On the efficiency of combining different methods for acceleration of iterations at the solution of PDEs by the method of collocations and least residuals

The preconditioner, multigrid algorithm, and the Krylov method are applied for accelerating the iteration process of solving the Navier–Stokes equations by the method of collocations and least residuals (CLR). These methods have been used simultaneously in their combination and separately. Their capabilities and efficiency have been verified by a considerable number of numerical experiments. To find the parameters of a preconditioner proposed in the work a relatively simple problem of minimizing the condition number of the system of linear algebraic equations to the solution of which the solution of the Navier–Stokes equation is reduced is solved. The original criterion of the degeneration degree of the Krylov subspace basis enables an automatic reduction of the subspace basis without a computer code restart in the region of small residuals of the PDE solution thereby improving the stability of iteration process in the above region. A combined simultaneous application of all three techniques for accelerating the iterative process of solving the boundary-value problems for two-dimensional Navier–Stokes equations has reduced the CPU time of their solution on computer by the factors up to 160 as compared to the case when none of them was applied. The proposed combination of the techniques for speeding up the iteration processes may be implemented also at the use of other numerical methods for solving the PDEs.