| The study of computational method has been puse into the forefront along with the development of computer.In fact.science compution has been the third method after experiment and theory, and it is driving the technological revolution with the speed that had never used in the past. In this revolution the computer and the computational method dependence each other.distribute each other.The advance of calculation capability depends on the development in these two aspects.The problems in science and engineer raised in practice.such as oil.the development and the prospect for precious deposits.....etc.Mathematic modes of themall belong to the large-scale differential equation.Hierachical basis method is a new method in age of 90 s.There are a broad developments in this aspect.In fact.steffen matrix is the Gram matrix made of basis function;.here pi is the node basis of trial function Sh. that is to say that every corresponds to a basis function ,satisfieshere n is the dimension of is called node basis.The matrix structured with the node basis is symmetrical and sparse.The character of node basis(l.l) decides that the diameter of support is 0(h)But the matrix made of node basis has obvious shortcoming:condition number is O(h2).and the calculation is difficult.Based on the way of Multi-Splitting of finite element space.H.Yserententbright forward the concept of Hierarchical basis.H.\serentent proved the condition number of steffen matrix that constructed by Hierarchical basis just is 0((log-)2) order in the case of two dimension.And he neither gives the imitate consistent assume of dissection.By means of the solution of linear equation system, only using several operations we can make the energy error less than f.Because of giving up the assume of imitate consistent.Hierachical basis is easer to deal with singular problems.The content mostly discussed in this papers is the alternate direction iterative method for solving the 2-dimention ellipic equations.Hierachical basis if fi is a polygon in plane.J/js a trangular for dissection fi.This means thatQis a set of triangular cell in J/,.where any two cells have the common side or common peak.or the intersection is an empty set.The finite element nodes are not classified.But in calculation the making of nodes is made in class.About the one demension problem,set ,is a gird dissection for .Giving the original dissection J0 in /.the nodes in ./o areThen set out from J0.build a set of nesting dissections here JA-+I is made of JA- in way of following:find the midpoint of /f in order to divide one cell into two.Let A~A. express the set of whole cell nodes in . express the function space of which every element is continuous.and is linear at every patch about in .For any Then defining the Hierachical basis by recursive.At first.the Hierachical basis in ,S'o is the node bases in this space:Secondly.let Hierachical basis in .S. has beendefined. Hierachical basis in ,s't and the node liases in .Compare \vith the coefficient matrix of node basis .4.the condition number of Hierachical basis matrix A is improved,and A is not sparsity enough and is not band.We recure to the (Transformation matrix G in finite element space ,S',.Due to the function of G.the matrix A and .4 satisfy the ralationwhere x,y are the arbitrary vectors in R",ThenIn this way.if node basis system isthe coefficients of node basis.then the Hierachical basis system is5 the coefficient matrix for Hierachical basis.is the coefficient of Hierachical basis.whereimproved Hierachical basis We improve the property of the matrix Hi-erachical basis by using a preconditioner P: letchange the Hierachical basis system to a preconditioned form as following :where .A" is three-diagonal matrix.Obviouslythe condition number of A is independent of h.letbe the coefficients of improved Hierachical basis.thenHierachical basis method for solving the 2-dimention ellipic equationsConsider the 2-dimention elliptic boundry value problem:where, a are continuous function on fi. |
