Error estimates for Raviart-Thomas interpolation of any order on anisotropic tetrahedra

Abstract:

We prove optimal order error estimates for the Raviart-Thomas interpolation of arbitrary order under the maximum angle condition for triangles and under two generalizations of this condition, namely, the so-called three-dimensional maximum angle condition and the regular vertex property, for tetrahedra.

Our techniques are different from those used in previous papers on the subject, and the results obtained are more general in several aspects. First, intermediate regularity is allowed; that is, for the Raviart-Thomas interpolation of degree $k\ge 0$, we prove error estimates of order $j+1$ when the vector field being approximated has components in $W^{j+1,p}$, for triangles or tetrahedra, where $0\le j \le k$ and $1\le p \le \infty$. These results are new even in the two-dimensional case. Indeed, the estimate was known only in the case $j=k$. On the other hand, in the three-dimensional case, results under the maximum angle condition were known only for $k=0$.