Towards Polyalgorithmic Linear System Solvers for Nonlinear Elliptic Problems

We investigate the performance of several preconditioned Conjugate-Gradient-like algorithms and a standard stationary iterative method (block-line-SOR) on linear systems of equations as they arise from a nonlinear elliptic ame sheet problem simulation. The nonlinearity forces a pseudo-transient continuation process that makes the problem parabolic and thus compacts the spectrum of the Jacobian matrix so that simple relaxation methods are viable in the initial stages of the solution process. However, because of the transition from parabolic to elliptic character as the time step is increased in pursuit of the steady-state solution, the performance of the candidate linear solvers spreads as the domain of convergence of Newton's method is approached. In numerical experiments over the course of a full nonlinear solution trajectory, short-recurrenceor optimal Krylov algorithms combined with a Gauss-Seidel preconditioning yield better execution times with respect to the standard block-line-SOR techniques, but SOR performs competitively at a smaller storage cost until the nal stages. Block-incomplete factorization preconditioned methods, on the other hand, require nearly a factor of two more storage than SOR and are uniformly less e ective during the pseudo-transient stages. The advantage of Gauss-Seidel preconditioning is partly attributable to the exploitation of a dominant convection direction in our examples; nevertheless, a multidomain version of Gauss-Seidel with streamwise coupling lagged at rows between adjacent subdomains incurs only a modest penalty.