研究针对紧度量图上由平移Kirchhoff-Laplacian分数次幂控制的分数阶扩散方程，建立严格数学框架，提出结合向后欧拉与有限元的完全离散格式，采用有理近似实现高效求解，推导误差估计并通过实验验证收敛性。
中文摘要 AI 辅助
我们研究紧度量图上的分数阶扩散方程，其中非局部动力学由平移Kirchhoff-Laplacian的分数次幂控制。基于度量图上分数阶算子分析的最新进展，我们建立了严格的数学框架，并提出了一种完全离散格式，该格式基于向后欧拉时间步进和有限元离散化。为逼近分数阶算子的作用，我们采用有理近似，将问题简化为一系列稀疏椭圆求解以实现高效计算。我们推导了时间、空间和有理离散化的误差估计，并通过数值实验验证了收敛性。
英文摘要
We study fractional diffusion equations on compact metric graphs, where the nonlocal dynamics is governed by fractional powers of the shifted Kirchhoff-Laplacian. Building on recent advances in the analysis of fractional operators on metric graphs, we establish a rigorous mathematical framework and propose a fully discrete scheme based on backward Euler time-stepping and finite element discretization. To approximate the action of the fractional operator, we employ rational approximations, reducing the problem to a sequence of sparse elliptic solves for efficient implementation. We derive error estimates for the temporal, spatial, and rational discretizations, and confirm convergence through numerical experiments.
