  We present in this paper a rigorous theoretical framework to show stability,
convergence and accuracy of improved edge-based and face-based smoothed finite
element methods (bESFEM and bFS-FEM) for nearly-incompressible elasticity
problems. The crucial idea is that the space of piecewise linear polynomials
used for the displacements is enriched with bubble functions on each element,
while the pressure is a piecewise constant function. The meshes of triangular
or tetrahedral elements required by these methods can be generated
automatically. The enrichment induces a softening in the bilinear form allowing
the weakened weak (W2)procedure to produce a high-quality solution, free from
locking and that does not oscillate. We prove theoretically that both methods
confirm the uniform inf-sup and convergence conditions. Four numerical examples
are given to validate the reliability of the bES-FEM and bFS-FEM.
