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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An H(div)-conforming Finite Element Method for Biot’s Consolidation Model
An H(div)-conforming finite element method for the Biot’s consolidation mo-
del is developed, with displacements and fluid velocity approximated by elements from
BDM_k space. The use of H(div)-conforming elements for flow variables ensures the local
mass conservation. In the H(div)-conforming approximation of displacement, the tan-
gential components are discretised in the interior penalty discontinuous Galerkin frame-
work,and the normal components across the element interfaces are continuous. Having
introduced a spatial discretisation, we develop a semi-discrete scheme and a fully dis-
crete scheme,prove their unique solvability and establish optimal error estimates for
each variable.
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- NSF-PAR ID:
- 10092054
- Date Published:
- Journal Name:
- East Asian Journal on Applied Mathematics
- Volume:
- 9
- Issue:
- 3
- ISSN:
- 2079-7370
- Page Range / eLocation ID:
- 558-579
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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