Preconditioning by Fast Direct Methods for Nonself-Adjoint Nonseparable Elliptic Equations

We consider the use of fast direct methods as preconditioners for iterative methods for computing the numerical solution of nonself-adjoint elliptic boundary value problems. We derive bounds on convergence rates that are independent of discretization mesh size. For two-dimensional problems on rectangular domains, discretized on an n × n grid, these bounds lead to asymptotic operation counts of O(n2 log n log ε-1) to achieve relative error ε and O(n2(log n)2) to reach truncation error.