Ariel L. Lombardi

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 $k=0$.

Blanca Ayuso de Dios, Ariel Lombardi, Paola Pietra et al.

We consider an exponentially fitted discontinuous Galerkin method and propose a robust block solver for the resulting linear systems.

Thomas Apel, Ariel L. Lombardi, Max Winkler

The paper is concerned with the finite element solution of the Poisson equation with homogeneous Dirichlet boundary condition in a three-dimensional domain. Anisotropic, graded meshes from a former paper are reused for dealing with the singular behaviour of the solution in the vicinity of the non-smooth parts of the boundary. The discretization error is analyzed for the piecewise linear approximation in the H^1(Ω)- and L^2(Ω)-norms by using a new quasi-interpolation operator. This new interpolant is introduced in order to prove the estimates for L^2(Ω)-data in the differential equation which is not possible for the standard nodal interpolant. These new estimates allow for the extension of certain error estimates for optimal control problems with elliptic partial differential equation and for a simpler proof of the discrete compactness property for edge elements of any order on this kind of finite element meshes.