An Extended-matrix Preconditioner for Nonself-adjoint Nonseparable Elliptic Equations∗

In a previous paper we proposed a preconditioning technique that generalizes an idea by M. Griebel. In this generalization, although all considerations are presented in a very general algebraic framework, the main idea behind them is to use as preconditioner a rectangular matrix constructed with the transfer operators between successive discretization levels of the initial problem. In this way we get an extended linear system, which is no more invertible, but still consistent such that any of its solutions generates the unique one of the initial system. For an appropriate choice of the preconditioner, this extended system is mesh-independent well conditioned, thus many classes of iterative solvers can be successfuly used. We prove this in the present paper for picewise linear finite element discretization of two types of nonself-adjoint nonseparable elliptic boundary value problems. Numerical experiments on 1D and 2D versions of the considered problems are also presented for CGN and Kaczmarz iterations.