Supporting Information 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. Nonetheless, challenges arise when addressing diffusion problems within the complex geometries of real minerals and under varying boundary conditions. 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, our numerical diffusion experiments highlight the method’s adaptability for addressing diffusion under these complexities. They also demonstrate that the concentration surface exhibits a pronounced sensitivity to geometric irregularities, particularly when the diffusion length is small. 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.