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The paper analyzes the divergence stability of the p-version of the finite element method with the applications to the Stokes problem and elasticity problem with nearly uncompressible material. Many problems in continuum mechanics involve an incompressibility condition, usually in the form of a divergence constraint. The numerical discretization of such a constraint presents some interesting problems with regard to stability. As an important example we consider the two dimensional Stokes equations. The lack of divergence stability affects the accuracy of the pressure approximation much more drastically, and a certain postprocessing (filtering) of the pressures may be necessary as p approaches infinity. (jhd)