Abstract
在這篇論文中,我們要做對於有大的係數變動的二度空間對稱正定二次的 橢圓形問題的史密斯求解法之穩定性;而已知史密斯求解法在係數變動不 大時,收斂速率是很均勻的!但至今我們仍不知道如果係數變動很大時,收 斂情況會如何?從我們的數值結果來看,情況數並沒有總是均勻收斂的;我 們為會對於一般情況下所有分割的情況數的變化有兩個推測,而我們必須 聲明這和史瓦茲方法的推測不同! In this theis, we carry out some numerical experiments in two dimensionsabout Smith's vertex algorithm for elliptic finite element problem which comefrom the discretizations of second order, positive definite and symmetric elliptic partial equations with large jump coefficients. The correspondinglinear systems are solved by preconditioned conjugate gradient method with respect to some special inner product. It is well-known that smith's algorithm has a uniform convergence rate which is independent of all mesh-size parameters if the variation in coefficients of original differential equation is notlarge. However, if we allow very large variations in coefficients, its convergence behavior is not clear until now. From our numerical data, condition numbers are not always uniformlly bounded with respect to thesejumps in coefficients. We will establish some conjuctures about how its condition numbers varies with all mesh-size parameters in the general case.We remark that these conjectures are different from those about two levelSchwarz methods problems with large-jump coefficients.