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Towards polyalgorithmic linear system solvers for nonlinear elliptic problems. (English) Zbl 0807.65027

The authors compare the performance of several iterative methods for the solution of nonsymmetric systems of linear equations on a problem that arises in a nonlinear elliptic axisymmetric flame sheet simulation. The linear solvers considered are the classical block-line successive overrelaxation (SOR) and two Krylov space based methods, generalized minimal residual (GMRES) and biconjugate gradient stabilized (Bi-CGSTAB), each with the set of three preconditioners, namely block-line Gauss- Seidel (GS), symmetric block-line GS (SGS), and a block incomplete LU decomposition (ILU). The blocks are induced by splitting the unknowns to groups by nodes.
The systems of linear equations are solved to evaluate the action of the inverse of the Jacobian that arises in the Newton iterations that treat the nonlinearity. A pseudotransient formulation is used to produce a parabolic-in-time problem. Results of systematic numerical experiments show that most efficient strategy exploits different solvers at different stages of a full nonlinear solution trajectory. Apart from numerical experiments, the authors collect relevant theoretical results to give yet another guide to assess the performance of the methods considered and to propose their optimal combination.

MSC:

65F10 Iterative numerical methods for linear systems
65F35 Numerical computation of matrix norms, conditioning, scaling
65H10 Numerical computation of solutions to systems of equations
65Y20 Complexity and performance of numerical algorithms
35J65 Nonlinear boundary value problems for linear elliptic equations
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
80A32 Chemically reacting flows

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