Numerical Methods for Partial Differential Equations
Spring 2010
Instructor: Jan Mandel <jan.mandel@ucdenver.edu>
Web: http://math.ucdenver.edu/~jmandel/classes/7824s10
Text: 1. C.
Johnson, Numerical
solution of partial differential equations by the finite element method,
Dover, 2009 ISBN 978-0486469003 2. R.
J. LeVeque, Finite Difference Methods for Ordinary
and partial Differential Equations, SIAM 2007 ISBN 978-0-898716-29-0
Time: Monday and Wednesday 2:30-3:45
Approach: Mostly practical and
algorithm oriented, with rigorous analysis only when a simple analysis is
feasible. Otherwise, the analysis will often rely on approximate arguments and
computational experiments. The course will use Matlab
for assignments and projects.
Purpose: Basics of the finite element
methods as generally expected of students in computational mathematics who do
not specialize in finite elements. Finite difference methods as used in
practical finite difference codes. Prerequisite methods for ODEs
are included but in less detail than usually covered in Numerical analysis II
and Numerical ODE courses.
Description: Solving partial differential
equations numerically is one of the most frequent applications of mathematics
and computing in science and engineering, and it constitutes the vast majority
of applications running on supercomputers. This class will provide an introduction
into numerical methods for stationary and time-dependent problems from a
mathematical perspective. It is suitable as a complement to a finite elements
engineering course, the first course in the subject for a numerical analyst, as
well as an overview for mathematics and engineering graduate students.
Prerequisites: Working knowledge of normed vector spaces and some computer programming
experience. Previous courses in analysis, numerical analysis, or partial
differential equations, are helpful but not required.
Tentative topics: Finite element
methods - variational formulation of
Laplace equation, Sobolev spaces, basic error estimates, construction of
finite element spaces,
approximation properties, implementation, sparse matrices, applications
to elasticity and structures. Finite difference methods - 5-point scheme for
the Laplace equation, truncation error, solution by Fast Fourier Transform,
methods for time-dependent problems (Euler, Ruge-Kutta, Crank-Nicholson,
leapfrog), A-stability, heat equation, nonlinear problems,
consistence, stability, convergence, methods for advection problems, upwinding,
applications to fluid dynamics and level set methods.