A crack-adapted Schwarz smoothing multigrid method with incrementally updated Galerkin projections for implicit quasi-static crack propagation in peridynamics

This work develops a crack-adapted Schwarz smoothing geometric multigrid method with incrementally updated Galerkin projections (MG-CAS) for sequences of linear systems, which arise from the implicit quasi-static crack propagation in peridynamics. Standard geometric multigrid may suffer from degraded convergence behavior as crack propagation introduces discontinuities, while reconstructing a crack-aware coarse space is complex to implement and increases the cumulative setup cost. To address this issue, MG-CAS keeps the multigrid transfer operators fixed. The Galerkin coarse-level operators are updated incrementally using low-rank modifications of stiffness matrices at a very low cost, thereby reducing the cumulative setup cost. To compensate for crack-induced local errors that are not well captured by the fixed coarse space, a crack-adapted Schwarz smoother is introduced. This smoother adaptively generates patches near the crack, performs local residual correction on these patches, and applies weighted Jacobi smoothing elsewhere, thereby reducing the iteration count and the overall solve time. Numerical experiments show that MG-CAS maintains relatively stable iteration behavior in crack propagation simulations and generally achieves higher overall efficiency than the baseline methods.