Overlapping Schwarz methods for Fekete and Gauss-Lobatto spectral elements

The classical overlapping Schwarz algorithm is here extended to the triangular/tetrahedral spectral element (TSEM) discretization of elliptic problems. This discretization, based on Fekete nodes, is a generalization to nontensorial elements of the tensorial Gauss–Lobatto–Legendre quadrilateral spectral elements (QSEM). The overlapping Schwarz preconditioners are based on partitioning the domain of the problem into overlapping subdomains, solving local problems on these subdomains, and solving an additional coarse problem associated with either the subdomain mesh or the spectral element mesh. The overlap size is generous, i.e., one element wide, in the TSEM case, while it is minimal or variable in the QSEM case. The results of several numerical experiments show that the convergence rate of the proposed preconditioning algorithm is independent of the number of subdomains $N$ and the spectral degree $p$ in case of generous overlap; otherwise it depends inversely on the overlap size. The proposed preconditioners are also robust with respect to arbitrary jumps of the coefficients of the elliptic operator across subdomains.