Finite element de Rham complexes and finite element Stokes complexes with varying degrees of smoothness in three dimensions are systematically constructed in this paper. Smooth scalar finite elements in three dimensions are derived through a non-overlapping decomposition of the simplicial lattice. H(div)-conforming finite elements and H(curl)-conforming finite elements with varying degrees of smoothness are devised based on these smooth scalar finite elements. The finite element de Rham complexes with corresponding smoothness and commutative diagrams are induced by these elements. The div stability of the H(div)-conforming finite elements is established, and the exactness of these finite element complexes is proven.
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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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- 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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Abstract We construct new families of
direct serendipity anddirect mixed finite elements on general planar, strictly convex polygons that areH 1andH (div) conforming, respectively, and possess optimal order of accuracy for any order. They have a minimal number of degrees of freedom subject to the conformity and accuracy constraints. The name arises because the shape functions are defineddirectly on the physical elements, i.e., without using a mapping from a reference element. The finite element shape functions are defined to be the full spaces of scalar or vector polynomials plus a space of supplemental functions. The direct serendipity elements are the precursors of the direct mixed elements in a de Rham complex. The convergence properties of the finite elements are shown under a regularity assumption on the shapes of the polygons in the mesh, as well as some mild restrictions on the choices one can make in the construction of the supplemental functions. Numerical experiments on various meshes exhibit the performance of these new families of finite elements.
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