Simple efficient solvers for certain ill-conditioned systems of linear equations, including H(div) problems

Simple, accurate, and efficient iterative methods for the solution of a nearly singular variational problem are described. The systems considered arise, e.g., when seeking to determine the flux of second order elliptic partial differential equations. Each (outer) iteration uses a robust and efficient solver, which may be a direct or iterative method, and only a few (outer) iterations are required to obtain an approximate solution. Reduced rank extrapolation may be applied to speed up the convergence.