TOWARDS POLYALGORITHMIC LINEAR-SYSTEM SOLVERS FOR NONLINEAR ELLIPTIC PROBLEMS

The authors investigate the performance of several preconditioned conjugate gradient-like algorithms and a standard stationary iterative method (block-line successive overrelaxation (SOR)) on linear systems of equations that arise from a nonlinear elliptic flame sheet problem simulation. The nonlinearity forces a pseudotransient 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 timestep 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 recurrence or optimal Krylov algorithms combined with a Gauss-Seidel (GS) 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 final stages. Block-incomplete factorization preconditioned methods, on the other hand, require nearly a factor of two more storage than SOR and are uniformly less effective during the pseudotransient stages. The advantage of GS preconditioning is partly attributable to die exploitation of a dominant convection direction in the examples; nevertheless, a multidomain version of GS with streamwise coupling lagged at rows between adjacent subdomains incurs only a modest penalty.