The parallel AGE method for the elliptic problem in two dimensions (Q1180611)

For the Dirichlet problem in a rectangle for the equation \(-\Delta u=0\) the simplest difference scheme leads to the usual system \((\Lambda_ 1+\Lambda_ 2)u=f\) where the matrix \(-h^ 2\Lambda_ s\) is the approximation of \(\partial^ 2/\partial x^ 2_ s\). The author uses a representation \(\Lambda_ s=\Lambda_ s^{(1)}+\Lambda_ s^{(2)}\) where any symmetric matrix \(\Lambda_ s^{(k)}\) becomes block diagonal with blocks \(2\times 2\) and \(1\times 1\) under suitable numeration of the grid points. If \(r>0\) then \((rI+\Lambda_ s^ (k))^{-1}\) is easy to find and a system \((rI+\Lambda_ s^{(k)})v=g\) splits into separate independent subsystems. The author suggests to use an iterative method of ADI-type with 4 intermediate steps on each iteration to solve the above mentioned subsystems. The single iterative parameter \(r>0\) is obtained numerically. On grids \(12\times 12 - 48\times 48\) the number of iterations is \(9-25\).