ELLIPTIC EQUATIONS WITH SINGULARITIES: A PRIORI ANALYSIS AND NUMERICAL APPROACHES

Open Access
Author:
Li, Hengguang
Graduate Program:
Mathematics
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
April 21, 2008
Committee Members:
  • Victor Nistor, Committee Chair
  • Ludmil Tomov Zikatanov, Committee Chair
  • Jinchao Xu, Committee Member
  • Anna L Mazzucato, Committee Member
  • Corina Stefania Drapaca, Committee Member
Keywords:
  • the finite element method
  • weighted sobolev spaces
  • singularities
  • elliptic equations
  • a priori analysis
  • the multigrid method
Abstract:
Elliptic equations in a two- or three-dimensional bounded domain may have singular solutions from the non-smoothness of the domain, changes of boundary conditions, and discontinuities, singularities of the coefficients. These singularities give rise to various difficulties in the theoretical analysis and in the development of numerical 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 understandings 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 corresponding 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 Schroedinger operator with singular 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.