Preconditioning the Mass Matrix for High Order Finite Element Approximation on Tetrahedra

Abstract

A preconditioner for the mass matrix for high order finite element discretization on tetrahedra is presented and shown to give a condition number that is independent of both the mesh size and the polynomial order of the elements. The preconditioner is described in terms of a new, high order basis which has the usual property whereby individual functions are associated with distinct geometric entities of the tetrahedron. It is shown that the basis enjoys the novel property that the resulting mass matrix is spectrally equivalent to its own diagonal with constants independent of h and p. Although the exposition is based on an explicit basis, the preconditioner can be applied to any choice of basis. In particular, the basis can be used to specify a basis-independent additive Schwarz method, meaning that, in order to apply the preconditioner to an alternative basis, one only need to implement an appropriate change of basis.