Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgements
- Symbols
- Part I Numerical Linear Algebra
- Part II Constructive Approximation Theory
- Part III Nonlinear Equations and Optimization
- Part IV Initial Value Problems for Ordinary Differential Equations
- Part V Boundary and Initial Boundary Value Problems
- 23 Boundary and Initial Boundary Value Problems for Partial Differential Equations
- 24 Finite Difference Methods for Elliptic Problems
- 25 Finite Element Methods for Elliptic Problems
- 26 Spectral and Pseudo-Spectral Methods for Periodic Elliptic Equations
- 27 Collocation Methods for Elliptic Equations
- 28 Finite Difference Methods for Parabolic Problems
- 29 Finite Difference Methods for Hyperbolic Problems
- Appendix A Linear Algebra Review
- Appendix B Basic Analysis Review
- Appendix C Banach Fixed Point Theorem
- Appendix D A (Petting) Zoo of Function Spaces
- References
- Index
27 - Collocation Methods for Elliptic Equations
from Part V - Boundary and Initial Boundary Value Problems
Published online by Cambridge University Press: 29 September 2022
- Frontmatter
- Contents
- Preface
- Acknowledgements
- Symbols
- Part I Numerical Linear Algebra
- Part II Constructive Approximation Theory
- Part III Nonlinear Equations and Optimization
- Part IV Initial Value Problems for Ordinary Differential Equations
- Part V Boundary and Initial Boundary Value Problems
- 23 Boundary and Initial Boundary Value Problems for Partial Differential Equations
- 24 Finite Difference Methods for Elliptic Problems
- 25 Finite Element Methods for Elliptic Problems
- 26 Spectral and Pseudo-Spectral Methods for Periodic Elliptic Equations
- 27 Collocation Methods for Elliptic Equations
- 28 Finite Difference Methods for Parabolic Problems
- 29 Finite Difference Methods for Hyperbolic Problems
- Appendix A Linear Algebra Review
- Appendix B Basic Analysis Review
- Appendix C Banach Fixed Point Theorem
- Appendix D A (Petting) Zoo of Function Spaces
- References
- Index
Summary
Collocation methods for elliptic problems are discussed here. We begin by providing their definition. For their analysis we first introduce a weighted weak formulation of the problem, and show that it is well posed. Then, we introduce and analyze a Galerkin approximation for this problem, where the subspace consists of polynomials that vanish sufficiently fast at the boundary. Next, a scheme with quadrature is proposed, and its analysis is provided using the theory of variational crimes and Strang lemmas. For its implementation and analysis the discrete cosine and Chebyshev transforms are introduced and analyzed. The phenomenon of aliasing is briefly discussed. Finally, we connect the weighted Galerkin approximation with quadrature to collocation methods, thus providing an analysis of collocation schemes.
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- Information
- Classical Numerical AnalysisA Comprehensive Course, pp. 742 - 773Publisher: Cambridge University PressPrint publication year: 2022