Abstract 

Multigrid methods have been implemented in many scientific libraries for solving matrices derived from partial differential equations due to its mesh-size-independent convergence rate and optimal complexity. We develop a fault-tolerant geometric multigrid method to address the loss of numerical data on faulty processors. Prior studies of fault-tolerant multigrid methods have focused on structured grids. In this research, unstructured grids distributed across multiple processors may manifest as local hierarchical grids with unaligned boundaries. This misalignment can cause divergence in the local multigrid recovery process. We address the recovery convergence issue from two distinct angles. Adjusting the mesh and adjusting the numerical algorithm. Our focus of the talk is on the latter. We propose a novel algorithm that modifies the classical multigrid V-cycle algorithm based on energy control criterion. We give a proof of the convergence of the new algorithm by using smoothing and approximation properties defined in non-nested subspaces. Additionally, we introduce a near-boundary treatment and a local energy criterion to improve the convergence rate and shorten the runtime of local recovery, respectively. If time allows, we will talk about an asynchronous recovery strategy that allows the healthy processors to continue computation while the faulty processor recovers.