Polynomial Approximation of Differential Equations

312 Pages·1992·11.35 MB·English

by  Daniele Funaro (auth.)

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Daniele Funaro
Polynomial Approximation
of Differential Equations
Springer-Verlag
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Author
DanieleFunaro
Dipartimentodi Matematica, Universitadegli Studi
StradaNuova,65,1-27100Pavia, Italy
ISBN 3-540-55230-8Springer-Verlag Berlin Heidelberg New York
ISBN0-387-55230-8Springer-Verlag NewYork Berlin Heidelberg
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PREFACE
This book is devoted to the analysis ofapproximate solution techniques for differential
equations, based on classical orthogonal polynomials. These techniques are popularly
known as spectral methods. In the last few decades, there has been a growing interest
in this subject. As a matter offact, spectral methods provide a competitive alternative
to other standard approximation techniques, for a large variety ofproblems. Initial ap
plications were concerned with the investigationofperiodic solutions ofboundary value
problems using trigonometric polynomials. Subsequently, the analysis was extended to
algebraic polynomials. Expansions in orthogonal basis functions were preferred, due to
their high accuracy and flexibility in computations.
The aim ofthis book is to present a preliminarymathematical backgroundfor be
ginners who wish to study and perform numericalexperiments, or who wish to improve
their skill in order to tackle more specific applications. In addition, it furnishes a com
prehensivecollectionofbasicformulas and theorems that are usefulfor implementations
at any level of complexity. We tried to maintain an elementary exposition so that no
experience in functional analysis is required.
The book is divided into thirteen chapters. In the first chapter, the reader canfind
definitions and properties relative to different families of polynomials. Orthogonality,
zeroes, Gaussian quadratureformulas, basis transformations, and manyother algebraic
relationsarestudiedinchapters 2, 3, 4. With the helpofthe short introductiontofunc
tionalanalysisgiveninchapter5, asurveyofresultsinapproximationtheory isprovided
in chapter 6. Chapter 7 is devoted to the study of the discretization of derivative op-
VI Polynomial Approximation ofDifferential Equations
erators. The eigenvalues ofthe corresponding matrices are analyzed in chapter 8. The
study of the differential equations begins in chapter 9. Ordinary differential equations
and time-dependent partial differential equations in one space variable are examined
in chapters 9 and 10 respectively. Domain-decomposition methods and related solu
tion techniques are considered in chapter 11. Numerical experiments on four upgraded
model problems in one dimension are presented in chapter 12. Finally, suggestions for
the treatment of problems in two variables are presented in the last chapter.
Acknowledgments
Theauthorisgratefulfor thehelpfulscientificsupportofClaudioCanutoandhiscritical
revision of the entire manuscript. Useful comments and suggestions were provided by
David Gottlieb for chapter 10 and Alfio Quarteroni for chapter 11.
Special thanks are due to the editors, YousuffHussaini and Gordon Erlebacher,for
the uncommoneffort ofmaking the exposition morefluent.
Final thanks go to the Istituto di Analisi Numerica del C.N.a., Pavia, for funding
this research in part.
Daniele Funaro
Pavia, December 1991
CONTENTS
1. Special families ofpolynomials
1.1 Sturm-Liouville problems. 1
1.2 The Gammafunction 2
1.3 Jacobi polynomials 4
1.4 Legendre polynomials 7
1.5 Chebyshev polynomials 9
1.6 Laguerre polynomials 12
1.7 Hermite polynomials 16
2. Orthogonality
2.1 Inner products and norms 21
2.2 Orthogonal functions 23
2.3 Fourier coefficients 27
2.4 The projection operator 30
2.5 The maximum norm 31
2.6 Basis transformations 32
3. Numerical integration
3.1 Zeroes oforthogonal polynomials 35
3.2 Lagrange polynomials 42
3.3 The interpolation operators 46
VIII Polynomial Approximation ofDifferential Equations
3.4 Gauss integration formulas . . . . 47
3.5 Gauss-Lobatto integration formulas 51
3.6 Gauss-Radau integration formulas 54
3.7 Clenshaw-Curtis integrationformulas 55
3.8 Discrete norms . . . . . 56
3.9 Discrete maximum norms 60
3.10 Scaled weights 62
4. Transforms
4.1 Fourier transforms 65
4.2 Aliasing . . . . . 68
4.3 Fast Fourier transform 70
4.4 Other fast methods 76
5. Functional spaces
5.1 The Lebesgue integral . . . . 77
5.2 Spaces ofmeasurablefunctions 81
5.3 Completeness 82
5.4 Weak derivatives 84
5.5 Transformation ofmeasurable functions in R 85
5.6 Sobolev spaces in R . . . 87
5.7 Sobolev spaces in intervals 90
6. Results in approximation theory
6.1 The problem ofbest approximation 93
6.2 Estimates for the projection operator 96
6.3 Inverse inequalities 104
6.4 Other projection operators 107
6.5 Convergence ofthe Gaussianformulas 111
6.6 Estimates for the interpolation operator 115
6.7 Laguerre and Hermite functions 119
6.8 Refinements ... . . . . . 123
Contents IX
7. Derivative Matrices
7.1 Derivative matrices in the frequency space 125
7.2 Derivative matrices iJl the physical space 129
7.3 Boundary conditions in the frequency space 135
7.4 Boundary conditions in the physical space 138
7.5 Derivatives ofscaled functions 146
7.6 Numerical solution 148
8. Eigenvalue analysis
8.1 Eigenvalues offirst derivative operators 151
8.2 Eigenvalues ofhigher-orderoperators 157
8.3 Condition number 165
8.4 Preconditioners for second-order operators 168
8.5 Preconditioners for first-order operators 172
8.6 Convergence ofeigenvalues 174
8.7 Multigrid method 179
9. Ordinary differential equations
9.1 General considerations 181
9.2 Approximation oflinear equations 183
9.3 The weak formulation 189
9.4 Approximation ofproblemsin the weak form 197
9.5 Approximation in unbounded domains 207
9.6 Other techniques 210
9.7 Boundary layers 211
9.8 Nonlinear equations 214
9.9 Systems ofequations 217
9.10 Integral equations 218
10. Time-dependent problems
10.1 The Gronwall inequality 221
10.2 Approximation ofthe heat equation 222
x Polynomial Approximation ofDifferential Equations
10.3 Approximation oflinear first-order problems 229
10.4 Nonlinear time-dependent problems 235
10.5 Approximation ofthe wave equation 237
10.6 Time discretization ..... 243
11. Domain-decomposition methods
11.1 Introductory remarks .... 249
11.2 Non-overlapping multidomain methods 250
11.3 Solution techniques . 260
11.4 Overlappingmultidomain methods 263
12. Examples
12.1 A free boundary problem . . . . . 265
12.2 An example in an unbounded domain 271
12.3 The nonlinear Schrodinger equation 274
12.4 Zeroes of Bessel functions . . 278
13. An example in two dimensions
13.1 Poisson's equation . . . . . . . . . . 281
13.2 Approximation by the collocation method 282
13.3 Hints for the implementation .. . . . 287
13.4 The incompressible Navier-Stokes equations 290
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
Index .............................. 305