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Open Hengguanglidissertation.Pdf The Pennsylvania State University The Graduate School ELLIPTIC EQUATIONS WITH SINGULARITIES: A PRIORI ANALYSIS AND NUMERICAL APPROACHES A Dissertation in Mathematics by Hengguang Li c 2008 Hengguang Li Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2008 The dissertation of Hengguang Li was reviewed and approved∗ by the following: Victor Nistor Professor of Mathematics Dissertation Advisor, Chair of Committee Ludmil T. Zikatanov Professor of Mathematics Dissertation Advisor Jinchao Xu Distinguished Professor of Mathematics Anna L. Mazzucato Professor of Mathematics Corina S. Drapaca Professor of Engineering John Roe Professor of Mathematics Head of the Department of Mathematics ∗Signatures are on file in the Graduate School. Abstract Elliptic equations in a two- or three-dimensional bounded domain may have singu- lar solutions from the non-smoothness of the domain, changes of boundary condi- tions, and discontinuities, singularities of the coefficients. These singularities give rise to various difficulties in the theoretical analysis and in the development of nu- merical algorithms for these equations. On the other hand, most of the problems arising from physics, engineering, and other applications have singularities of this form. In addition, the study on these elliptic equations leads to good understand- ings of other types of PDEs and systems of PDEs. This research, therefore, is not only of theoretical interest, but also of practical importance. This dissertation includes a priori estimates (well-posedness, regularity, and Fredholm property) for these singular solutions of general elliptic equations in weighted Sobolev spaces, as well as effective finite element schemes and correspond- ing multigrid estimates. Applications of this theory to equations from physics and engineering will be mentioned at the end. This self-contained work develops systematic a priori estimates in weighted Sobolev spaces in detail. It establishes the well-posedness of these equations and proves the full regularity of singular solutions between suitable weighted spaces. Besides, the Fredholm property is discussed carefully with a calculation of the index. For the numerical methods for singular solutions, based on a priori analysis, this work constructs a sequence of finite element subspaces that recovers the optimal rate of convergence for the finite element solution. In order to efficiently solve the algebraic system of equations resulting from the finite element discretization on these finite subspaces, the method of subspace corrections and properties of weighted Sobolev spaces are used to prove the uniform convergence of the multigrid method for these singular solutions. To illustrate wide extensions of this theory, a Schr¨odinger operator with singular iii potentials and a degenerate operate from physics and engineering are studied in Chapter 6 and Chapter 7. It shows that similar a priori estimates and finite element algorithms work well for equations with a class of singular coefficients. The last chapter contains a brief summary of the dissertation and plans for possible work in the future. iv Table of Contents List of Figures viii List of Tables x List of Symbols xi Acknowledgments xii Chapter 1 Introduction 1 1.1 A Brief Review . 2 1.1.1 Singular Solutions of EBVPs . 2 1.1.2 Numerical Methods for Singular Solutions . 4 1.2 Contents . 5 1.3 Preliminaries and Notations . 7 1.3.1 Sobolev Spaces and the Weak Solution . 8 1.3.2 The Finite Element Method . 10 1.3.3 Multigrid (MG) Methods . 12 1.3.4 Corner Singularities . 15 Chapter 2 A Priori Estimates for Elliptic Operators on the Infinite Domain 18 2.1 The Mellin Transform . 18 2.2 A Priori Estimates . 20 2.2.1 The Infinite Domain . 20 2.2.2 The Weighted Sobolev Space . 29 2.3 Some Lemmas for Weighted Sobolev Spaces . 36 2.3.1 Notation . 36 v 2.3.2 Lemmas . 38 Chapter 3 A Priori Analysis in Weighted Sobolev Spaces 43 3.1 Formulation of the Problem and Main Results . 47 3.1.1 The Domain . 47 3.1.2 The Equation . 50 3.1.3 Estimates for Smooth Interfaces . 54 3.2 Estimates of Neumann Problems and Singular Transmission Problems 57 3.2.1 The Laplace Operator . 57 3.2.2 Transmission Problems . 62 3.3 Domains with a Polygonal Structure . 64 3.3.1 The Unfolded Boundary and Closure . 65 3.3.2 Domains with a Polygonal Structure . 66 3.3.3 Smooth Functions . 68 3.3.4 The Canonical Weight Function . 69 3.4 Properties of Sobolev Spaces with Weights . 71 3.4.1 Differential Operators . 72 3.4.2 Inhomogeneous Weighted Sobolev Spaces . 72 3.5 Proofs of the Three Main Theorems . 77 Chapter 4 The Finite Element Method for Singular Solutions 80 4.1 Analysis of the FEM . 80 4.1.1 Approximation Away from the Vertices . 81 4.1.2 Approximation Near the Vertices . 84 4.2 Numerical Tests . 89 4.2.1 Domains with Cracks and Neumann-Neumann Vertices . 91 4.2.2 Domains with Artificial Vertices . 93 4.2.3 Transmission Problems . 94 Chapter 5 The Multigrid Method on Graded Meshes 97 5.1 Preliminaries and Notation . 99 5.2 Weighted Sobolev Spaces and the Method of Subspace Corrections . 100 5.2.1 Weighted Sobolev Spaces and Graded Meshes . 100 5.2.2 The Method of Subspace Corrections . 101 5.3 Uniform Convergence of the MG Method on Graded Meshes . 106 5.4 Numerical Illustration . 113 vi Chapter 6 Application I: a Schr¨odingerType Operator 115 6.1 Weighted Sobolev Spaces . 120 6.2 The Well-posedness and Regularity of the Solution . 123 6.3 Estimates for the Finite Element Method . 128 6.3.1 Approximation Away from the Singular Set V . 129 6.3.2 Construction of the Finite Element Spaces . 131 6.4 Numerical Results . 137 Chapter 7 Application II: an Operator Degenerate on a Segment 142 m 7.1 Weighted Sobolev Spaces Ka . 145 7.1.1 Notation . 145 7.1.2 Lemmas . 147 7.2 The Well-posedness and Regularity of the Solution . 155 7.3 The Finite Element Method . 159 7.3.1 Estimate for the Interpolation . 160 7.3.2 Construction of the Mesh . 162 7.4 Numerical Results . 167 Chapter 8 Conclusions 172 8.1 Summary . 172 8.2 Future Work . 174 Bibliography 176 vii List of Figures 1.1 Domain Ω with a re-entrant vertex Q. 15 2.1 The infinite conical domain C with the vertex O at the origin. 19 2.2 The resulting domain D from the Mellin transform on the t-θ plane. 20 3.1 The domain Ω and the interface Γ . 44 3.2 Domain with two vertices touching a side. 48 3.3 Domain with ramified crack. 50 3.4 A vertex touching a smooth side. The picture actually represents uΩ. The map κ identifies X and Y ................... 68 4.1 One refinement of the triangle T with vertex Q, κ = l1=l2 . 85 4.2 Domain with crack: initial triangles (left); triangulation after one refinement, κ = 0:2 (right) . 91 4.3 Initial triangles for a Neumann-Neumann vertex Q (left); the tri- angulation after one refinement, κ = 0:2 (right). 92 4.4 The numerical solution for the mixed problem and a Neumann- Neumann vertex. 92 4.5 Domain with artificial vertex: initial triangles (left); triangulation after four refinements, κ = 0:2 (right). 94 4.6 The transmission problem: initial triangles (left); triangulation af- ter four refinements, κ = 0:2 (right). 95 4.7 The numerical solution for the transmission problem. 96 5.1 Initial triangles with vertex Q0 (left); layer D0 and D1 after one refinement (right), κ = 0:2. 107 5.2 Crack: initial triangulation (left) and the triangulation after one refinement (right), κ = 0:2. 113 6.1 One refinement of triangle T with vertex QI 2 M, κ = l1=l2. 133 6.2 Initial triangles for Ω^ (left); the mesh after four refinements , κ = 0:2 (right). 138 viii ^ 6.3 Initial triangles for Ω1 and boundary conditions (left); the mesh after one refinement , κ = 0:2 (right) . 139 7.1 Domain Ω and the position of X. 146 7.2 Initial mesh with three triangles(left); Triangulation after one re- finement N = 1; κ = 0:5 (right). 163 7.3 Initial mesh with three triangles (left); the mesh after one refine- ment κ = 0:2 (right). 169 7.4 The mesh after 5 levels of refinement for κ = 0:2. 169 ix List of Tables 4.1 Convergence history in the case of a crack domain. 92 4.2 Convergence history in the case of a Neumann-Neumann vertex. 93 4.3 Convergence history in the case of an artificial vertex. 94 4.4 Convergence history for the transmission problem . 96 5.1 Asymptotic convergence factors (rΩMG) for the MG V(1,1)-cycle applied to problem (5.6) with Gauss Seidel smoother . 114 6.1 Interior singularities . 141 6.2 Boundary singularities . 141 7.1 Convergence history. 170 x List of Symbols Ω A bounded polygonal domain in R2, possible with curved boundaries V The set of \vertices", containing geometric vertices and possible artificial vertices Qi The i-th \vertex" of the domain, Qi 2 V Si A small neighborhood of Qi, Si ⊂ Ω Hm The usual Sobolev spaces with square-integratable weak derivatives up to order m m Ka The weighted Sobolev space of order m and index a m K! The inhomogeneous weighted Sobolev space a @uΩ The unfolded boundary of Ω uΩ The unfolded closure of Ω T The triangulation of Ω Γ The interface of in the transmission problem where the coefficients jump xi Acknowledgments I am very grateful to those, in particular, my dissertation committee members, who once helped me during my Ph.D.
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