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Learn a novel computational method for bounding high-order multi-dimensional polynomials in finite element approximations through this 34-minute conference talk from the MFEM Workshop 2025. Discover how to precompute optimal piecewise-linear bounding boxes for polynomial basis functions that can efficiently bound any combination of these functions locally. Explore the method's universal applicability across different element and basis types at any approximation order, with efficient on-the-fly evaluation capabilities for large-scale scientific simulations. Examine recent developments in high-order mesh optimization within the MFEM framework, including tangential relaxation techniques for curved boundaries, mesh validity detection through Jacobian determinant bounds of nodal position functions, and PDE-constrained r-adaptivity approaches that enhance mesh quality to minimize solution errors. Gain insights into advanced computational techniques for modular finite element methods from LLNL's research team, presented as part of the fifth annual MFEM community workshop that brings together users and developers to review software features, development roadmaps, and cutting-edge applications in high-performance scientific computing.
