2. Continuous Element for 1D Problems

  1. Pavel Šolín

Published Online: 30 NOV 2005

DOI: 10.1002/0471764108.ch2

Partial Differential Equations and the Finite Element Method

Partial Differential Equations and the Finite Element Method

How to Cite

Šolín, P. (2005) Continuous Element for 1D Problems, in Partial Differential Equations and the Finite Element Method, John Wiley & Sons, Inc., Hoboken, NJ, USA. doi: 10.1002/0471764108.ch2

Publication History

  1. Published Online: 30 NOV 2005
  2. Published Print: 4 NOV 2005

ISBN Information

Print ISBN: 9780471720706

Online ISBN: 9780471764106

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Summary

Chapter 2 presents an introduction to the finite element method in one spatial dimension. We begin with the Galerkin method and prove its convergence on the general level, using the Cea's lemma. The finite element Galerkin subspaces are constructed using piecewise-linear and then also piecewise-polynomial functions. Efficient higher-order Gaussian numerical quadrature rules, including adaptive quadrature, are discussed. Introduced are both the classical (nodal) concept of finite elements where the shape functions are associated with suitable (Gauss-Lobatto, Chebyshev) point sets, and the modern hierarchic concept where the shape functions are chosen with orthogonality structure to reduce the condition number of the stiffness matrix. Conditioning comparisons are presented. Data structures and assembling algorithms are presented. Discussed is the sparse structure of the stiffness matrix and its efficient representation. Explained is the implementation of various types of nonhomogeneous boundary conditions. Presented are several ways of interpolation on finite elements with different ratios of quality and computational cost.