弱条件下二阶椭圆问题求解的数值积分格式与收敛性分析.pdfVIP

第39卷第4期 郑 州大 学 学报(理 学版) Vol_39 No．4 2007年 12月 J．of Zhengzhou Univ．(Nat．Sci．Ed．) Dec．2007 A New Discrete Scheme to Solve Second Order Elliptic Problem with Numerical Integration and Convergence Analysis Under Weak Conditions SONG Shi—cang， CHEN Shao—chun (Department of Mathematics，Zhengzhou University，Zhengzhou 450001，China) Abstract：Compared with the traditional finite element method to solve second order elliptic problem，a new discrete scheme is established by using numerical integration．Under weaker conditions，some better convergence results are obtained，and two numerical examples are given． Key words：finite element；elliptic problem；numerical integration CLC number：O 241．8 Article ID：1671—6841(2007)04～O010—04 0 Intr0ducti0n Consider the second order elliptic problem △“一_，’in n r1、 1 “一O on an where n is a convex bounded polygonal domain in R ．The corresponding variational form is to find u∈ H (n)，such that a(“， )一厂( )，V v∈H (． ) (2) r r where以( ， )一I · vdxdy，_厂( )：I fvdxdy．Based on the regularity of differential equations[1]， VfEL (n)，the problem(2)has unique solution uffH。(n)and ll u ll 2，n≤C ll_厂ll ． For a quasi uniform triangulation to n，define shape function space V^一{v^l v^∈C。(n)，v^I加一0；VhIK∈Pl(K)，V K∈ )． The finite element method is to find u^∈V^，such that a(u^，v^)一厂(v^)，V v^∈V^ (3) Th