Numerical methods for solving linear elliptic PDEs: Direct solvers and high order accurate ...

229 Pages·2015·5.93 MB·English

Numerical methods for solving linear elliptic PDEs:
Direct solvers and high order accurate discretizations
by
Sijia Hao
B.S.,University of Science and Technology of China, 2009
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Department of Applied Mathematics
2015
This thesis entitled:
Numerical methods for solving linear elliptic PDEs:
Direct solvers and high order accurate discretizations
written by Sijia Hao
has been approved for the Department of Applied Mathematics
Per-Gunnar Martinsson
Gregory Beylkin
James Bremer
Bengt Fornberg
Zydrunas Gimbutas
Date
The final copy of this thesis has been examined by the signatories, and we find that both the
content and the form meet acceptable presentation standards of scholarly work in the above
mentioned discipline.
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Hao, Sijia (Ph.D., Applied Mathematics)
Numerical methods for solving linear elliptic PDEs:
Direct solvers and high order accurate discretizations
Thesis directed by Professor Per-Gunnar Martinsson
This dissertation describes fast and highly accurate techniques for solving linear elliptic
boundary value problems associated with elliptic partial differential equations such as Laplace’s,
Helmholtz equations and time-harmonic Maxwell’s equation. It is necessary to develop efficient
methods to solve these problems numerically.
The techniques we develop in this dissertation are applicable to most linear elliptic PDEs,
especially to the Helmholtz equation. This equation models frequency domain wave-propagation,
and has proven to be particularly difficult to solve using existing methodology, in particular in
situations where the computational domain extends over dozens or hundreds of wave-lengths. One
of the difficulties is that the linear systems arising upon discretization are ill-conditioned that
have proven difficult to solve using iterative solvers. The other is that errors are aggregated over
each wave-length such that a large number of discretization nodes are required to achieve certain
accuracy. The objective of this dissertation is to overcome these difficulties by exploring three sets
of ideas: 1) high order discretization, 2) direct solvers and 3) local mesh refinements.
In terms of “high-order” discretization, the solutions to the PDEs are approximated by
high-order polynomials. We typically use local Legendre or Chebyshev grids capable of resolving
polynomials up to degree between 10 and 40. For solving the linear system formed upon high-order
discretization, there exist a broad range of schemes. Most schemes are “iterative” methods such
as multigrid and preconditioned Krylov methods. In contrast, we in this thesis mainly focus on
“direct” solvers in the sense that they construct an approximation to the inverse of the coefficient
matrix. Such techniques tend to be more robust, versatile and stable compared to the iterative
techniques. Finally, local mesh refinement techniques are developed to effectively deal with sharp
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gradients, localized singularities and regions of poor resolution. A variety of two-dimensional and
three-dimensional problems are solved using the three techniques described above.
Dedication
I dedicate this dissertation to my family, especially my parents and my sister, who have
encouraged and supported me for so many years.
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Acknowledgements
First and foremost, I would like to thank my advisor Gunnar Martisson, who has supported
and advised me through my time at University of Colorado. Without his knowledge and help, this
dissertation would never have been possible.
Next, Iwouldliketothankallmycommitteemembersfortheirtimeandinteresttomywork.
A special thank you to Gregory Beylkin who has provided valuable suggestions on my dissertation.
The faculty in the Department of Applied Mathematics are awesome. The Grandview Gang,
especially Tom Manteuffel and Steve McCormick have provided guidance in my first two years of
study.
During my six years life at Boulder, I am fortunate to make some great friends who helped
ease the stress. Kuo Liu and Lei Tang are generous to share their experience. Ying Zhao, Qian Li
and Liang Zhang have been so supportive of me and for that, I am grateful.
Finally, thankyoutomysisterandparentswhowerealwaystheretoencourageme.
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Contents
Chapter
1 Introduction 1
1.1 Theme I: High order methods for boundary integral equations in the plane. . . . . . 4
1.1.1 High-order Nystr¨om discretization and quadrature schemes . . . . . . . . . . 5
1.1.2 Local mesh refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.3 Direct solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Theme II: Acoustic scattering involving rotationally symmetric scatterers . . . . . . 7
1.2.1 A single rotationally symmetric scatter. . . . . . . . . . . . . . . . . . . . . . 7
1.2.2 Multibody scattering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Theme III: Scattering in variable wave-speed media in 2D and 3D . . . . . . . . . . 10
1.3.1 Direct solver via composite spectral discretization . . . . . . . . . . . . . . . 10
1.3.2 Local mesh refinement scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.3 Accelerated 3D direct solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Structure of dissertation and overview of principle contributions . . . . . . . . . . . . 12
2 High-order accurate methods for Nystr¨om discretization of integral equations on smooth
curves in the plane 17
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1.1 Overview of Nystr¨om discretization . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.2 Types of singular quadrature schemes . . . . . . . . . . . . . . . . . . . . . . 22
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2.1.3 Related work and schemes not compared . . . . . . . . . . . . . . . . . . . . 24
2.2 A brief review of Lagrange interpolation . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 Nystr¨om discretization using the Kapur-Rokhlin quadrature rule . . . . . . . . . . . 25
2.3.1 The Kapur–Rokhlin correction to the trapezoidal rule . . . . . . . . . . . . . 25
2.3.2 A Nystr¨om scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 Nystr¨om discretization using the Alpert quadrature rule . . . . . . . . . . . . . . . . 27
2.4.1 The Alpert correction to the trapezoidal rule . . . . . . . . . . . . . . . . . . 27
2.4.2 A Nystr¨om scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5 Nystr¨om discretization using modified Gaussian quadrature . . . . . . . . . . . . . . 30
2.5.1 Modified Gaussian quadratures of Kolm–Rokhlin . . . . . . . . . . . . . . . . 31
2.5.2 A Nystr¨om scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.6 Nystr¨om discretization using the Kress quadrature rule. . . . . . . . . . . . . . . . . 34
2.6.1 Product quadratures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.6.2 The Kress quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.6.3 A Nystr¨om scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.7 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.7.1 A 1D integral equation example. . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.7.2 Combined field discretization of the Helmholtz equation in R2 . . . . . . . . . 40
2.7.3 Effect of quadrature scheme on iterative solution efficiency . . . . . . . . . . 43
2.7.4 The Laplace BVP on axisymmetric surfaces in R3 . . . . . . . . . . . . . . . 46
2.8 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3 Asimplifiedtechniquefortheefficientandhighlyaccuratediscretizationofboundaryintegral
equations in 2D on domains with corners 50
3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2 A linear algebraic observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3 Matrix skeletons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
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3.4 Outline of the solution process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.5 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.6 Numerical illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.7 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4 A high-order Nystr¨om discretization scheme for Boundary Integral Equations Defined on
Rotationally Symmetric Surfaces 61
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.1.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.1.2 Applications and prior work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.1.3 Kernel evaluations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.1.4 Principal contributions of present work . . . . . . . . . . . . . . . . . . . . . 64
4.1.5 Asymptotic costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.1.6 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2 Fourier Representation of BIE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2.1 Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2.2 Truncation of the Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3 Nystr¨om discretization of BIEs on the generating curve . . . . . . . . . . . . . . . . 69
4.3.1 Quadrature nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.3.2 A simplistic Nystr¨om scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.3.3 High-order accurate Nystr¨om discretization . . . . . . . . . . . . . . . . . . . 70
4.4 The full algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.4.2 Cost of computing the coefficient matrices . . . . . . . . . . . . . . . . . . . . 71
4.4.3 Computational Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.5 Accelerations for the Single and Double Layer Kernels Associated with Laplace’s
Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
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4.5.1 The Double Layer Kernels of Laplace’s Equation . . . . . . . . . . . . . . . . 74
4.5.2 Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.5.3 Evaluation of Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.6 Fast Kernel Evaluation for the Helmholtz Equation . . . . . . . . . . . . . . . . . . . 79
4.6.1 Rapid Kernel Calculation via Convolution . . . . . . . . . . . . . . . . . . . . 79
4.6.2 Application to the Helmholtz Equation . . . . . . . . . . . . . . . . . . . . . 81
4.7 Fast evaluation of fundamental solutions in cylindrical coordinates . . . . . . . . . . 83
4.8 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.8.1 Laplace’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.8.2 Helmholtz Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5 Anefficientandhighlyaccuratesolverformulti-bodyacousticscatteringproblemsinvolving
rotationally symmetric scatterers 93
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2 Mathematical formulation of the scattering problem . . . . . . . . . . . . . . . . . . 96
5.3 Discretization of rotationally symmetric scattering bodies . . . . . . . . . . . . . . . 97
5.3.1 Nystr¨om discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.3.2 A single rotationally symmetric scatterer . . . . . . . . . . . . . . . . . . . . 99
5.3.3 Multibody scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.4 A block-diagonal pre-conditioner for the multibody scattering problem . . . . . . . . 102
5.5 Accelerated multibody scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.6 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.6.1 Laplace’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.6.2 Helmholtz Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.6.3 Accelerated scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.7 Conclusions and Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119