p-FEM quadrature error analysis on tetrahedra

In this paper we consider the p-FEM for elliptic boundary value problems on tetrahedral meshes where the entries of the stiffness matrix are evaluated by numerical quadrature. Such a quadrature can be done by mapping the tetrahedron to a hexahedron via the Duffy transformation. We show that for tensor product Gauss-Lobatto-Jacobi quadrature formulas with q+1=p+1 points in each direction and shape functions that are adapted to the quadrature formula, one again has discrete stability for the fully discrete p-FEM. The present error analysis complements the work [Eibner/Melenk 2005] for the p-FEM on triangles/tetrahedra where it is shown that by adapting the shape functions to the quadrature formula, the stiffness matrix can be set up in optimal complexity