Optimized interface preconditioners for the FETI method

In the past two decades, the FETI method introduced in [10] and its variants have become a class of popular methods for the parallel solution of large-scale finite element problems, see e.g. [11], [9], [14], [15], [8]. A key ingredient in this class of methods is a good preconditioner for the dual Schur complement system whose operator is a weighted sum of subdomain Neumann to Dirichlet (NtD) maps. One choice is the so-called Dirichlet preconditioner, which is the primal Schur complement, i.e. a weighted sum of subdomain Dirichlet to Neumann (DtN) maps. The Dirichlet preconditioner is quasi-optimal in the sense that together with an appropriate coarse space, it leads to a polylogarithmic condition number in H/h, see e.g. [14]. However, in terms of total CPU time, often a cheaper alternative called the lumped preconditioner performs better [11, 8]. We show here that the lumped preconditioner can be further improved by introducing parameters into the tangential interface operator and optimizing them to get condition numbers as small as possible while keeping the cost of the preconditioner low. Since these preconditioners, like the lumped preconditioner, only involve computations along the interface, and no computations in the interior of subdomains, we call them interface preconditioners. We consider the model problem { −uxx−uyy = f , (x,y) ∈ R2 lim(x,y)→∞ u = 0,