ArXiv (open access)
Key ideas
- We consider functions defined on an open cover of a domain.
- The Čech-de Rham complex is obtained by combining
- differential operators such as the gradient, curl, and divergence.
- differences of functions on the overlaps.
- We endow this double complex with a Hilbert space structure to create the $L^2$ Čech-de Rham complex.
Main findings
- The $L^2$ Čech-de Rham complex is a closed Hilbert complex and a Fredholm complex.
- In turn, the Hodge-Laplace problems on the Čech-de Rham complex are well-posed.
- Depending on the choice of open cover, the Hodge-Laplace problem corresponds to certain coupled problems, including:
- (1D-1D) elastically attached strings.
- (3D-3D) multi-porosity flow systems.
- (1D-3D) flow around wells in the subsurface and arteries in biophysical applications.
