Math 621  Numerical Analysis II
Spring 2006  Matthias K. Gobbert
Class Presentations of the Final Projects
Thursday, May 18, 2004, 01:00 p.m., MP 401

01:0501:25
Convergence Study of the Finite Element Solution of the
Bioheat Equation Using FEMLAB
Maher D. Salloum
The fundamental bioheat equation is solved using the finite element method
on a tapered cylindrical domain. The goal of this paper is to study the
convergence of the finite element solution. Both the transient and steady state
equation are studied in terms of the
L^{2} and
L^{inf} errors
as a function of mesh size. All the computations were done using the
scripting interface of FEMLAB 3.1i software.
When the L^{2} error norm is used,
the results show good agreement between the
calculated and theoretical values of the order of convergence.
The results also show a similar behavior between
L^{2} and
L^{inf} in terms of convergence order.

01:3001:50
Boundary Condition Homogenization for the Immersed Boundary Method
Alex Szatmary
The Immersed Boundary Method, initially developed by Peskin, has been
successful in simulations of fluidstructure interaction, especially
in biological applications. The Stokes equation is solved by a
spectral projection method, using Fourier series as basis functions.
When applying nonperiodic boundary conditions, the discrepancy
between the basis functions and the boundary conditions leads to error
and discontinuities. A method is proposed for applying a linear
velocity gradient while avoiding the disadvantages of the resulting
nonperiodic boundary conditions. The new technique is demonstrated to
decrease error by a factor of at least 1000, while eliminating the
discontinuities previously observed near the boundaries.

01:5502:15
Efficient Leaping Methods for Stochastic Chemical Systems
Ioana Cipcigan
Well stirred chemical reaction systems which involve small
numbers of molecules for some species have a stochastic behavior and can be
modeled by a continuous time, discrete state Markov process. An exact
method for simulating the time evolution of the system is the Stochastic
Simulation Algorithm, but this method is extremely slow for realistic
biological systems. We present some efficient numerical methods for
simulating these systems: an adaptive leaping method based on local error
formulas and an implicit method, and we compare these methods with the exact
method using some numerical examples.
This is joint work with my advisor, Dr. Muruhan Rathinam, Department of
Mathematics and Statistics, UMBC.

02:2002:40
A Review of Discontinuous Galerkin Methods
for Hyperbolic and Elliptic Problems
Alen Agheksanterian
The Discontinuous Galerkin Method (DGM) has become
increasingly popular in solving problems with discontinuities.
DGMs have been developed
and studied for hyperbolic, parabolic, and elliptic
problems. In this paper we conduct a survey of
DGM by examining the applications of DGM in several problems.