A 2-complex containing Sobolev spaces of matrix fields

TL;DR

This paper introduces and analyzes new Sobolev spaces of matrix fields using 2-complexes, revealing their properties, dualities, and potential applications in finite element methods and variational problems.

Contribution

It defines novel Sobolev spaces of matrix fields via 2-complexes, explores their properties, dualities, and decompositions, and discusses implications for finite element analysis.

Findings

Spaces have very weak second-order derivatives

Stable decompositions into more regular functions established

Duality relationships between Sobolev spaces discovered

Abstract

Using a generalization of complexes, called 2-complexes, this paper defines and analyzes new Sobolev spaces of matrix fields and their interrelationships within a commuting diagram. These spaces have very weak second-order derivatives. An example is the space of matrix fields of square-integrable components whose row-wise divergence followed by yet another divergence operation yield a function in a standard negative-order Sobolev space. Similar spaces where the double divergence is replaced by a curl composed with divergence, or a double curl operator (the incompatibility operator), are also studied. Stable decompositions of such spaces in terms of more regular component functions (which are continuous in natural norms) are established. Appropriately ordering such Sobolev spaces with and without boundary conditions (in a weak sense), we discover duality relationships between them. Motivation to study such Sobolev spaces, from a finite element perspective and implications for weak well-posed variational formulations are pointed out.