Fast Galerkin Multigrid Method for Unstructured Meshes

We present a novel multigrid solver framework that significantly advances the efficiency of physical simulation for unstructured meshes. While multi-grid methods theoretically offer linear scaling, their practical implementation for deformable body simulations faces substantial challenges, particularly on GPUs. Our framework achieves up to 6.9× speedup over traditional methods through an innovative combination of matrix-free vertex block Jacobi smoothing with a Full Approximation Scheme (FAS), enabling both piecewise constant and linear Galerkin formulations without the computational burden of dense coarse matrices. Our approach demonstrates superior performance across varying mesh resolutions and material stiffness values, maintaining consistent convergence even under extreme deformations and challenging initial configurations. Comprehensive evaluations against state-of-the-art methods confirm our approach achieves lower simulation error with reduced computational cost, enabling simulation of tetrahedral meshes with over one million vertices at approximately one frame per second on modern GPUs.