Applications of algebraic topology to compatible spatial discretizations.
- Los Alamos National Laboratory, Los Alamos, NM
We provide a common framework for compatible discretizations using algebraic topology to guide our analysis. The main concept is the natural inner product on cochains, which induces a combinatorial Hodge theory. The framework comprises of mutually consistent operations of differentiation and integration, has a discrete Stokes theorem, and preserves the invariants of the DeRham cohomology groups. The latter allows for an elementary calculation of the kernel of the discrete Laplacian. Our framework provides an abstraction that includes examples of compatible finite element, finite volume and finite difference methods. We describe how these methods result from the choice of a reconstruction operator and when they are equivalent.
- Research Organization:
- Sandia National Laboratories
- Sponsoring Organization:
- USDOE
- DOE Contract Number:
- AC04-94AL85000
- OSTI ID:
- 969879
- Report Number(s):
- SAND2005-3331C
- Country of Publication:
- United States
- Language:
- English
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