and satisfying the energy compatibility condition is shown to be an ultimate key to obtaining optimal tress modes. By using compatible isoparametric bilinear (Q4) displacements and 5-parameter energy ompatible stresses of the combined hybrid nite element CH(0-1), a robust 4-node plane stress element ECQ4 is derived. Equivalence to another hybrid stress element LQ6 with 9-parameter complete linear stresses based on a modi ed Hellinger–Reissner principle is established. A convergence analysis is given and numerical experiments show that elements ECQ4\LQ6 have high performance, i.e. are accurate at coarse meshes, insensitive to mesh distortions and free from locking. Copyright? 2003 John Wiley & Sons, Ltd.

A hybrid stress quadrilateral macro-element HQM is proposed. Compatible linear displacements are used on its two triangular sub-domains, and a 5-parameter incomplete linear stress mode is suggested. Equivalence to another quadrilateral element HQ4 with compatible isoparametric bilinear displacements is proven. Due to elimination of stress parameters at the element level, the computational cost of HQM/HQ4 is as same as that of Q4. Numerical tests show that the element is accurate, insensitive to mesh distortions, and free from Poisson locking.

A 4-node hybrid stress quadrilateral element is presented with compatible isoparametric bilinear displacements and a new version of 5-parameter stresses which satisfies both self-equilibrium equations and an energy orthogonal relation between stress terms and Wilson incompatible strains. Based on different element formulations, another two new elements are also derived and shown to be identical to the first element. Numerical benchmark experiments show that the three equivalent elements possess high accuracy at coarse meshes, are frame invariant and free from Poisson locking at the nearly incompressible limit. Since the 5-parameter stress mode used does not have uncoupled constant terms, the elements do not pass the patch test.

In this paper, we consider 2D and 3D Darcy-Stokes interface problems. These equations are related to Brinkman model that treats both Darcy’s law and Stokes equations in a single form of PDE but with strongly discontinuous viscosity coefficient and zeroth- order term coefficient. We present three different methods to construct uniformly stable finite element approximations. The first two methods are based on the original weak formulations of Darcy-Stokes-Brinkman equations. In the first method we consider the existing Stokes elements. We show that a stable Stokes element is also uniformly stable with respect to the coefficients and the jumps of Darcy-Stokes-Brinkman equations if and only if the discretely divergence-free velocity implies almost everywhere divergence-free one. In the second method we construct uniformly stable elements by modifying some well-known H(div)-conforming elements. We give some new 2D and 3D elements in a unified way. In the last method we modify the original weak formulation of Darcy-Stokes- Brinkman equations with a stabilization term. We show that all traditional stable Stokes elements are uniformly stable with respect to the coefficients and their jumps under this new formulation.

A nonconforming finite element method of finite difference streamline diffusion type is proposed to solve the time-dependent linearized Navier-Stokes equations. The backward Euler scheme is used for time discretization. Crouzeix-Raviart nonconforming finite element approximation, namely, nonconforming (P1)2-P0 element, is used for the velocity and pressure fields with the streamline diffusion technique to cope with usual instabilities caused by the convection and time terms. Stability and error estimates are derived with suitable norms.