6,095 research outputs found
Nonconforming tetrahedral mixed finite elements for elasticity
This paper presents a nonconforming finite element approximation of the space
of symmetric tensors with square integrable divergence, on tetrahedral meshes.
Used for stress approximation together with the full space of piecewise linear
vector fields for displacement, this gives a stable mixed finite element method
which is shown to be linearly convergent for both the stress and displacement,
and which is significantly simpler than any stable conforming mixed finite
element method. The method may be viewed as the three-dimensional analogue of a
previously developed element in two dimensions. As in that case, a variant of
the method is proposed as well, in which the displacement approximation is
reduced to piecewise rigid motions and the stress space is reduced accordingly,
but the linear convergence is retained.Comment: 13 pages, 2 figure
Weakly imposed symmetry and robust preconditioners for Biot's consolidation model
We discuss the construction of robust preconditioners for finite element
approximations of Biot's consolidation model in poroelasticity. More precisely,
we study finite element methods based on generalizations of the
Hellinger-Reissner principle of linear elasticity, where the stress tensor is
one of the unknowns. The Biot model has a number of applications in science,
medicine, and engineering. A challenge in many of these applications is that
the model parameters range over several orders of magnitude. Therefore,
discretization procedures which are well behaved with respect to such
variations are needed. The focus of the present paper will be on the
construction of preconditioners, such that the preconditioned discrete systems
are well-conditioned with respect to variations of the model parameters as well
as refinements of the discretization. As a byproduct, we also obtain
preconditioners for linear elasticity that are robust in the incompressible
limit.Comment: 21 page
Stabilized mixed finite element methods for linear elasticity on simplicial grids in
In this paper, we design two classes of stabilized mixed finite element
methods for linear elasticity on simplicial grids. In the first class of
elements, we use - and
- to approximate the stress
and displacement spaces, respectively, for , and employ a
stabilization technique in terms of the jump of the discrete displacement over
the faces of the triangulation under consideration; in the second class of
elements, we use - to
approximate the displacement space for , and adopt the
stabilization technique suggested by Brezzi, Fortin, and Marini. We establish
the discrete inf-sup conditions, and consequently present the a priori error
analysis for them. The main ingredient for the analysis is two special
interpolation operators, which can be constructed using a crucial
bubble function space of polynomials on each
element. The feature of these methods is the low number of global degrees of
freedom in the lowest order case. We present some numerical results to
demonstrate the theoretical estimates.Comment: 16 pages, 1 figur
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