A right-inverse for the divergence operator in spaces of piecewise polynomials

SummaryIn the first part of this paper we study in detail the properties of the divergence operator acting on continuous piecewise polynomials on some fixed triangulation; more specifically, we characterize the range and prove the existence of a maximal right-inverse whose norm grows at most algebraically with the degree of the piecewise polynomials.In the last part of this paper we apply these results to thep-version of the Finite Element Method for a nearly incompressible material with homogeneous Dirichlet boundary conditions. We show that thep-version maintains optimal convergence rates in the limit as the Poisson ratio approaches 1/2. This fact eliminates the need for any “reduced integration” such as customarily used in connection with the more standardh-version of the Finite Element Method.