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半线性椭圆问题Petrov-Galerkin逼近及亏量迭代

司红颖1, 陈绍春2   

  1. 1. 商丘师范学院数学系, 河南商丘 476000;
    2. 郑州大学数学系, 郑州 450052
  • 收稿日期:2013-11-15 出版日期:2014-08-15 发布日期:2014-09-28
  • 基金资助:

    国家自然科学基金(No.11071226);河南省自然科学基金(No.132300410272).

司红颖, 陈绍春. 半线性椭圆问题Petrov-Galerkin逼近及亏量迭代[J]. 计算数学, 2014, 36(3): 316-324.

Si Hongying, Chen Shaochun. PETROV-GALERKIN APPROXIMATION OF THE SEMI-LINEAR ELLIPTIC AND THE DEFECT ITERATION[J]. Mathematica Numerica Sinica, 2014, 36(3): 316-324.

PETROV-GALERKIN APPROXIMATION OF THE SEMI-LINEAR ELLIPTIC AND THE DEFECT ITERATION

Si Hongying1, Chen Shaochun2   

  1. 1. Department of Mathematics, Shangqiu Normal University, Shangqiu 476000, Henan, China;
    2. Department of Mathematics, Zhengzhou University, Zhengzhou 450052, China
  • Received:2013-11-15 Online:2014-08-15 Published:2014-09-28
本文考虑了二阶半线性椭圆问题的Petrov-Galerkin逼近格式, 用双二次多项式空间作为形 函数空间,用双线性多项式空间作为试探函数空间, 证明了此逼近格式与标准的二次有限元逼近格式 有同样的收敛阶. 并且根据插值算子的逼近性质, 进一步证明了半线性有限元解的亏量迭代序列收敛到 Petrov-Galerkin解.
In this paper we introduce a Petrov-Galerkin approximation model to semi-linear elliptic boundary value problems in which biquadratic polynomial space and bilinear polynomial space are used as the shape function space and the test function space, respectively. We prove that the approximation order of the standard quadratic finite element can be abtained by this Petrov-Galerkin model. Based on the so-called "contractivity" of the interpolation operator, we further prove that the defect iterative sequence of the semi-linear finite element solution converge to the proposed Petrov-Galerkin approximate solution.

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[2] 张新建. W_2~m空间中样条插值算子与线性泛函的最佳逼近[J]. 计算数学, 2002, 24(2): 129-136.
[3] 张新建,黄建华. W_2~m空间中样条插值算子与最佳逼近算子的一致性[J]. 计算数学, 2001, 23(4): 385-392.
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