##
**Finite elements and approximation.**
*(English)*
Zbl 0582.65068

A Wiley-Interscience Publication. New York: John Wiley & Sons, Inc. XIII, 328 p. (1983).

This book is designed as an undergraduate text in finite difference and finite element methods designed primarily for civil engineering and physics majors. However, it is probably one of the best introductory texts on the subject yet written for any audience.

Chapter 1 surveys the use of finite difference techniques to solve boundary value problems for ordinary and partial differential equations. Model problems are introduced which are used later to illustrate various approximation methods.

Chapter 2 introduces a finite linear combination \({\hat \phi}\) of M independent trial functions satisfying a completeness requirement to approximate the desired solution \(\phi\) in a region \(\Omega\) bounded by a closed curve \(\Gamma\) where the approximation is exact on \(\Gamma\). The constants \b{a} in the linear combination are determined by the weighted residual method which requires \(\int_{\Omega}W_ l(\phi -{\hat \phi})d\Omega =0,\quad l=1,2,...,\) for an independent set of weighting functions \(W_ l\). A linear system K\b{a}\(=f\) may be obtained by: (1) requiring satisfaction of the residual equation at M distinct points of \(\Gamma\) ; (2) requiring the integrated error to be zero on M subregions; (3) requiring the weight and trial functions to be the same, i.e., the Galerkin method. Examples are given of solutions of linear differential equations \(L\phi +p=0\), where L is a linear differential operator and p is independent of \(\phi\). The authors discuss the weak form of weighted residuals as well as boundary solution processes. A nonlinear ordinary differential equation is solved by iteration.

Chapter 3 surveys piecewise defined trial functions and the finite element method. Three physical analogues are used to illustrate the process of assembling the coefficient matrix from the individual finite elements. A two-dimensional heat conduction problem is solved using triangular and rectangular elements. Finite difference equations are derived using the finite element method.

In chapter 4, linear, quadratic, and cubic shape elements with \(C^ 0\) continuity are used to solve a boundary value problem for \(\phi ''=\phi\) on [0,1]. Hierarchical polynomials are introduced with nearly orthogonal form which lead to sparse coefficient matrices. Products of one- dimensional Lagrange polynomials give two-dimensional rectangular shape elements. Two-dimensional shape functions for triangles are derived.

Chapter 5 uses mapping to extend the variety of possible shape elements. To evaluate the elements of the coefficient matrix, Gaussian quadrature is considered for single, double, and triple integrals.

Chapter 6 derives a variational principle for a boundary value problem involving a symmetric, positive definite operator. The Rayleigh-Ritz method applied to the variational form is shown to lead to the same set of linear equations as the Galerkin method. Boundary conditions are treated as constraints on \(\phi\) by using Lagrange multipliers and penalty numbers.

Chapter 7 is concerned with partial discretization and time dependent problems. Applied to problems of the form \(L\phi +p-\alpha \partial \phi /\partial t-\beta \partial^ 2\phi /\partial t^ 2=0\), where L is a linear operator involving space derivatives and p,\(\alpha\),\(\beta\) are prescribed functions of position and time, this method can reduce the problem to the solution of a system of ordinary differential equations solvable by analytic theory. Solution of nonlinear problems is accomplished by using finite elements to represent the time domain with the conditions at the end of each element used as initial conditions for the text element. Stability characteristics of 2- and 3-level difference methods for solving systems of first and second order equations are derived.

In chapter 8 the energy norm is used to compare the error in several examples, including the finite element analysis of a gravity dam. Each chapter has suitable problems.

Chapter 1 surveys the use of finite difference techniques to solve boundary value problems for ordinary and partial differential equations. Model problems are introduced which are used later to illustrate various approximation methods.

Chapter 2 introduces a finite linear combination \({\hat \phi}\) of M independent trial functions satisfying a completeness requirement to approximate the desired solution \(\phi\) in a region \(\Omega\) bounded by a closed curve \(\Gamma\) where the approximation is exact on \(\Gamma\). The constants \b{a} in the linear combination are determined by the weighted residual method which requires \(\int_{\Omega}W_ l(\phi -{\hat \phi})d\Omega =0,\quad l=1,2,...,\) for an independent set of weighting functions \(W_ l\). A linear system K\b{a}\(=f\) may be obtained by: (1) requiring satisfaction of the residual equation at M distinct points of \(\Gamma\) ; (2) requiring the integrated error to be zero on M subregions; (3) requiring the weight and trial functions to be the same, i.e., the Galerkin method. Examples are given of solutions of linear differential equations \(L\phi +p=0\), where L is a linear differential operator and p is independent of \(\phi\). The authors discuss the weak form of weighted residuals as well as boundary solution processes. A nonlinear ordinary differential equation is solved by iteration.

Chapter 3 surveys piecewise defined trial functions and the finite element method. Three physical analogues are used to illustrate the process of assembling the coefficient matrix from the individual finite elements. A two-dimensional heat conduction problem is solved using triangular and rectangular elements. Finite difference equations are derived using the finite element method.

In chapter 4, linear, quadratic, and cubic shape elements with \(C^ 0\) continuity are used to solve a boundary value problem for \(\phi ''=\phi\) on [0,1]. Hierarchical polynomials are introduced with nearly orthogonal form which lead to sparse coefficient matrices. Products of one- dimensional Lagrange polynomials give two-dimensional rectangular shape elements. Two-dimensional shape functions for triangles are derived.

Chapter 5 uses mapping to extend the variety of possible shape elements. To evaluate the elements of the coefficient matrix, Gaussian quadrature is considered for single, double, and triple integrals.

Chapter 6 derives a variational principle for a boundary value problem involving a symmetric, positive definite operator. The Rayleigh-Ritz method applied to the variational form is shown to lead to the same set of linear equations as the Galerkin method. Boundary conditions are treated as constraints on \(\phi\) by using Lagrange multipliers and penalty numbers.

Chapter 7 is concerned with partial discretization and time dependent problems. Applied to problems of the form \(L\phi +p-\alpha \partial \phi /\partial t-\beta \partial^ 2\phi /\partial t^ 2=0\), where L is a linear operator involving space derivatives and p,\(\alpha\),\(\beta\) are prescribed functions of position and time, this method can reduce the problem to the solution of a system of ordinary differential equations solvable by analytic theory. Solution of nonlinear problems is accomplished by using finite elements to represent the time domain with the conditions at the end of each element used as initial conditions for the text element. Stability characteristics of 2- and 3-level difference methods for solving systems of first and second order equations are derived.

In chapter 8 the energy norm is used to compare the error in several examples, including the finite element analysis of a gravity dam. Each chapter has suitable problems.

### MSC:

65Nxx | Numerical methods for partial differential equations, boundary value problems |

74S05 | Finite element methods applied to problems in solid mechanics |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |

35J65 | Nonlinear boundary value problems for linear elliptic equations |

35K60 | Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations |

34B15 | Nonlinear boundary value problems for ordinary differential equations |