TOWARDS POLYALGORITHMIC LINEAR-SYSTEM SOLVERS FOR NONLINEAR ELLIPTIC PROBLEMS

The authors investigate the performance of several preconditioned conj ugate gradient-like algorithms and a standard stationary iterative met hod (block-line successive overrelaxation (SOR)) on linear systems of equations that arise from a nonlinear elliptic flame sheet problem sim ulation. The nonlinearity forces a pseudotransient continuation proces s that makes the problem parabolic and thus compacts the spectrum of t he Jacobian matrix so that simple relaxation methods are viable in the initial stages of the solution process. However, because of the trans ition from parabolic to elliptic character as the timestep is increase d in pursuit of the steady-state solution, the performance of the cand idate linear solvers spreads as the domain of convergence of Newton's method is approached. In numerical experiments over the course of a fu ll nonlinear solution trajectory, short recurrence or optimal Krylov a lgorithms combined with a Gauss-Seidel (GS) preconditioning yield bett er execution times with respect to the standard block-line SOR techniq ues, but SOR performs competitively at a smaller storage cost until th e final stages. Block-incomplete factorization preconditioned methods, on the other hand, require nearly a factor of two more storage than S OR and are uniformly less effective during the pseudotransient stages. The advantage of GS preconditioning is partly attributable to die exp loitation of a dominant convection direction in the examples; neverthe less, a multidomain version of GS with streamwise coupling lagged at r ows between adjacent subdomains incurs only a modest penalty.