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二阶椭圆问题的弱迦辽金四边形谱元方法

潘佳佳1,2, 李会元1   

  1. 1. 中国科学院软件研究所, 北京 100190;
    2. 中国科学院大学, 北京 100190
  • 收稿日期:2020-03-24 出版日期:2021-12-15 发布日期:2021-12-07
  • 基金资助:
    国家自然科学基金(No.11871455,No.11971016)资助.

潘佳佳, 李会元. 二阶椭圆问题的弱迦辽金四边形谱元方法[J]. 数值计算与计算机应用, 2021, 42(4): 303-322.

Pan Jiajia, Li Huiyuan. WEAK GALERKIN QUADRILATERAL SPECTRAL ELEMENT METHODS FOR SECOND ORDER ELLIPTIC EQUATIONS[J]. Journal on Numerica Methods and Computer Applications, 2021, 42(4): 303-322.

WEAK GALERKIN QUADRILATERAL SPECTRAL ELEMENT METHODS FOR SECOND ORDER ELLIPTIC EQUATIONS

Pan Jiajia1,2, Li Huiyuan1   

  1. 1. Institute of Software, Chinese Academy of Science, Beijing 100190, China;
    2. University of Chinese Academy of Science, Beijing 100190, China
  • Received:2020-03-24 Online:2021-12-15 Published:2021-12-07
本文对二阶椭圆方程特征值问题的弱伽辽金谱元方法开展相关数值研究.与弱有限元方法类似,弱伽辽金谱元方法的逼近函数空间包括各个单元上的独立内部分量、并辅以各单元边界分量作为单元与单元间的联系.本文聚焦任意凸四边形网格剖分下的弱伽辽金四边形谱元方法,弱逼近函数中的各内部分量与边界分量分别由参考正方形单元与参考单元边界上的正交多项式通过双线性变换来构造;而弱梯度逼近空间则由参考正方形上的正交多项式通过Piola变换构造.在此基础上,本文提出了二阶椭圆方程特征值问题的弱伽辽金四边形谱元方法逼近格式和实现算法,并通过对离散弱梯度核空间的系统研究,具体分析了逼近格式的适定性.通过大量的数值实验,本文具体分析了弱伽辽金四边形谱元方法的精度和收敛性,特别是逼近函数空间与离散弱梯度空间中多项式次数的不同搭配对精度和收敛性的影响.研究表明,p-型弱伽辽金四边形谱元方法承袭了谱方法的指数阶收敛性质;h-型弱伽辽金四边形谱元方法不但具有h-型方法在通常意义上的满阶收敛性,而且完全可以通过逼近空间多项式次数的灵活匹配达到超收敛.
Numerical studies on the weak Galerkin spectral element method for eigenvalue problems of second order elliptic equations are carried out in this paper. In analogy to weak Galerkin finite element methods, the approximation space in a weak Galerkin spectral element method contains independently interior components on subdomains, and boundary components at subdomain interfaces are supplemented to ensure information interchange between subdomains. The weak Galerkin quadrilateral spectral element method with arbitrarily convex quadrilateral meshes is the focus of this paper. The interior and boundary components of the weak approximation function on each subdomain are constructed by orthogonal polynomials on the reference square and its edges via the bilinear mapping, respectively. While approximation spaces for weak gradients on each subdomain are established by orthogonal polynomials on the reference square via the Piola transform. In such circumstances, a weak Galerkin quadrilateral spectral element approximation scheme together with its implementation algorithm is then proposed for eigenvalue problems of second order elliptic equations, and the wellposedness of the approximation scheme is analyzed in detail after a systematic investigation of nullities of discrete weak gradients. Abundant numerical experiments are performed to elaborately analyze the accuracy and convergence of the weak Galerkin quadrilateral spectral element method, especially the impact of the diverse match of polynomial degrees in the approximation function space and the discrete weak gradient space upon the accuracy and convergence. Numerical results show that the p-version weak Galerkin quadrilateral spectral element method inherits the exponential orders of convergence of spectral methods, while the h-version weak Galerkin quadrilateral spectral element method not only possesses full-order convergence of h-version methods but also achieves a super convergence by properly matching polynomial degrees in the approximation spaces.

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