Description
After the discretization of a nonsymmetric elliptic problem, the resulting system of linear equations has to be solved to obtain the numerical solution. To achieve that, several iterative methods can be used, but their performance may highly depend on the coefficients of the PDE. In this talk, we consider the Conjugate Gradient method applied to the normal equation (CGN) and the Generalized Conjugate Residual method (GCR),
and we compare their rate of convergence both numerically and theoretically using the linear and superlinear convergence estimates, which can in fact describe the convergence quite accurately. The numerical experiments suggest that for our special class of problems, depending on the coefficients of the PDE, we can forecast (or at least explain) which method will converge faster. Apart from this comparison, the estimates' continuous dependence on the domain will be illustrated as well, which suggests a similar continuous dependence of the residual norms.
I will also present a recent result about an improved superlinear convergence estimate.