  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(\Omega)- and L^2(\Omega)-norms by using a new
quasi-interpolation operator. This new interpolant is introduced in order to
prove the estimates for L^2(\Omega)-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.
