On a Right-Inverse for the Divergence Operator in Spaces of Continuous Piecewise Polynomials

Abstract

This paper is concerned with the norm of a right-inverse for the divergence operator between spaces of piecewise polynomials on triangular elements. More specifically, one tries to construct a right-inverse acting from the space W of continuous piecewise polynomials of degree p into the space V of R2-valued piecewise polynomials of degree p+1. Our results are as follows. Assume that we are dealing with a quasiuniform family of triangulations {Mh} satisfying two additional hypotheses: Mh can be transformed into a quadriangular mesh by grouping its element two by two and Mh has no boundary vertex shared by only one or exactly three elements. In that context, one proves that the divergence has a right-inverse with an operator norm growing at most like [Formula: see text] when both W and V are equipped with the H1-norm and at most like [Formula: see text] if W is equipped with the L2-norm only. An application of the first of these two results is the approximation of the thermoelasticity equations by the p-version of the finite element methods. One also shows how the second result can be used in the context of higher-order Hood–Taylor method for Stokes problem.