The Earth system presents some of the most demanding challenges for simulation and prediction, spanning a vast range of scales in space and time, involving tightly coupled multiphysics processes, and requiring integration of massive observational datasets. As climate, seismic, hydrological, and subsurface models evolve toward higher fidelity, the need for extreme-scale computing and advanced numerical methods becomes more pressing. This minisymposium shall connect researchers at the intersection of geosciences, computational mathematics, and high-performance computing to present recent advances and emerging trends. We aim to highlight progress in areas including (but not limited to) discretization techniques, robust solvers for multiphysics and multiscale systems, multigrid and domain decomposition methods, uncertainty quantification, and machine learning integration in forward and inverse problems. Of particular interest are innovations enabling simulations on state-of-the-art exascale architectures, e.g., via performance-portable implementations, improved algorithms or simulation software design. Applications span the solid Earth, ocean and atmosphere dynamics, ice sheet evolution, groundwater flow, and coupled Earth system modeling. We welcome contributions focused on algorithmic development, software frameworks, mathematical analysis, and interdisciplinary case studies that push the boundaries of what is computationally and scientifically feasible in the geosciences.