Ricardo G. Durán

G. Acosta, Th. Apel, R. G. Durán et al.

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 $k=0$.

Maria E. Cejas, Ricardo G. Duran, Maria I. Prieto

We analyze the approximation by mixed finite element methods of solutions of equations of the form $-\mbox{div\,} (a\nabla u) = g$, where the coefficient $a=a(x)$ can degenerate going to cero or infinity. First, we extend the classic error analysis to this case provided that the coefficient $a$ belongs to the Muckenhoupt class $A_2$. The analysis developed applies to general mixed finite element spaces satisfying the standard commutative diagram property, whenever some stability and interpolation error estimates are valid in weighted norms. Next, we consider in detail the case of Raviart-Thomas spaces of lowest order, obtaining optimal order error estimates for general regular elements as well as for some particular anisotropic ones which are of interest in problems with boundary layers. Finally we apply the results to a problem arising in the solution of the fractional Laplace equation.

Irene Drelichman, Ricardo Durán, Ignacio Ojea

We consider the approximation of Poisson type problems where the source is given by a singular measure and the domain is a convex polygonal or polyhedral domain. First, we prove the well-posedness of the Poisson problem when the source belongs to the dual of a weighted Sobolev space where the weight belongs to the Muckenhoupt class. Second, we prove the stability in weighted norms for standard finite element approximations under the quasi-uniformity assumption on the family of meshes.