Graduate Numerical Analysis Courses

Regular Courses

• Numerical Analysis
• Numerical Methods for Partial Differential Equations
• Iterative Techniques
• Computational Linear Algebra
• Mathematical Theory of Finite Element Methods

Special Courses

• Technique of Finite Element Methods
• Special Topics in Finite Element
• Seminar in Numerical Methods for Stokes and Navier-Stokes Equations
• VIGRE Seminar in Numerical Methods and Algorithms
• VIGRE Seminar in Introduction to Scientific Computing
• The Construction and Analysis of Preconditioners
• Multigrid Methods
• Mixed Finite Element Methods
• Computational Software for Large-Scale PDE Solvers
• Seminar in Numerical Methods for Maxwell's Equations
• Numerical Methods for Fluid Flows
• Upscaling/Multiscale Methods for Partial Differential Equations and Applications to Porous Media Flows

MATH 609. Numerical Analysis (Credit 4) (Fall every year)
Interpolation, numerical evaluation of definite integrals, and solution of ordinary differential equations, stability and convergence of methods and error estimates.
Prerequisite: Knowledge of computer programming (C or FORTRAN).

MATH 610. Numerical Methods in Partial Differential Equations (Credit 4) (Spring every year)
Introduction to finite difference and finite element methods for solving partial differential equations; stability and convergence of methods and error bounds.
Prerequisite: MATH 417 or MATH 609 or their equivalent.

MATH 639. Iterative Techniques (Credit 4) (Spring 1998, Spring 2000, Spring 2002, Spring 2004)
Numerical methods for solving linear and nonlinear equations and systems of equations; eigenvalue problems.
Prerequisite: Elementary linear algebra and knowledge of computer programming (C or FORTRAN).

MATH 660. Computational Linear Algebra (Credit 3) (Spring 1997, Fall 1999, Fall 2001, Fall 2003, Fall 2005)
Techniques in matrix computation; elimination methods, matrix decomposition, generalized inverses, orthogonalization and least-squares, eigenvalue problems and singular value decomposition, iterative methods and error analysis. Cross-listed with CPSC 660.
Prerequisites: MATH 417 or equivalent or CPSC 442 or equivalent.


MATH 661. Mathematical Theory of Finite Element Methods (Credit 3 or 4) (Fall 1999, Fall, 2001, Fall 2003, Fall 2005)
Basic mathematical theory of finite element methods; construction of finite element spaces, polynomial approximation in Sobolev space, interpolation error estimates, n-dimensional variational problems, finite element convergence estimates, interpolated boundary conditions, nonconforming finite element methods and numerical integration.

MATH 663. Technique of Finite Element Methods (Credit 3) (Spring 2003 Bramble)
The course is concerned with some of the basic mathematical tools which are useful in the understanding and development of numerical methods for partial differential equations. The course is intended primarily for mathematics graduate students specializing in scientific computation and mathematically sophisticated engineers and scientists.
Prerequisites: MATH 609 & 610 or equivalent.

MATH 689. Preconditioning Techniques in Finite Element Methods (Credit 3) (Spring 2002 Bramble)
Preconditioning is a critical aspect for the development of efficient computational algorithms for large scale scientific computations. This course will cover a broad range of techniques for the construction and analysis of preconditioners for the discrete systems which arise from finite element approximations of boundary value problems.
Prerequisites: MATH 609 & 610 or equivalent.

MATH 663. Numerical Methods for Stokes and Navier-Stokes Equations (Credit 3) (Spring 2001 Bramble)
This course will cover for the most part the basic mathematical theory of the linear stationary Stokes equations and the finite element methods for their numerical approximation. It is intended primarily for mathematics graduate students specializing in scientific computation and mathematically sophisticated engineers and scientists.
Prerequisites: Approval of instructor.

MATH 663. VIGRE Seminar in Numerical Methods and Algorithms (Credit 3) (Fall 2000 Lazarov, Pasciak)
This course will be focused on numerical algorithms and large scale scientific computation. The goal is to first study classical numerical algorithms as presented in Numerical Recipes. Our aim will be to survey many classical algorithms along with enough background to understand the basic principles on which they are based. In particular, we plan to cover many of the "so-called" algorithms of the 20'th Century (SIAM NEWS 33 #4, 2000). The second phase of the course is more project oriented. Students will be required to undertake a more detailed study in some computational algorithm or problem. Alternatively, more advanced students can suggest their own projects or tailor their projects in a direction more consistent with their thesis research. The projects will supplement the material of the class and students will be required to give a description, background information and present their results.
Prerequisite: The course is designed to be accessible to undergraduates so only advanced calculus and linear algebra are required. In addition, the project aspect of the course requires some programming background. The preferred languages, from most to least, are C++, C, Fortran and Matlab.

MATH 663. VIGRE Seminar in Introduction to Scientific Computing (Credit 3) (Spring 2001 Schoberl, Lazarov, Pasciak)
This course will provide an introduction to Scientific Computing methods for the solution of partial differential equations. The main applications are in computational science and engineering although some examples are also applicable in finance, economics, biology, etc. There will be three major aspects of the course. The first part illustrates the modeling of problems from physics and engineering in terms of partial differential equations. Next, basic approximation methods utilizing finite difference, finite element, and spectral approximation will be studied. Finally, various software tools and programming methodologies will be surveyed. Further applications will be introduced as project reports given by students. We should point out the differences between this course and MATH 610. This course is more of a technology course. While it is necessary to cover some mathematical concepts in the development of the algorithms, an in depth theoretical development will be avoided. Such a theoretical approach is taken in MATH 610, a course which is focused on preparing students for the numerical analysis qualifying exam.

MATH 689. The Construction and Analysis of Preconditioners (Credit 3) (Spring 2000 Pasciak)
Algebraic preconditioners including modified incomplete factorization, hierarchical multigrid, and algebraic multilevel. Analytic preconditioners based on non-overlapping domain decomposition, Schwartz methods and two level methods.
Prerequisites: MATH 610, MATH 639 or approval of instructor.

MATH 689. Multigrid Methods (Credit 4) (Fall 1998 Bramble, Fall 2000 Pasciak)
This course will include multilevel methods, basic iterative methods, interpolation spaces, smoothing operators, multigrid algorithms (for nested and non-nested spaces, V-cycle, W-cycle), anisotropic problems, multilevel characterization of Sobolev spaces, full multigrid, multigrid for non-nested spaces application to various types of boundary value problems.

MATH 689. Mixed Finite Element Methods (Credit 4 Spring 1999 Pasciak, Credit 3 Spring 2005 Pasciak)
This course will consider the approximation of boundary value problems in partial differential equations by mixed finite element methods. Such techniques can be used on quadratic minimization problems with linear constraints which include a number of engineering applications such as incompressible flows, nearly incompressible elasticity, thin plates modeling, etc.

MATH 664 + MATH 694. Computational Software for Large-Scale PDE Solvers (Credit 3+1) (Spring 2006 Bangerth)
Software for the finite element simulation of realistic phenomena and with a complexity corresponding to today's state of the art tends to be very large. Typical codes have several 100,000 lines of code, offer fully adaptive meshes in 2d and 3d, many different finite elements, and support for complex solvers. Writing such codes is beyond what a student can do within her graduate time, but there are open source libraries available that offer all this functionality and allow to quickly build sophisticated finite element solvers for new equations. This course is an introduction to the use of the deal.II software library (see http://www.dealii.org/) that provides all the building blocks for complex, large scale finite element software. An important part of the course is to give students hands-on experience in programming with this software, through projects that are related to their graduate work. The goal is to learn how to use a modern software environment and to develop applications that are applicable to a student's graduate project also beyond the end of this course.
Prerequisites: MATH 609 & 610 or equivalent, knowledge of computer programming in C++.

MATH 664. Seminar in Numerical Methods for Maxwell's Equations (Credit 3) (Fall 2006 Bramble, Pasciak)
The subject of the seminar will be Maxwell's equations with an emphasis on matters related to their numerical approximation. This could include scattering problems for electromagnetic and acoustic waves and issues of absorbing boundary conditions to approximate conditions at infinity.

MATH 689. Numerical Methods for Fluid Flows (Credit 3) (Fall 2006 Guermond)
The objective of this course is to introduce advanced techniques for approximating fluid flow problems. This is a graduate class in mathematics and students taking this class should have notions of fluid mechanics and some knowledge of numerical analysis. This is a proof-based class. Algorithms will be introduced and convergence theorems will be given (and possibly proved). The course will be organized as follows. The conservation laws of fluid mechanics will be recalled and model problems will be identified. The model problems that will be studied in this course are:
1) Time-independent incompressible Stokes equations. The velocity/pressure coupling will be analyzed and the notion of compatible approximation spaces will be introduced. These notions will be illustrated using finite differences and finite elements.
2) Time-dependent compressible Navier-Stokes. The notion of operator splitting and velocity/pressure uncoupling will be investigated. The notions of semi-discrete, fully discrete, fully implicit, semi-implicit, and explicit schemes will be introduced.
3) Transport equations. Up-winding and artificial viscosity methods will be analyzed. More elaborate higher order methods will also be studied: method of characteristics, stabilization methods (Discontinuous Galerkin, Galerkin/Least-Square).
4) Nonlinear conservation laws. The notion of weak solutions and entropy solutions will be introduced. Elementary one-dimensional schemes will be discussed and convergence to entropy solutions will be proven.
Prerequisites: An undergraduate course in fluid mechanics, MATH 417 (or equivalent), MATH 412 (or equivalent).

MATH 689. Upscaling/Multiscale Methods for Partial Differential Equations and Applications to Porous Media Flows (Credit 3) (Spring 2007 Efendiev)
The course will cover analytical and numerical techniques for multiscale problems. Homogenization methods and their applications to numerical simulations of multiscale problems will be discussed. Various finite element/volume based numerical homogenization/multiscale methods as well as multiscale numerical methods for problems without scale separation will be covered in this course.
Prerequisites: Instructor's approval.