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Leveraging the power of nowadays graphics processing units for robust power grid simulation remains a challenging task. Existing preconditioned iterative methods that require incomplete matrix factorizations cannot be effectively accelerated on graphics processing unit (GPU) due to its limited hardware resource as well as data parallel computing. This paper presents an efficient GPU-based multigrid preconditioning algorithm for robust power grid analysis. By combining the fast geometric multigrid solver with the robust Krylov-subspace iterative solver, power grid DC and transient analysis can be performed efficiently on GPU without loss of accuracy (largest errors <;0.5 mV). Unlike previous GPU-based algorithms that rely on good power grid regularities, the proposed algorithm can be applied for more general power grid structures. Additionally, we also propose an accuracy-aware GPU performance modeling and optimization framework to automatically obtain the best power grid simulation configurations. Experimental results show that the DC and transient analysis on GPU can achieve more than 25X speedups over the best available CPU-based solvers.