Superlinearly convergent PCG algorithms for some nonsymmetric elliptic systems

The conjugate gradient method is a widespread way of solving nonsymmetric linear algebraic systems, in particular for large systems arising from discretized elliptic problems. A celebrated property of the CGM is superlinear convergence, see the book [2] where a comprehensive summary is given on the convergence of the CGM. For discretized elliptic problems, the CGM is mostly used with suitable preconditioning [2], which often provides mesh independent convergence. Moreover, it has been shown in [6] that the preconditioned CGM can be competitive with multigrid methods. The mesh independence property is a basic reason to involve underlying Hilbert space theory in the study of the CGM. Linear convergence results for such PCG methods are treated in the rigorously described framework of equivalent operators in Hilbert space [6, 14], which provides mesh independence for the condition numbers of the discretized problems. The CGM for nonsymmetric equations in Hilbert space has been studied in the author’s papers [3, 4]: in the latter superlinear convergence has been proved in Hilbert space and, based on this, mesh independence of the superlinear estimate has been derived for FEM discretizations of elliptic Dirichlet problems. The numerical realization of this method has been demonstrated in [12]. The goal of this paper is to extend the mesh independent superlinear convergence results of [4] from a single equation to systems. An important advantage of the obtained preconditioning method for systems is that one can define decoupled preconditioners, hence the size of the auxiliary systems remains as small as for a single equation, moreover, parallelization of the auxiliary systems is available.