||Solutions to Partial Di erential Equations (PDEs) are often computed by discretizing the domain into a collection of computational elements referred to as a mesh. This solution is an approximation with an error that decreases as the mesh spacing decreases. However, decreasing the mesh spacing also increases the computational requirements. Adaptive mesh re nement (AMR) attempts to reduce the error while limiting the increase in computational requirements by re ning the mesh locally in regions of the domain that have large error while maintaining a coarse mesh in other portions of the domain. This approach often provides a solution that is as accurate as that obtained from a much larger xed mesh simulation, thus saving on both computational time and memory. However, historically, these AMR operations often limit the overall scalability of the application. Adapting the mesh at runtime necessitates scalable regridding and load balancing algorithms. This dissertation analyzes the performance bottlenecks for a widely used regridding algorithm and presents two new algorithms which exhibit ideal scalability. In addition, a scalable space- lling curve generation algorithm for dynamic load balancing is also presented. The performance of these algorithms is analyzed by determining their theoretical complexity, deriving performance models, and comparing the observed performance to those performance models. The models are then used to predict performance on larger numbers of processors. This analysis demonstrates the necessity of these algorithms at larger numbers of processors. This dissertation also investigates methods to more accurately predict workloads based on measurements taken at runtime. While the methods used are not new, the application of these methods to the load balancing process is. These methods are shown to be highly accurate and able to predict the workload within 3% error. By improving the accuracy of these estimations, the load imbalance of the simulation can be reduced, thereby increasing the overall performance.