Stabilized mixed finite element methods for linear elasticity on simplicial grids in Rn\mathbb{R}^{n}

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 $\boldsymbol{H}(\mathbf{div}, \Omega; \mathbb{S})$-$P_k$ and $\boldsymbol{L}^2(\Omega; \mathbb{R}^n)$-$P_{k-1}$ to approximate the stress and displacement spaces, respectively, for $1\leq k\leq n$, 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 $\boldsymbol{H}_0^1(\Omega; \mathbb{R}^n)$-$P_{k}$ to approximate the displacement space for $1\leq k\leq n$, 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 $\boldsymbol{H}(\mathbf{div})$ 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

Finite element approximation of the three field formulation of the Stokes problem using arbitrary interpolations

The stress-displacement-pressure formulation of the elasticity problem may suffer from two types of numerical instabilities related to the finite element interpolation of the unknowns. The first is the classical pressure instability that occurs when the solid is incompressible, whereas the second is the lack of stability in the stresses. To overcome these instabilities, there are two options. The first is to use different interpolation for all the unknowns satisfying two inf-sup conditions. Whereas there are several displacement-pressure interpolations that render the pressure stable, less possibilities are known for the stress interpolation. The second option is to use a stabilized finite element formulation instead of the plain Galerkin approach. If this formulation is properly designed, it is possible to use arbitrary interpolation for all the unknowns. The purpose of this paper is precisely to present one of such formulations. In particular, it is based on the decomposition of the unknowns into their finite element component and a subscale, which will be approximated and whose goal is to yield a stable formulation. A singular feature of the method to be presented is that the subscales will be considered orthogonal to the finite element space. We describe the design of the formulation and present the results of its numerical analysis. &nbsp

An extended finite element method with smooth nodal stress

The enrichment formulation of double-interpolation finite element method (DFEM) is developed in this paper. DFEM is first proposed by Zheng \emph{et al} (2011) and it requires two stages of interpolation to construct the trial function. The first stage of interpolation is the same as the standard finite element interpolation. Then the interpolation is reproduced by an additional procedure using the nodal values and nodal gradients which are derived from the first stage as interpolants. The re-constructed trial functions are now able to produce continuous nodal gradients, smooth nodal stress without post-processing and higher order basis without increasing the total degrees of freedom. Several benchmark numerical examples are performed to investigate accuracy and efficiency of DFEM and enriched DFEM. When compared with standard FEM, super-convergence rate and better accuracy are obtained by DFEM. For the numerical simulation of crack propagation, better accuracy is obtained in the evaluation of displacement norm, energy norm and the stress intensity factor