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Adaptive Mesh Refinement (AMR) is a computational technique used to reduce the amount of computation and memory required in scientific simulations. Geosimulations are scientific simulations using geographic data, routinely used to predict outcomes of urbanization in urban studies. However, the lack of support for AMR techniques with geosimulations limits exploring prediction outcomes at multiple resolutions. In this paper, we propose an adaptive mesh refinement framework FUTURES-AMR, based on static user-defined policies to enable multi-resolution geosimulations. We develop a prototype for the cellular automaton based urban growth simulation FUTURES by exploiting static and dynamic mesh refinement techniques in conjunction with the Patch Growing Algorithm (PGA). While, the static refinement technique supports a statically defined fixed resolution mesh simulation at a location, the dynamic refinement technique supports dynamically refining the resolution based on simulation outcomes at runtime. Further, we develop two approaches - asynchronous AMR and synchronous AMR, suitable for parallel execution in a distributed computing environment with varying support for solution integration of the multi-resolution results. Finally, using the FUTURES-AMR framework with different policies in an urban study, we demonstrate reduced execution time, and low memory overhead for a multi-resolution simulation.