With analysis of water balance in elements controlled by nodes,a stable computational scheme is proposed similar to FEM of Galerkin.It holds mass conservation automatically.A nonlinear phreatic water problem is solved conveniently by this scheme.Numerical calculation provides good result in solving a Theis problem.

In the essay, the finite element algebraic equations by element assembling is replaced by node assembling, and the right term is dealt with by the idea of lumped mass, and the most compactly-stored finite element method is obtained.Thus, the total matrix elements are reduced to less than three times of unknown node, and the stored-units are even less than which the rectangle grid finite-difference method needs. The program structure is more rational and simpler, meanwhile the calculation quantity is reduced and the computational effciency is improded. All these become more obvious in process of solving nonlinear problems.

Using the concepts of element order and order matrix, some practical problems are discussed in which the traditional preconditioning methods ICCG and MICCG are adopted. If the fill-in number is fixed, why the method of ICCG(s,t) becomes the most efficient when (s,t) is successively (1,1), (1,2), (1,3), (2, 4), (3, 5),..? Why the number of iterations didn't decrease when m is larger than 3 for MICCG(m)? Is it possible to improve the fill-in method of MICCG? Is it always true that MICCG is better than ICCG? It tries to give a preliminary discussion on these problems in here. From the way of high order approximate LU decomposition, a method is introduced which improves and systematizes the ICCG and MICCG. An estimation of the condition number of ICCG is given based on the discussion of the order matrix for the error matrix. It is also pointed out that there was a trouble in selecting the parameter for MICCG. A reasonable way to select the parameter is given. Thus the number of iterations of MICCG decreases when the order of MICCG increases.