On Combining Different Acceleration Techniques at the Iterative Solution of PDEs by the Method of Collocations and Least Residuals

Abstract

In the work, we consider the problem of accelerating the iteration process of the numerical solution of boundary-value problems for partial differential equations (PDE) by the method of collocations and least residuals (CLR). To solve this problem, it is proposed to combine simultaneously three techniques of the iteration process acceleration: the preconditioner, the multigrid algorithm, and the correction of the PDE solution at the intermediate iterations in the Krylov subspace. The influence of all three techniques of the iteration acceleration was investigated both individually for each technique and at their combination. Each of the above techniques is shown to make its contribution to the quantitative figure of iteration process speed-up. The algorithm which employs the Krylov subspaces makes the most significant contribution. The joint simultaneous application of all three techniques for accelerating the iterative solution of specific boundary-value problems enabled a reduction of the CPU time of their solution on computer by a factor of up to 230 in comparison with the case when no acceleration techniques were applied. A two-parameter preconditioner was investigated. It is proposed to find the optimal values of its parameters by the numerical solution of a computationally inexpensive problem of minimizing the condition number of the system of linear algebraic equations. The problem is solved by the CLR method and it is modified by the preconditioner. It is shown that it is sufficient to restrict oneself in the multigrid version of the CLR method only to a simple solution prolongation operation on the multigrid complex to reduce substantially the CPU time of the boundary-value problem solution.Numerous computational examples are presented, which demonstrate the efficiency of the approaches proposed for accelerating the iterative processes of the numerical solution of the boundary-value problems for the two-dimensional Navier–Stokes equations. It is pointed out that the proposed combination of the techniques for accelerating the iteration processes may be also implemented within the framework of other numerical techniques for the solution of PDEs.

Keywords

• preconditioning
• krylov subspaces
• multigrid
• gauss–seidel iterations
• navier–stokes equations
• the method of collocations and least residuals