ABSTRACT

Structure-preserving numerical methods have had a transformative impact on the numerical analysis of partial differential equations. These methods reproduce the fundamental mathematical structures of numerous partial differential equations exactly at the numerical level. This talk gives an introduction and overview of structurepreserving finite element methods via "finite element exterior calculus” (FEEC) and explores some new directions in the field. The fundamental mathematical structures in FEEC are differential complexes and their cohomology, preserved at the numerical level. This connection between geometry, topology, and classical numerical analysis provides a unified perspective on finite element methods in vector analysis. We address two recent research developments: mixed boundary conditions in FEEC and finite element methods over manifolds. We furthermore discuss future directions in numerical electromagnetism, elasticity, and relativity.

Registration URL

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