University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Doctoral Dissertations Graduate School 8-2013 Finite Difference and Discontinuous Galerkin Finite Element Methods for Fully Nonlinear Second Order Partial Differential Equations Thomas Lee Lewis [email protected] Follow this and additional works at: https://trace.tennessee.edu/utk_graddiss Part of the Numerical Analysis and Computation Commons Recommended Citation Lewis, Thomas Lee, "Finite Difference and Discontinuous Galerkin Finite Element Methods for Fully Nonlinear Second Order Partial Differential Equations. " PhD diss., University of Tennessee, 2013. https://trace.tennessee.edu/utk_graddiss/2446 This Dissertation is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a dissertation written by Thomas Lee Lewis entitled "Finite Difference and Discontinuous Galerkin Finite Element Methods for Fully Nonlinear Second Order Partial Differential Equations." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the equirr ements for the degree of Doctor of Philosophy, with a major in Mathematics. Xiaobing H. Feng, Major Professor We have read this dissertation and recommend its acceptance: Suzanne M. Lenhart, Ohannes Karakashian, Alfredo Galindo-Uribarri Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) University of Tennessee, Knoxville Trace: Tennessee Research and Creative Exchange Doctoral Dissertations Graduate School 8-2013 Finite Difference and Discontinuous Galerkin Finite Element Methods for Fully Nonlinear Second Order Partial Differential Equations Thomas Lee Lewis [email protected] This Dissertation is brought to you for free and open access by the Graduate School at Trace: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of Trace: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a dissertation written by Thomas Lee Lewis entitled "Finite Difference and Discontinuous Galerkin Finite Element Methods for Fully Nonlinear Second Order Partial Differential Equations." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy, with a major in Mathematics. Xiaobing H. Feng, Major Professor We have read this dissertation and recommend its acceptance: Suzanne M. Lenhart, Ohannes Karakashian, Alfredo Galindo-Uribarri Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official student records.) Finite Difference and Discontinuous Galerkin Finite Element Methods for Fully Nonlinear Second Order Partial Differential Equations A Dissertation Presented for the Doctor of Philosophy Degree The University of Tennessee, Knoxville Thomas Lee Lewis August 2013 c by Thomas Lee Lewis, 2013 All Rights Reserved. ii This dissertation is lovingly dedicated to my parents who have always encouraged and supported me. iii Acknowledgements I would like to thank all of the people that have contributed to my doctoral research. I first would like to express my sincerest gratitude to my advisor, Professor Xiaobing Feng, for his guidance and encouragement throughout my graduate studies. Dr. Feng involved me in research early in my graduate career, and he has always shared his knowledge and experiences so that I could better understand the rewards of pursuing an academic career. Dr. Feng's willingness to share his wisdom has both made this work possible and laid a foundation for a future in academics. I would also like to thank the remaining members of my dissertation committee, Professor Ohannes Karakashian, Professor Susanne Lenhart, and Dr. Alfredo Galindo-Uribarri, for their willingness to serve on my committee and answer any questions that I had. Dr. Karakashian's leadership in the classroom contributed to much of my interest and background in numerical analysis. Dr. Lenhart's advice was indispensable when preparing research talks, and Dr. Galindo-Uribarri's encouragement at ORNL first planted the seed for an academic research career. Finally, I would like to thank my friends and family for their patience and encouragement along the way. My friends were always available for a distraction and understanding during my absence. My parents were always able to help me forget the stresses associated with my doctoral research. My parents-in-law took me in, always offering encouragement and support. Most importantly, I cannot express the thankfulness I have for my wife, Beth, and the love she has provided me. iv Abstract The dissertation focuses on numerically approximating viscosity solutions to second order fully nonlinear partial differential equations (PDEs). The primary goals of the dissertation are to develop, analyze, and implement a finite difference (FD) framework, a local discontinuous Galerkin (LDG) framework, and an interior penalty discontinuous Galerkin (IPDG) framework for directly approximating viscosity solutions of fully nonlinear second order elliptic PDE problems with Dirichlet boundary conditions. The developed frameworks are also extended to fully nonlinear second order parabolic PDEs. All of the proposed direct methods are tested using Monge-Amp`ereproblems and Hamilton-Jacobi-Bellman (HJB) problems. Due to the significance of HJB problems in relation to stochastic optimal control, an indirect methodology for approximating HJB problems that takes advantage of the inherent structure of HJB equations is also developed. First, a FD framework is developed that guarantees convergence to viscosity solutions when certain properties concerning admissibility, stability, consistency, and monotonicity are satisfied. The key concepts introduced are numerical operators, numerical moments, and generalized monotonicity. One class of FD methods that fulfills the framework provides a direct realization of the vanishing moment method for approximating second order fully nonlinear PDEs. Next, the emphasis is on extending the FD framework using DG methodologies. In particular, some nonstandard LDG and IPDG methods that utilize key concepts from the FD framework are formulated. Benefits of the DG methodologies over the FD methodology include the ability to v handle more complicated domains, more freedom in the design of meshes, higher potential for adaptivity, and the ability to use high order elements as a means for increased accuracy. Last, a class of indirect methods for approximating HJB equations using the vanishing moment method paired with a splitting formulation of the HJB problem is developed and tested numerically. The proposed methodology is well- suited for both continuous and discontinuous Galerkin methods, and it complements the direct methods developed in the dissertation. vi Table of Contents 1 Introduction1 1.1 Prelude..................................1 1.2 PDE Solution Concepts and Facts....................2 1.2.1 Viscosity Solutions........................5 1.2.2 Existence and Uniqueness....................9 1.3 Overview of Numerical Methods for Fully Nonlinear PDEs...... 10 1.3.1 Indirect Methods......................... 12 1.3.2 Direct Methods.......................... 15 1.4 Selected Applications of Second Order Fully Nonlinear PDEs..... 16 1.4.1 Monge-Amp`ereEquations.................... 17 1.4.2 Hamilton-Jacobi-Bellman Equations............... 19 1.5 Dissertation Organization........................ 22 1.6 Mathematical Software and Implementation.............. 23 2 Finite Difference Methods 24 2.1 Difference Operators........................... 25 2.2 The Finite Difference Framework of Crandall and Lions for Hamilton- Jacobi Problems.............................. 32 2.3 A New Finite Difference Framework for One-Dimensional Second Order Elliptic Problems............................. 36 2.3.1 Definitions............................. 37 vii 2.3.2 Convergence Analysis....................... 40 2.3.3 Examples of Numerical Operators................ 46 2.3.4 Verification of the Consistency, G-Monotonicity, Stability, and Admissibility of Lax-Friedrichs-like Finite Difference Methods 51 2.3.5 Numerical Experiments...................... 55 2.4 Extensions of the New FD Framework to Second Order Parabolic Problems in One Spacial Dimension................... 69 2.4.1 Formulation of the Fully Discrete Framework.......... 69 2.4.2 Numerical Experiments...................... 72 2.5 Extensions of the New FD Framework to Second Order Elliptic Problems in Higher Dimensions..................... 79 2.5.1 Formulation of the Framework.................. 79 2.5.2 Numerical Experiments...................... 84 3 Local Discontinuous Galerkin Methods 91 3.1 Notation.................................. 92 3.2 A Monotone Framework for Second Order Elliptic Problems..... 95 3.2.1 Motivation............................. 95 3.2.2 Element-Wise Formulation of the LDG Methods........ 99 3.2.3 Whole Domain Formulation