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非线性抛物方程混合有限元方法的高精度分析

王俊俊1, 李庆富1, 石东洋2   

  1. 1. 平顶山学院 数学与统计学院, 平顶山 467000;
    2. 郑州大学 数学与统计学院, 郑州, 450001
  • 收稿日期:2017-11-03 出版日期:2019-06-15 发布日期:2019-05-18
  • 基金资助:

    国家自然科学基金(11271340),平顶山学院博士启动基金(PXY-BSQD-2019001),平顶山学院培育基金(PXY-PYJJ-2019006).

王俊俊, 李庆富, 石东洋. 非线性抛物方程混合有限元方法的高精度分析[J]. 计算数学, 2019, 41(2): 191-211.

Wang Junjun, Li Qingfu, Shi Dongyang. SUPERCONVERGENCE ANALYSIS OF A MIXED FINITE ELEMENT METHOD FOR NONLINEAR PARABOLIC EQUATION[J]. Mathematica Numerica Sinica, 2019, 41(2): 191-211.

SUPERCONVERGENCE ANALYSIS OF A MIXED FINITE ELEMENT METHOD FOR NONLINEAR PARABOLIC EQUATION

Wang Junjun1, Li Qingfu1, Shi Dongyang2   

  1. 1. School of Mathematics and Statistics, Pingdingshan University, Pingdingshan 467000, China;
    2. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, China
  • Received:2017-11-03 Online:2019-06-15 Published:2019-05-18
采用双线性元及零阶Raviart-Thomas元(Q11+Q10×Q01)对非线性抛物方程讨论了一种H1-Galerkin混合有限元方法.提出一个线性化的二阶格式,利用数学归纳法有技巧的导出了原始变量uH1(Ω)模意义下及流量p=▽uL2(Ω)模意义下的Oh2+τ2)阶超逼近性质.引入一个有关初始点的时间离散方程,并利用其得到了▽ ·在L2(Ω)模意义下的Oh2+τ2)阶的超逼近结果.同时利用插值后处理技巧得到整体超收敛.最后,数值算例结果验证了理论分析(其中,h是剖分参数,τ是时间步长).
An H1-Galerkin mixed finite element method is discussed for nonlinear parabolic equations with the bilinear element and the zero-order Raviart-Thomas element (Q11+Q10×Q01). A linearized second order fully-discrete scheme is proposed. The superclose results with O(h2 + τ2) of original variant u in H1-norm and flux variant p in L2-norm are derived technically. A time semi-discrete equation at the starting point is introduced and the superclose property of ▽·p in L2-norm is reduced. Furthermore, the corresponding global superconvergence results are obtained by the interpolated postprocessing technique. At last, numerical results are presented to illustrate the feasibility of the proposed method (Here, h is the subdivision parameter, and τ, the time step).
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