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Finite Element Methods with Higher Order Polynomials

In: Exploring Mathematical Analysis, Approximation Theory, and Optimization

Author

Listed:
  • Konstantina C. Kyriakoudi

    (University of Ioannina)

  • Michail A. Xenos

    (University of Ioannina)

Abstract

The Finite Element Method (FEM) has recently been implemented in the fluid mechanics field to solve the instabilities that arise as a result of the equations’ non-linearities. For this reason, novel formulations of FEM were introduced, including the use of orthogonal polynomials and high-order polynomials. In this review, the focus rests on studying and analysing the aforementioned formulations and describing their improvements over the classical method. Initially, a theoretical background of FEM is introduced, with an emphasis on evaluating the basis of the function space. Additionally, the p-version of FEM is analysed, using Legendre polynomials. A comparison of the classical h-version and the p-version in terms of convergence. Moreover, other formulations that yield, using higher-order polynomials, such as hp-FEM and Spectral Element Method, are briefly reviewed. Finally, applications on FEM are presented, revealing the effects of the increase in the degree of the polynomials when solving a fluid mechanics problem.

Suggested Citation

  • Konstantina C. Kyriakoudi & Michail A. Xenos, 2023. "Finite Element Methods with Higher Order Polynomials," Springer Optimization and Its Applications, in: Nicholas J. Daras & Michael Th. Rassias & Nikolaos B. Zographopoulos (ed.), Exploring Mathematical Analysis, Approximation Theory, and Optimization, pages 161-176, Springer.
    • Handle: RePEc:spr:spochp:978-3-031-46487-4_10
    DOI: 10.1007/978-3-031-46487-4_10
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