Error estimates for Raviart-Thomas interpolation of any order on anisotropic tetrahedra
For researchers in finite element methods, this provides more general error estimates for Raviart-Thomas interpolation, though the improvement is incremental over existing results.
The paper proves optimal order error estimates for Raviart-Thomas interpolation of any order under the maximum angle condition for triangles and its generalizations for tetrahedra, allowing intermediate regularity (0 ≤ j ≤ k) for vector fields in W^{j+1,p}, which is new even in 2D and extends 3D results beyond k=0.
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 $k=0$.