This paper is concerned with the derivation and properties of differential complexes arising from a variety of problems in differential equations, with applications in continuum mechanics, relativity, and other fields. We present a systematic procedure which, starting from well-understood differential complexes such as the de Rham complex, derives new complexes and deduces the properties of the new complexes from the old. We relate the cohomology of the output complex to that of the input complexes and show that the new complex has closed ranges, and, consequently, satisfies a Hodge decomposition, Poincaré-type inequalities, well-posed Hodge–Laplacian boundary value problems, regular decomposition, and compactness properties on general Lipschitz domains.
Bibliographical noteFunding Information:
The work of the first author was supported by NSF Grant DMS-1719694 and Simons Foundation Grant 601937, DNA.
© 2021, SFoCM.
- BGG resolution
- Differential complex
- Finite element exterior calculus
- Hilbert complex
- de Rham complex