Conditioning of hierarchic p-version Nedelec elements on meshes of curvilinear quadrilaterals and hexahedra

The conditioning of a set of hierarchic basis functions for p-version edge element approximation of the space H(curl) is studied. Theoretical bounds are obtained on the location of the eigenvalues and on the growth of the condition numbers for the mass, curl-curl, and stiffness matrices that naturally arise from Galerkin approximation of Maxwell's equations. The theory is applicable to meshes of curvilinear quadrilaterals or hexahedra in two and three dimensions, respectively, including the case in which the local order of approximation is nonuniform. Throughout, the theory is illustrated with numerical examples that show that the theoretical asymptotic bounds are sharp and are attained within the range of practical computation.