**ENM 540. Topics in Computational Science and Engineering (Spring 2007)**
**Lecture: M.W. 3 PM. – 4:30 PM **
**Music 303 **
• **Instructor:**
Howard Hu,
Towne 241, Phone: 898-8504,
E-mail: __hhu@seas__
Office hours: T.R.: 3-4 pm / by appointment
• **Prerequisite knowledge and/or skills**: Background in ordinary and partial differential equations; proficiency in a programming language such as MATLAB, C, or Fortran.
• **Description**: This course focused on techniques for numerical solutions of ordinary and partial differential equations. The content will include: algorithms and their analysis for ODEs; finite element analysis for elliptic, parabolic and hyperbolic PDEs; approximation theory and error estimates for FEM.
• **Evaluation**:
Learning of the course material will be evaluated by grading of homework, and two independent projects (midterm and final). The projects and each homework assignment will be given a numerical grade that is combined to form a cumulative score. The cumulative score is based on the following weights
50% Homework
25% Midterm Exam (or Project)
25% Final Exam (or Project)
• **References**:
E.B. Becker, G.F. Carey, and J.T. Oden, *Finite Elements: An Introduction*, vol.I, Prentice Hall, Englewood Cliffs, 1981.
J. Stoer, R. Bulirsch, *Introduction to Numerical Analysis*, Springer-Verlag, New York, 1983.
T.J.R. Hughes, *the Finite Element Method, Linear Static and Dynamic Finite Element Analysis; *Prentice-Hall, 1987.
Y. Saad, ``*Iterative methods for Sparse Linear Systems*'', PWS Publishing Comp., 1996.
Class Notes
• **Course Outline:**
week 1,2
Error analysis
first order ODEs, initial value problem, uniqueness, existence
Euler method, higher order methods, Runge-Kutta methods
Convergence, error estimate, influence of rounding errors
week 3
classification of PDEs motivation, examples
BCs, maximum-minimum principle, well-posed problems
week 4
finite difference method, discretization
consistency, stability, convergence rate, efficiency
week 5,6
vector space, inner product space, Hilbert space,
strong/weak formulation, positive definite operators, Lax-Milgram theorem
a model problem
week 7
one-dimensional finite element methods
Galerkin’s method and extremal principles, finite element approximation
Piecewise Lagrange approximation, interpolation errors, matrix formulation
week 8
multi-dimensional finite element approximations
function spaces and approximation, finite element spaces
week 9, 10
approximation theorems,
convergence theory
error estimates for FEM, mesh quality
adaptive FEM method
week 11
sparse matrix, linear iterative solvers
preconditioning, matrix free methods
week 12
FEM for parabolic problems
semi-discrete Galerkin method
space-time finite element methods
convection-diffusion systems
week 13
FEM for hyperbolic problems
flow problems and upwind weighting
artificial diffusion, streamline weighting
discontinuous Galerkin method |