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A finite element elasticity complex on tetrahedral meshes and the corresponding commutative diagram are devised. The H 1 H^1 conforming finite element is the finite element developed by Neilan for the velocity field in a discrete Stokes complex. The symmetric div-conforming finite element is the Hu-Zhang element for stress tensors. The construction of an H ( inc ) H(\operatorname {inc}) -conforming finite element of minimum polynomial degree 6 6 for symmetric tensors is the focus of this paper. Our construction appears to be the first H ( inc ) H(\operatorname {inc}) -conforming finite elements on tetrahedral meshes without further splitting. The key tools of the construction are the decomposition of polynomial tensor spaces and the characterization of the trace of the inc \operatorname {inc} operator. The polynomial elasticity complex and Koszul elasticity complex are created to derive the decomposition. The trace of the inc \operatorname {inc} operator is induced from a Green’s identity. Trace complexes and bubble complexes are also derived to facilitate the construction. Two-dimensional smooth finite element Hessian complex and div div \operatorname {div}\operatorname {div} complex are constructed.
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On Perrot’s index cocycles
We give a simplified version of a construction due to Denis Perrot that recovers the Todd class of the complexified tangent bundle of a smooth manifold from a JLO-type cyclic cocycle. The construction takes place within an algebraic framework, rather than the customary functional-analytic frame- work of the JLO theory. The series expansion for the exponential function is used in place of the heat kernel from the functional-analytic theory; the Dirac operator chosen is far from elliptic; and a remarkable new trace discovered by Perrot replaces the operator trace. In its full form, Perrot’s theory constitutes a wholly new approach to index theory. The account presented here covers most but not all of his approach.
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- Award ID(s):
- 1952669
- PAR ID:
- 10412583
- Editor(s):
- Connes, A.; Consani, C.; Dundas, B.; Khalkhali, M.; Moscovici, H.
- Date Published:
- Journal Name:
- Proceedings of symposia in pure mathematics
- Volume:
- 105
- ISSN:
- 2324-707X
- Page Range / eLocation ID:
- 29-62
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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