Numerical Verification for Elliptic Boundary Value Problem with Nonconforming \(\mathcal {P}_1\) Finite Elements

详细信息    查看全文

• 关键词：Nakao’s method ; Numerical verification ; Elliptic boundary value problem ; Nonconforming \(\mathcal {P}_1\) ; finite element
• 刊名：Lecture Notes in Computer Science
• 出版年：2016
• 出版时间：2016
• 年：2016
• 卷：9553
• 期：1
• 页码：269-279
• 全文大小：274 KB
• 参考文献：1.Babuška, I., Suri, M.: On locking and robustness in the finite element method. SIAM J. Numer. Anal. 29(5), 1261–1293 (1992)MathSciNet CrossRef MATH
  2.Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Society for Industrial and Applied Mathematics, Philadelphia (2002)CrossRef MATH
  3.Dupont, T., Scott, L.: Polynomial approximation of functionals in sobolev spaces. Math. Comput. 34, 441–463 (1979)MathSciNet CrossRef MATH
  4.Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Classics in Applied Mathematics. SIAM, Boston (1985)MATH
  5.Lee, C.O., Lee, J., Sheen, D.: A locking-free nonconforming finite element method for planar linear elasticity. Adv. Comput. Math. 19, 277–291 (2003)MathSciNet CrossRef MATH
  6.Liu, X.: Analysis of error constants for linear conforming and nonconforming finite elements. Ph.D. thesis, University of Tokyo, March 2009
  7.Nakao, M.T.: A numerical approach to the proof of existence of solutions for elliptic problems. Jpn J. Appl. Math. 5(2), 313–332 (1988)MathSciNet CrossRef MATH
  
• 作者单位：Tomoki Uda (16)
  
  16. Department of Mathematics, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto, 606-8502, Japan
  
• 丛书名：Scientific Computing, Computer Arithmetic, and Validated Numerics
• ISBN：978-3-319-31769-4
• 刊物类别：Computer Science
• 刊物主题：Artificial Intelligence and Robotics
  Computer Communication Networks
  Software Engineering
  Data Encryption
  Database Management
  Computation by Abstract Devices
  Algorithm Analysis and Problem Complexity
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1611-3349

文摘

We propose a numerical method with the nonconforming \(\mathcal {P}_1\) FEM to verify the existence of solutions to an elliptic boundary value problem. Formulating the boundary value problem as a fixed-point problem on the sum space of the nonconforming \(\mathcal {P}_1\) finite element space with the Sobolev space of 1st order with zero Dirichlet condition, we construct the numerical verification method based on the Schauder fixed-point theorem. We show a constructive inequality for a boundary integral that appears due to the discontinuity of a nonconforming \(\mathcal {P}_1\) finite element function. Finally, we present a numerical example to show our proposed method works well.