AUGMENTED MATCHED INTERFACE AND BOUNDARY(AMIB) METHOD FOR SOLVING INTERFACE AND BOUNDARY VALUE PROBLEMS by HONGSONG FENG SHAN ZHAO, COMMITTEE CHAIR DAVID HALPERN WEI ZHU MOJDEH RASOULZADEH XIAOWEN WANG A DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics in the Graduate School of The University of Alabama TUSCALOOSA, ALABAMA 2021 Copyright HONGSONG FENG 2021 ALL RIGHTS RESERVED ABSTRACT This dissertation is devoted to the development of the augmented matched interface and bound- ary(AMIB) method and its applications for solving interface and boundary value problems. We start with a second order accurate AMIB introduced for solving two-dimensional (2D) el- liptic interface problems with piecewise constant coefficients, which illustrates the theory of AMIB illustrated in details. AMIB method is different from its ancestor matched interface and boundary (MIB) method in employing fictitious values to restore the accuracy of central differences for in- terface and boundary value problems by approximating the corrected terms in corrected central differences with these fictitious values. Through the augmented system and Schur complement, the total computational cost of the AMIB is about O(N log N) for degree of freedom N on a Carte- sian grid in 2D when fast Fourier transform(FFT) based Poisson solver is used. The AMIB method achieves O(N log N) efficiency for solving interface and boundary value problems, which is a sig- nificant advance compared to the MIB method. Following the theory of AMIB in chapter 2, chapter 3 to chapter 6 cover the development of AMIB for a high order efficient algorithm in solving Poisson boundary value problems and a fourth order algorithm for elliptic interface problems as well as efficient algorithm for parabolic interface problems. The AMIB adopts a second order FFT-based fast Poisson solver in solving elliptic interface problems. However, high order FFT-based direct Poisson solver is not available in the literature, which imposes a grand challenge in designing a high order efficient algorithm for elliptic interface problems. The AMIB method investigates efficient algorithm of Poisson boundary value problem (BVP) on rectangular and cubic domains by converting Poisson BVP to an immersed boundary problem, based on which a high order FFT algorithm is proposed. This naturally allows for fulfilling a fourth order fast algorithm for solving elliptic interface problems. Besides the FFT algorithm, a multigrid method is also considered to achieve high efficiency in solving parabolic ii interface problems. Extensive numerical results are included in each chapter of the concerned problem, and are used to show the robustness and efficiency of AMIB method. Keyword: Elliptic interface problem; Parabolic interface problem; Corrected differences; High order accuracy; Fast Fourier transform(FFT); O(N log N) algorithm; Augmented Matched interface and boundary (AMIB); Augmented system; Schur complement; Multigrid. iii DEDICATION To my parents, mentors, friends and all those who have ever helped me in my life. iv ACKNOWLEDGMENTS First and foremost, I am gratefully indebted to my advisor Professor Shan Zhao, and wish to express my deepest appreciation to him for his persistent guidance, patience, and encouragement on my research over the past five years of my PhD study. His integrity, enthusiasm and profession- alism inspire me to pursue academic advances, and to become an educator in the future. I would like to acknowledge helpful suggestions from my committee members: Dr. David Halpern, Dr. Wei Zhu, Dr. Mojdeh Rasoulzadeh, and Dr. Xiaowen Wang. I thank them for their time, support and expertise to improve my dissertation work. Many faculty and staff in the math department of The University of Alabama gave me great assistance and encouragement in various ways during the course of my PhD study. I really appre- ciate all I have learnt from the professors in classes. I am especially grateful to Professor David Halpern, Professor David Cruz-Uribe and Ms. Natalie Lau for their continuous assistance during the long time of my study at UA. Especially, I would like to thank Professor Runchang Lin as my master advisor during my study at Texas A&M International University. His patient guidance led me to my academic life, and I appreciate his continuous support over these years since I came to USA. Last but not least, I am grateful to my parents and sister for their blessings and selfless support in my path of academic pursuance and journey of life. v CONTENTS ABSTRACT . ii DEDICATION . iv ACKNOWLEDGMENTS . v LIST OF TABLES . xi LIST OF FIGURES . xii CHAPTER 1 INTRODUCTION . 1 CHAPTER 2 SECOND ORDER AMIB FOR ELLIPTIC INTERFACE PROBLEM . 10 2.1 AMIB Theory and algorithm . 10 2.1.1 Correcting finite difference for Laplacian . 11 2.1.2 Reconstructing the Cartesian derivative jumps . 16 2.1.3 Augmented system . 24 2.2 Reducing condition number . 28 2.2.1 Traditional MIB . 29 2.2.2 Shifting fictitious point . 29 2.2.3 Shifting real point . 30 2.3 Matrix structure from 1D MIB . 32 2.3.1 Coefficients derivation for Case L = 1.................... 33 2.3.2 Coefficients derivation for Case L = 2.................... 34 2.3.3 Coefficients redistribution . 37 2.4 Alternative to jump-corrected differences . 40 2.5 Numerical experiments . 43 2.5.1 Numerical accuracy . 44 vi 2.5.2 Cartesian jump reconstruction . 51 2.5.3 Computational efficiency . 53 2.5.4 Comparison of AMIB-CFD and AMIB-CFP . 55 CHAPTER 3 AMIB FOR POISSON BOUNDARY VALUE PROBLEM . 57 3.1 Preliminary . 57 3.2 Theory and algorithm . 58 3.2.1 Fast Poisson solver . 60 3.2.2 Immersed boundary problem and corrected differences . 66 3.2.3 Augmented system . 76 3.3 Extensions . 78 3.3.1 An extension on high order algorithms via FFT . 78 3.3.2 An extension on high order corrected differences. 79 3.4 Numerical experiments . 83 3.4.1 Accuracy studies . 84 3.4.2 Computational efficiency . 88 CHAPTER 4 AMIB FOR POISSON BOUNDARY VALUE PROBLEM ON STAG- GERED GRIDS . 92 4.1 Preliminary . 92 4.2 Theory and algorithm . 93 4.2.1 Immersed boundary problem . 93 4.2.2 Reconstructing the Cartesian derivative jumps . 93 4.2.3 Augmented system . 99 4.3 Numerical experiments . 99 4.3.1 The AMIB method for staggered boundaries . 100 4.3.2 Comparison of AMIB-S and AMIB-NS . 102 4.3.3 Combined AMIB method . 106 vii 4.3.4 Comparison of two AMIB methods . 107 4.3.5 AMIB for problem with low regularities . 110 CHAPTER 5 FOURTH ORDER AMIB FOR ELLIPTIC INTERFACE PROBLEM . 113 5.1 Preliminary . 113 5.2 Theory and algorithm . 114 5.2.1 Immersed boundary . 115 5.2.2 Laplacian approximation . 118 5.2.3 Fictitious values formulation . 120 5.2.4 Formulation of augmented system . 124 5.3 Numerical experiments . 125 5.3.1 Accuracy studies . 126 5.3.2 Computational efficiency . 141 CHAPTER 6 AMIB FOR PARABOLIC INTERFACE PROBLEM . 143 6.1 Theory and algorithm . 143 6.1.1 Temporal discretization . 149 6.1.2 Spatial discretization . 149 6.1.3 Formulation of augmented system . 150 6.2 Numerical experiments . 160 6.2.1 Numerical accuracy . 160 CHAPTER 7 CONCLUSIONS . 177 REFERENCES . 180 viii LIST OF TABLES 2.1 Example 1 – Numerical error analysis. Here β+ = 10; β− = 2: . 46 2.2 Example 2 – Numerical error analysis. 46 2.3 Example 3A – Numerical error analysis. Here β− = 10; β+ = 1: . 48 2.4 Example 3B – Numerical error analysis. Here β− = 1000; β+ = 1: . 49 2.5 Example 4A – Numerical error analysis. Here β+ = 100; β− = 1: . 50 2.6 Example 4B – Numerical error analysis. Here β+ = 1; β− = 100: . 50 2.7 Example 5A – Numerical error analysis. Here β+ = 1000; β− = 1: . 51 2.8 Example 5B – Numerical error analysis. Here β+ = 1; β− = 1000: . 52 2.9 Reconstructed Cartesian derivative jumps for Example 1 and 3A. 52 2.10 Numerical error analysis. Here β+ = 100; β− = 1: . 55 2.11 Numerical error analysis. Here β+ = 1; β− = 100: . 56 3.1 Numerical errors of the FFT and LU methods for Example 1 in 1D. 65 3.2 Numerical errors of the FFT and LU methods for Example 2 in 1D. 65 3.3 Example 1 –Second and fourth order numerical error analysis. 85 3.4 Example 2 –Fourth order numerical error analysis of AMIB vs MIB. 85 3.5 Example 3 –High order numerical error analysis. 87 3.6 Example 4 –Fourth order numerical error analysis for the Helmholtz equation . 87 3.7 Example 5 –Numerical error analysis of the AMIB4 on 3D Poisson’s equation. 88 3.8 Computational efficiency comparison. 90 4.1 Example 1 –High order numerical error analysis. 101 4.2 Example 2 –Numerical error analysis of the AMIB4-S on 3D Poisson’s equation. 102 4.3 Example 3 –Fourth order numerical error analysis of the AMIB4-S and AMIB4-NS. 103 4.4 Example 4–Fourth order numerical error analysis of the AMIB4-S. 104 ix 4.5 Example 5 –Fourth order numerical error analysis. 105 4.6 Example 6 –Numerical error analysis of AMIB4-S for mixed boundary conditions. 106 4.7 Example 7 –Fourth order numerical error analysis of the AMIB4. 107 4.8 Example for staggered mesh . 109 4.9 Example for traditional mesh . 110 4.10 Condition number on STAGGERED mesh . 110 4.11 Condition number on TRADITIONAL mesh . ..