Two-step Convergence Of The Multi-splitting Methods

In many fields of natural science and engineering science, we will encounter initial value and boundary value problems of differential equations, but only a very few simple equations whose analytical solution can be expressed.However since those practical problems are complicated differential equations, such as elliptic, parabolic and hyperbolic equations, we should find out the solution to these equations or the values of the function at the discrete points, we look for the numerical solution of the differential equations. When we seek the numerical solution of the elliptic boundary value by using the differential method for the solutions, it comes down to solving large sparse linear system finally.As we know, the methods of solving linear system include direct and iterative method, and the proportional of non-zero elements of the coefficient matrix generated by the large sparse linear system is small and there is great regularity in the distribution among them. The iterative method is not only easy to be carried out, but also saves the computer memory storage, so the iterative method is an important way to solve the differential equations. As for a large sparse matrix, the convergence rate of the selected iterative methods to solve linear system is extremely important, and there is only the convergent iterative methods work for the practical purpose. In this paper we study the convergence condition of the two-stage multisplitting method which is now discussed ardently.In reference[1], Jae Heon Yun considered the convergence of two-stage multisplitting method using H-compatible splitting as outer splitting and AOR splitting or SSOR splitting as inner splitting, and he discussed the convergence conditions of this method. In this paper, firstly we present the definition of the TOR multisplitting method who is more general than that of the AOR multisplitting method, secondly we discuss the convergence of the TOR multisplitting method and give the corresponding theoretic proof. Thirdly we prove that the AOR multisplitting is the special case of the TOR multisplitting, so we get the convergence theorem of the AOR multisplitting which discussed in reference [1], and we deduce a lot of new corollaries. Further more, in reference [1] Jae Heon Yun discussed convergence of the two-stage AOR multisplitting under the conditions of 0 <γ≤ωand 0 <ω<2(1+α)(ωis the relaxing parameter of iterative method,γis the accelerating parameter), however here we discuss the convergence of this method under the 0 <ω≤γcondition,and find out the precondition of the convergence.We not only extend the AOR multisplitting method to TOR multisplitting but also spread the condition 0 <γ≤ωto 0 <ω≤γ, so we enlarge the application scope of the two-stage multisplitting method, and there is a certain reference and practical value for the scholars or researchers who are engaged in the numerical calculations. It is of great significance to discuss the improvement and development of the existing conclusions to the convergence of two-stage multisplitting method.