Tensor preconditioners in decomposition methods

One of the most effective approach in solution of mesh and finite-element SLAEs Au = f, arrising in approximation of two-dimensional (or multi-dimensional) problems is the decomposition method. The essense of the method consists in special choice of easy-invertible linear transformation H and in a successive realization of iterating process uk+1 = uk – H-1(Auk – f). The transformation H is often taken from the reasons, reflecting special properties of the initial problem, which is approximated by SLAE. From the point  of linear algebra these reasons may be expressed with the help of the following example. Let us approximate the matrix A by the product H = A1·A2 of matrices  having  more  simple structure (upper- and lower band). The inversion of the product matrix H on an arbitrary vector is done easily with the help of the sweep Cholessky method. It causes high efficiency of iterative process. Matrix H can also be used as a preconditioner in the method of conjugate gradients, this additionally speeds up the solution of the problem.