分层基法对差分方程的应用

Application of Hierachical Basis Method to Defference Equations

设Ａｕ＝ｂ是二阶椭圆方程的差分逼近，熟知矩阵Ａ的条件数ｃｏｎｄ（Ａ）＝Ｏ（ｈ￣（－２））（ｈ→０）．将差分方程表为ＧＤＭ（广义差分法）形式，并利用分层基法将它化为等价方程Ｂｖ＝ｃ，使ｃｏｎｄ（Ｂ）＝Ｏ（（Ｉｇｈ￣（－１））￣２）．然后用某些迭比法（包括Ｒｉｃｈａｒｄｓｏｎ迭氏、共轭斜量法和Ｃｈｅｂｙｓｈｖ半迭代）解Ｂｖ＝ｃ。理论分析和数值试验证明有高敛速。

Let Au=b be the difference approximation to the second order elliptic equation. It is knownthat the conditional number of matrix A has the asymptotic relation cond(A)=O(h￣(-2))as h→0.Inthis paper,we express the difference schemes as the form of GDM(generalized difference methods,cf.[2] ),and transfer it into an equivalent system Bv=by the hierachical basis methods[1] so that thecond(B)equalstoO((Igh￣(-1))￣2).Finally,several iterated methods(including Richardson's iteration,conjugate gradient method and Chebyshev semi-iteration)are used to solve Bv=c,The higher conver-gence rate has been testified by the theoretical analysis and a numerical example.