Spectral element FETI-DP and BDDC preconditioners with multi-element subdomains

The discrete systems generated by spectral or hp-version finite elements are much more ill-conditioned than the ones generated by standard low-order finite elements or finite differences. This paper focuses on spectral elements based on Gauss–Lobatto–Legendre (GLL) quadrature and the construction of primal and dual non-overlapping domain decomposition methods belonging to the family of Balancing Domain Decomposition methods by Constraints (BDDC) and Dual-Primal Finite Element Tearing and Interconnecting (FETI-DP) algorithms. New results are presented for the spectral multi-element case and also for inexact FETI-DP methods for spectral elements in the plane. Theoretical convergence estimates show that these methods have a convergence rate independent of the number of subdomains and coefficient jumps of the elliptic operator, while there is only a polylogarithmic dependence on the spectral degree p and the ratio H/h   of subdomain and element sizes. Parallel numerical experiments on a Linux cluster confirm these results for tests with spectral degree up to p=32p=32, thousands of subdomains and coefficient jumps up to 8 orders of magnitude.