MATH 523 and MATH 524. Numerical Analysis

These courses form a two-semester graduate level introduction to numerical analysis. They will be mainly focused on the design and the analysis of classical as well as recently developed numerical algorithms and techniques, for the solution of variety of problems in mathematial analysis and algebra. In short, this course will provide an introduction to the basics of the mathematical theory behind scientific and engineering computing. The students who take this course should have a very good and stable knowledge of single and multivariable calculus, linear algebra and be familiar with basic facts from functional, real and complex analysis and the theory of partial differential equations.

1. Approximation and Interpolation; Numerical Quadrature; Direct Methods of Numerical Linear Algebra; Numerical solution of nonlinear systems and optimization
   • Polynomial Approximation
   • Lagrange Interpolation
   • Least Squares Polynomial Approximation
   • Piecewise polynomial approximation and interpolation
   • The Fast Fourier Transform
   • Multipole method for special dense matrix vector product
2. Numerical Quadrature
   • Basic quadrature
   • The Peano Kernel Theorem
   • Richardson Extrapolation
   • Asymptotic error expansions
   • Romberg Integration
   • Gaussian Quadrature
   • Adaptive quadrature
   • Monte Carlo methods for higher dimensional integrals.
3. Direct Methods of Numerical Linear Algebra
   • Triangular systems
   • Gaussian elimination and LU decomposition
   • Pivoting
   • Backward error analysis
   • Conditioning and roundoff errors.
4. Numerical solution of nonlinear systems and optimization
   • One-point iteration
   • Newton's method
   • Unconstrained minimization
   • Newton's method
   • Line search methods
   • Conjugate gradients

1. Numerical Solution of Ordinary Differential Equations
   • Euler's Method
   • Linear multistep methods
   • One step methods
   • Stiffness
2. Numerical Solution of Partial Differential Equations
   • BVPs for 2nd order elliptic PDEs
   • The five-point discretization of the Laplacian
   • Finite element methods
   • Difference methods for the heat equation
   • Difference methods for hyperbolic equations
   • Hyperbolic conservation laws
3. Some Iterative Methods of Numerical Linear Algebra
   • Classical iterations
   • Multigrid methods

1. Analysis of Numerical Methods, by Eugene Isaacson and Herbert Bishop Keller; Dover Publications 1994.
2. Introduction to Numerical Analysis, by J. Stoer and R. Bulirsch; Springer-Verlag 1980. ISBN 0-387-90420-4.

MATH/CSE 552. Numerical Solution of Partial Differential Equations

This is an introductory graduate course on numerical methods for partial differential equations. It is designed mainly for graduate students in department of mathematics. The course should also be appropriate for non-math graduate students who are good at advanced calculus and linear algebra. The students are required to do computing projects in addition to theoretical homework problmes.

All the graduate students in computational and applied math program are required to take this course.

• A review of basic methods for the Poisson Equations on regular domain (1 week)
  1. The finite difference method
  2. The finite element method
  3. The finite volume method
• Second order elliptic boundary value equations (5 weeks)
  • A review of qualitative properties (2 hour)
    1. Maximal principle
    2. Existence and uniqueness (existence of classical and weak solutions)
    3. Regularity (H² regularity of the solutions for smooth or convex domain)
  • The finite difference method (4 hours)
    1. Basic finite difference schemes
    2. Discrete maximal principle and M-matrices
    3. Error estimates
    4. Broundary treatments
  • The finite element method (5 hours)
    1. Linear finite element methods
    2. Error estimates: estimates in H¹ norm and L² norm
    3. Construction of more general finite element methods
  • The finite volume method (3 hours)
    1. Basic finite volume schemes
    2. Conservation properties
    3. Relation with finite element method
    4. Error estimates
    5. High order finite volume method
• Direct and iterative methods for solving the discrete systems (2 weeks)
  • Direct methods (1)
  • Sparse matrix data structure (1)
  • Basic iterative methods (2)
    1. Basic iterative methods
    2. Jacobi and Gauss-Seidel methods
  • The method of subspace corrections and its convergence properties (2)
  • Conjugate gradient methods and preconditioning (2)
• The multigrid method (2 weeks)
  • Introduction of the algorthm using one dimensional problem (2 hours)
  • Algorithmic details for /-cycle, \-cycle, (2 hours) V-cycle and W-cycle algorithms
  • Convergence analysis using the method of subspace corrections (2 hours)
• Parabolic and hyperbolic problems (4 weeks)
  • Model problems and stability estimates (2 hours)
  • Examples of the methods of lines (2 hours)
  • The Lax-Richtmyer equivalence theorem (1 hour)
  • Stability analysis (2 hours)
    1. Discrete Fourier series (.5)
    2. von Neumann stability analysis (.5)
    3. The Kreiss matrix theory (1)
  • Consistency, convergence and error estimates (1)
• Convection dominated problems (1 week)
  • The failure of standard discretization
  • Monotone schemes and Godunov theorem
  • Higher order methods
  • Nonlinear problems

There are many text books available, but there is no single one that would fit the aforementioned syllabus.

1. Hackbusch, W., Elliptic differential equations : theory and numerical treatment Berlin ; New York : Springer-Verlag, c1992. [good reference for elliptic problems]
2. Strikwerda, John C., Finite difference schemes and partial differential equations / John C. Strikwerda. Pacific Grove, Calif. : Wadsworth \& Brooks/Cole Advanced Books \& Software, c1989. [good reference for linear parabolic and hyperbolic problems]
3. Johnson, Claes, Numerical solution of partial differential equations by the finite element method / Claes Johnson. Cambridge [Cambridgeshire] ; New York: Cambridge University Press, c1987. [Overall good reference for finite element method for this course; but not enough materials for theoretical analysis]
4. Susanne Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, New York : Springer-Verlag, c1994. [Chapter 3 is a good reference on the construction of a finite element space; other parts of the book are too theoretical/technical for this course]
5. Jinchao Xu, Lecture notes for MATH/CSE 552 [covers materials that can not be found in the above three books and other major text books, especially good materials for iterative and multigrid methods, finite volume methods]

MATH/CSE 556. Finite Element Methods

This is a graduate course on finite element method and theory for partial differential equations. It is designed for graduate students in department of mathematics. The students who take the course should have a good knowlege of multivariable caclulus, real, comples and functional analysis, linear algebra and basic knowlege of partial differential equations.

Actual syllabus may depend on the instructor. The following syllabus is designed by Jinchao Xu.

• Preliminaries
  1. Linear algebra
  2. Sobolev spaces
  3. Bramble-Hilbert Lemma
  4. Babuska-Brezzi theory
• Definition of finite element triple
• H(id)=L² elements
• H(grad)=H¹ elements and second order elliptic boundary value problems
• H(curl) elements and Maxwell equations
• H(div) elements and and mixed finite element methods
• Exact sequence relating H(grad), H(curl), H(div) and H(id) elements; Helmholtz decomposition
• H² elements and biharmonic equations
• Non-conforming elements
• Generalized finite element method; Mortar elements
• Stokes equations and Navier-Stokes equations
• Multigrid methods and convergence analysis

1. P. Ciarlet, The finite element method for elliptic problems North-Holland, Amsterdam 1978 (it was recently published by SIAM) [Preliminaries, finite element triple, H(grad) and H² elements, non-conforming elements]
2. Vivette Girault, Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations : theory and algorithms, Berlin ; New York : Springer-Verlag, c1986. [H(curl) and H(div) elements]
3. Jinchao Xu, Lecture notes for MATH/CSE 552 [multigrid methods]