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

  1. 01:05-01:25
    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 L2 and Linf errors as a function of mesh size. All the computations were done using the scripting interface of FEMLAB 3.1i software. When the L2 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 L2 and Linf in terms of convergence order.

  2. 01:30-01:50
    Boundary Condition Homogenization for the Immersed Boundary Method
    Alex Szatmary
    The Immersed Boundary Method, initially developed by Peskin, has been successful in simulations of fluid-structure interaction, especially in biological applications. The Stokes equation is solved by a spectral projection method, using Fourier series as basis functions. When applying non-periodic 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 non-periodic 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.

  3. 01:55-02:15
    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.

  4. 02:20-02:40
    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.