Enm 540. Topics in Computational Science and Engineering (Spring 2007) Lecture: M. W. 3 Pm. – 4: 30 pm music 303

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ENM 540. Topics in Computational Science and Engineering (Spring 2007)
Lecture: M.W. 3 PM. – 4:30 PM

Music 303


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.


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)


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

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