Spectral Equivalence Properties of Higher-Order Tensor Product Finite Elements and Applications to Preconditioning.

Abstract

The focus of this study is on spectral equivalence results for higher‚order tensor product finite elements in the H (curl), H (div), and L function spaces. For certain choices of the higher‚order shape functions, the resulting mass and stiffness matrices are spectrally equivalent to those for an assembly of lowest‚order edge‚, face‚ or interior‚based elements on the associated Gauss‚Lobatto‚Legendre ̆GLL ̄ mesh. This equivalence will help enable the development of efficient domain decomposition or multigrid preconditioners. Specifically, preconditioners for the equivalent lowest‚ order linear system can be used for the higher‚order problem and avoid the demands of assembling a higher‚order coefficient matrix. Using assemblies of lowest‚order ̆linear ̄ elements for efficient preconditioning of higher‚order discretizations in the function space H is not new. We refer the interested reader to Section 7.1 of [10] or the introduction of [2] for a discussion of the pioneering work by Orszag [9], Deville and Mund [3, 8], Canuto [1] and others. We are, however, not aware of similar approaches for problems using higher‚order edge‚ ̆Nédélec ̄, face‚ ̆Raviart‚Thomas ̄ or interior‚based elements. We note for the case of nodal elements that the degrees of freedom ̆DOFs ̄ for a higher‚order element and its equivalent assembly of lowest‚order elements are nodal values in both cases. This natural one‚to‚one correspondence of DOFs can be realized for edge‚, face‚ and interior‚based elements by using shape functions ̆bases ̄ associated with integrals and introduced by Gerritsma [5]. For edge‚based elements, the DOFs for the shape functions are associated with integrals of tangential components of a vector field along each edge of the associated GLL mesh ̆see Figure 1 left ̄. Similarly, DOFs for face‚based elements correspond to integrals of the normal component of a vector field over individual faces of the GLL mesh ̆see Figure 1 right ̄. For completeness, we also present shape functions