Finite Element Method for Garnet Diffusion Chronometry

Abstract

Crystal geometries and boundary conditions significantly influence garnet diffusion chronometry results, yet they are frequently overlooked and pose challenges in the resolution of the diffusion equation under these complexities. To address these challenges, we introduced the Finite Element Method (FEM). We elucidated the method's rationale, from the derivation of the weak form to the formulation of the global linear system. We then evaluated the method's accuracy against the exact solution, revealing a relative error of ±3–4‰ under the specified settings, which is an order of magnitude lower than that of LA-ICP-MS, thus demonstrating the robustness of the method. Following that, our two- and three-dimensional numerical experiments showcased FEM's adaptability in modeling species diffusion across arbitrary geometries and both Dirichlet and Neumann boundary conditions. Finally, the crystal's geometric effects on the ultimate elemental concentration were examined, revealing that they hold particular significance when the diffusion length is small. We conclude that the FEM surpasses the geometric limitations of minerals while simultaneously accommodating a variety of boundary conditions, thus offering significant potential for broad applications in the field.