Colloquium: "Perspectives in Compatible Discretizations"

Abstract: Compatible discretizations have had a transformative impact on the numerical analysis of partial differential equations over the last decade. Many mathematical models in physics and engineering have qualitative features, such as conservation laws or physically relevant nullmodes. Such qualitative features need to be reproduced by any numerical discretization consistent with the original partial differential equations. This is accomplished by compatible discretizations. The research in compatible discretizations goes beyond classical topics in numerical analysis and relates differential geometry and algebraic topology to scientific computing.

In this colloquium talk, I will outline finite element exterior calculus as a major example of a compatible discretization. Finite element exterior calculus uses the language of differential forms to unify results in finite element methods for vector field equations. I will specifically focus on the construction of commuting projections from Sobolev de Rham complexes onto finite element de Rham complexes, which only recently has been fully understood. We will demonstrate the theory by applications in numerical electromagnetism. In particular, this discussion will showcase how finite element exterior calculus connects numerical analysis with various branches of mathematical research, such as geometric measure theory, algebraic topology, and analysis on manifolds.

Among the numerous research perspectives in finite element exterior calculus, I am going to focus on two in particular: on the one hand, we will see opportunities in connecting Sobolev de Rham complexes and geometric measure theory. On the other hand, we will discuss differential complexes in elasticity and relativity that are at the frontier of research in pure, applied, and numerical mathematics.

Host: Renato Feres

Tea will be served @ 3:30pm in room 200