Finite element spaces on a tetrahedron are constructed for div div -conforming symmetric tensors in three dimensions. The key tools of the con- struction are the decomposition of polynomial tensor spaces and the charac- terization of the trace operators. First, the div div Hilbert complex and its corresponding polynomial complexes are presented. Several decompositions of polynomial vector and tensor spaces are derived from the polynomial com- plexes. Second, traces for the divdiv operator are characterized through a Green’s identity. Besides the normal-normal component, another trace involving combination of first order derivatives of the tensor is continuous across the face. Due to the smoothness of polynomials, the symmetric tensor element is also continuous at vertices, and on the plane orthogonal to each edge. Besides, a finite element for sym curl-conforming trace-free tensors is constructed following the same approach. Putting all together, a finite element div div complex, as well as the bubble functions complex, in three dimensions is established.
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A finite element elasticity complex in three dimensions
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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- NSF-PAR ID:
- 10423049
- Date Published:
- Journal Name:
- Mathematics of Computation
- Volume:
- 91
- Issue:
- 337
- ISSN:
- 0025-5718
- Page Range / eLocation ID:
- 2095 to 2127
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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We construct conforming finite element elasticity complexes on Worsey–Farin splits in three dimensions. Spaces for displacement, strain, stress, and the load are connected in the elasticity complex through the differential operators representing deformation, incompatibility, and divergence. For each of these component spaces, a corresponding finite element space on Worsey–Farin meshes is exhibited. Unisolvent degrees of freedom are developed for these finite elements, which also yields commuting (cochain) projections on smooth functions. A distinctive feature of the spaces in these complexes is the lack of extrinsic supersmoothness at subsimplices of the mesh. Notably, the complex yields the first (strongly) symmetric stress finite element with no vertex or edge degrees of freedom in three dimensions. Moreover, the lowest order stress space uses only piecewise linear functions which is the lowest feasible polynomial degree for the stress space.
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Abstract We construct new families of
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We construct several smooth finite element spaces defined on three-dimensional Worsey–Farin splits. In particular, we construct C 1 C^1 , H 1 ( curl ) H^1(\operatorname {curl}) , and H 1 H^1 -conforming finite element spaces and show the discrete spaces satisfy local exactness properties. A feature of the spaces is their low polynomial degree and lack of extrinsic supersmoothness at subsimplices of the mesh. In the lowest order case, the last two spaces in the sequence consist of piecewise linear and piecewise constant spaces, and are suitable for the discretization of the (Navier-)Stokes equation.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1090/mcom/3739
