Neumann-Neumann-Schur complement methods for Fekete spectral elements

For the iterative solution of the Schur complement system associated with the discretization of an elliptic problem by means of a triangular spectral element method (TSEM), Neumann–Neumann (NN) type preconditioners are constructed and studied. The TSEM approximation, based on Fekete nodes, is a generalization to non-tensorial elements of the classical Gauss–Lobatto–Legendre quadrilateral spectral elements. Numerical experiments show that the TSEM Schur complement condition number grows linearly with the polynomial approximation degree, N, and quadratically with the inverse of the mesh size, h. NN preconditioners for the Schur complement allow to reduce the N-dependence of the condition number, by solving local Neumann problems on each spectral element, and to eliminate the h-dependence if an additional coarse solver is employed. Numerical results indicate that, in spite of the more severe ill-conditioning, the condition number of the TSEM preconditioned operator satisfies the same bound as that of the standard SEM, i.e., Ch −2(1 + log N)2 for one-level NN preconditioning and C(1 + log N)2 for two-level Balancing Neumann-Neumann (BNN) preconditioning.