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The paper is concerned with the finite element solution of the Poissonequation with homogeneous Dirichlet boundary condition in a three-dimensionaldomain. Anisotropic, graded meshes from a former paper are reused for dealingwith the singular behaviour of the solution in the vicinity of the non-smoothparts of the boundary. The discretization error is analyzed for the piecewiselinear approximation in the H^1(\Omega)- and L^2(\Omega)-norms by using a newquasi-interpolation operator. This new interpolant is introduced in order toprove the estimates for L^2(\Omega)-data in the differential equation which isnot possible for the standard nodal interpolant. These new estimates allow forthe extension of certain error estimates for optimal control problems withelliptic partial differential equation and for a simpler proof of the discretecompactness property for edge elements of any order on this kind of finiteelement meshes.

Anisotropic mesh refinement in polyhedral domains: error estimates with data in L^2(\Omega)