Coarse spaces for FETI-DP and BDDC Methods for heterogeneous problems: connections of deflation and a generalized transformation-of-basis approach. ETNA - Electronic Transactions on Numerical Analysis

In FETI-DP (Finite Element Tearing and Interconnecting) and BDDC (Balancing Domain Decomposition by Constraints) domain decomposition methods, the convergence behavior of the iterative scheme can be improved by implementing a coarse space using a transformation of basis and local assembly. This is an alternative to coarse spaces implemented by deflation or balancing. The transformation-of-basis approaches are more robust with respect to inexact solvers than deflation and therefore more suitable for multilevel extensions. In this paper, we show a correspondence of FETI-DP or BDDC methods using a generalized transformation-of-basis approach and of FETI-DP methods using deflation or balancing, where the deflation vectors are obtained from the transformation of basis. These methods then have essentially the same eigenvalues. As opposed to existing theory, this result also applies to general scalings and highly heterogeneous problems. We note that the new methods differ slightly from the classic FETI-DP and BDDC methods using a transformation of basis and that the classic theory has to be replaced. An important application for the theory presented in this paper are FETI-DP and BDDC methods with adaptive coarse spaces, i.e., where deflation vectors are obtained from approximating local eigenvectors. These methods have recently gained considerable interest