A Right-Inverse for the Divergence Operator in Spaces of Piecewise Polynomials. Application to the p-Version of the Finite Element Method

摘要

In the first part of this paper we study in detail the properties of the divergence operator acting on continuous piecewise polynomials; 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 the p-version of the Finite Element Method for a nearly incompressible material with homogeneous Dirichlet boundary conditions. We show that the p-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 standard h-version of the Finite Element Method. (Author)