Error Estimates for the Raviart–Thomas Interpolation Under the Maximum Angle Condition

Abstract

The classical error analysis for the Raviart-Thomas interpolation on triangular elements requires the so-called regularity of the elements, or equivalently, the minimum angle condition. However, in the lowest order case, optimal order error estimates have been obtained in [G. Acosta and R. G. Duran, SIAM J. Numer. Anal., 37 (2000), pp. 18-36] replacing the regularity hypothesis by the maximum angle condition, which was known to be sufficient to prove estimates for the standard Lagrange interpolation. In this paper we prove error estimates on triangular elements for the Raviart-Thomas interpolation of any order under the maximum angle condition. Also, we show how our arguments can be extended to the three-dimensional case to obtain error estimates for tetrahedral elements under the regular vertex property introduced in [G. Acosta and R. G. Duran, SIAM J. Numer. Anal., 37 (2000), pp. 18-36].