_{Supporting information and the FEM source code for "Finite element method for garnet diffusion chronometry}_"

Solving the diffusion equation for species diffusion in garnet provides insights into the duration and rate of geological processes. However, previous research has oversimplified garnet as a sphere, failing to accurately represent the intricate shapes of real minerals and varying boundary conditions, which limits its applicability. In this study, we introduce a numerical approach known as the Finite Element Method (FEM) to overcome these challenges. We thoroughly explain the rationale behind FEM, detailing the procedure for transforming the original diffusion equation into a final linear system for easier resolution. Subsequently, we implemented the method and conducted a comprehensive accuracy analysis involving lutetium (Lu) diffusion within a spherical garnet. Our accuracy analysis reveals that with finer discretization, the relative error is maintained within ±1‰, surpassing the accuracy of LA-ICP-MS by an order of magnitude. This finding underscores the robustness of the method. Furthermore, we conducted two- and three-dimensional numerical diffusion experiments across arbitrarily defined geometries and diverse boundary conditions, which highlight the method’s adaptability for addressing diffusion under these complexities. We conclude that FEM effectively eliminates the geometric constraints of minerals and allows for the application of diverse boundary conditions, thereby holding promise for widespread applications in the field.

求解石榴子石中组分扩散的扩散方程，可为地质作用的持续时长与速率提供关键认知依据。然而既往研究多将石榴子石简化为球体，无法精准还原天然矿物的复杂形貌与多变的边界条件，进而限制了该方法的适用范围。本研究引入有限单元法（Finite Element Method, FEM）以应对上述挑战，详细阐释了该方法的理论原理，并完整说明了将原始扩散方程转化为最终线性方程组以简化求解的具体步骤。随后，本研究完成了该方法的程序实现，并针对球形石榴子石内镥（Lu）的扩散过程开展了全面的精度验证分析。精度验证结果表明，当离散化程度更高时，相对误差可控制在±1‰以内，其精度比激光剥蚀电感耦合等离子体质谱法（LA-ICP-MS）高出一个数量级，充分证明了本方法的稳健性。此外，本研究针对任意自定义几何形貌与多样边界条件开展了二维与三维数值扩散模拟实验，结果充分体现了该方法在处理复杂条件下扩散问题的适配能力。本研究最终得出结论：有限单元法可有效突破矿物形貌的几何限制，并支持多种边界条件的应用，因此在该领域具备广泛的应用前景。