2. Continuous Element for 1D Problems
Published Online: 30 NOV 2005
Copyright © 2006 John Wiley & Sons, Inc. All rights reserved.
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
- Published Online: 30 NOV 2005
- Published Print: 4 NOV 2005
Print ISBN: 9780471720706
Online ISBN: 9780471764106
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.