MATH 610. Numerical Methods in Partial Differential Equations (Credit 4)
(Spring every year)
MATH 639. Iterative Techniques (Credit 4) (Spring 1998, Spring
2000, Spring 2002, Spring 2004)
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.
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 689. Preconditioning Techniques in Finite Element Methods (Credit
3) (Spring 2002 Bramble)
MATH 663. Numerical Methods for Stokes and Navier-Stokes Equations
(Credit 3) (Spring 2001 Bramble)
MATH 663. VIGRE Seminar in Numerical Methods and Algorithms (Credit
3) (Fall 2000 Lazarov, Pasciak)
MATH 663. VIGRE Seminar in Introduction to Scientific Computing
(Credit 3) (Spring 2001 Schoberl, Lazarov, Pasciak)
MATH 689. The Construction and Analysis of Preconditioners (Credit 3)
(Spring 2000 Pasciak)
MATH 689. Multigrid Methods (Credit 4) (Fall 1998 Bramble, Fall 2000 Pasciak)
MATH 689. Mixed Finite Element Methods
(Credit 4 Spring 1999 Pasciak, Credit 3 Spring 2005 Pasciak)
MATH 664 + MATH 694. Computational Software for Large-Scale PDE
Solvers (Credit 3+1)
(Spring 2006 Bangerth)
MATH 664. Seminar in Numerical Methods for Maxwell's Equations (Credit
3) (Fall 2006
Bramble, Pasciak)
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.
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.
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.
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.
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.
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.
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.
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++.
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.