Iterative alternating direction methods

The alternating direction method for solving the difference Dirichlet problem in a rectangle. So far we have considered the Dirichlet problem for Poisson s equation in a rectangle G = (0 / , a = 1, 2)  

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The numerical solution of problem (1) by means of iteration schemes can be done using the alternating direction scheme for the heat conduction 

The transition from the ith iteration to the (j + l)th iteration can be done by the elimination method along the rows as well as along the columns for the following three-point equations  

the users must perform 0(l/(/ j h )) arithmetic operations in calculating one iteration or 0(1) arithmetic operations at every node of the grid 

By analogy with the nonstationary heat conduction equation the iterative process (2) is referred to as the alternating direction method 

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In the two-dimensional case the iterative alternating direction method or the direct decomposition method turns out to be more economical, but for the multidimensional Dirichlet problem ATM is the most economical one among other available methods. This advantage is stipulated by the special structure of the operator A (see Chapter 4, Section 5) ... 

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