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Explore the intersection of numerical analysis and spacetime through this 56-minute conference talk that delves into high-order approximations of partial differential equations (PDE). Learn how physical laws elegantly described through PDE require sophisticated numerical methods that maintain consistency, stability, and convergence while preserving important underlying structures including geometrical, algebraic, topological, and homological properties. Discover the finite element approach for PDE and understand the mathematical theory of finite element exterior calculus, which enables the description of polynomial differential forms forming finite-dimensional subcomplexes of the deRham complex on simplicial or tensorial domains. Examine concrete examples in R^2 and R^3 before advancing to the design of high-order discretizations of PDEs in R^4 for space-time problems. Investigate families of conforming high-order finite elements on both simplicial elements and non-simplicial domains through explicit constructions that utilize techniques from finite element exterior calculus, presented as part of collaborative research with David Williams during Women in Math Day at the Centre de recherches mathématiques.
