Multigrid method as an accelerating procedure for solving systems of linear algebraic equations with a dissipative matrix (Q1128392)

Iterative methods are the main tool for solving large systems of linear algebraic equations (SLAE). These methods have a number of clear advantages compared with direct elimination methods, including ease of implementation on parallel or vector computers. Several simple basic iterative methods are known. All other iterative methods can be obtained from these basic methods by improvements of a certain kind (acceleration and (or) preconditioning). For progressively more complex problems, the basic iterative methods become inefficient or even inapplicable. It is therefore important to extend the applicability and to improve the efficiency of iterative methods for the solution of SLAE, and especially to reduce the number of iterations needed to achieve the required accuracy. Here two approaches are possible: 1) changing the original SLAE by preconditioning; 2) changing the iterative method by acceleration. In real-life problems, the two approaches are usually combined: first the system is preconditioned, and then the iterative method is accelerated. Several approaches to acceleration of iterative methods are available. These include the variational, the polynomial, the hybrid, and the multigrid approach. We consider the acceleration of the iterative method by a multigrid procedure.