Math 621 - Numerical Analysis II
Numerical Methods for
Partial Differential Equations
Fall 2003 - Matthias K. Gobbert
Project Presentations
This page can be reached via my homepage at
http://www.math.umbc.edu/~gobbert.
Program for Day 1
Monday, November 24, 2003, 05:30 p.m., MP 401
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05:35-05:45
The Finite Element Method for the
Elliptic Neumann Boundary-Value Problem
Bogdan Gavrea
The goal of this project is to perform some theoretical analysis
on the elliptic prototype problem with homogeneous Neumann boundary
conditions. The analysis will include the weak formulation of the
above problem, existence and uniqueness results and estimates for
the finite element solution. When homogeneous Neumann conditions
are used the estimates for the finite element solution are similar
to the ones obtained for the same problem with Dirichlet boundary
conditions. In the case of Neumann boundary conditions, in order to
prove existence and uniqueness of a weak solution,
some assumptions must be made on the domain and its boundary.
These needed assumptions are incorporated in a trace theorem
which will be presented as an auxiliary result that will be
used later in the proofs for existence and uniqueness.
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05:50-06:00
Convergence Proof of the Finite Element Method
for a Parabolic Prototype Problem
Kevin P. Allen
This project considers the heat equation with a homogeneous Dirichlet
boundary condition as parabolic prototype problem.
The equation is semi-discretized in space
by the finite element method. The error between the
approximation and the true solution to our equation is analyzed in the
L2-norm.
As a finer mesh is used, this error should tend to zero
demonstrating the convergence of the approximation to the true solution.
The convergence proof for the semi-discrete problem will be presented,
and the convergence theorem for the fully discretized problem
will also be discussed.
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06:05-06:15
Solution of a Simple System of Hyperbolic
Partial Differential Equations
Mark Breitenbach
The governing equations for the scalar conservation laws with Euler fluxes
can be written as a system of first order hyperbolic partial differential
equations (PDE). In this analysis, the 2-D Euler equations are presented
as a system of four PDE. Then, a coupled, linear, four-equation system in
2-D with far-field boundary conditions and an initial condition will be
presented and solved. The solution of the latter requires the simultaneous
diagonalization of the coefficient matrices for the spatial derivatives.
The aim of the discussion is to give the reader a sense of what physical
processes are modeled by scalar transport equations and how these systems
are solved.
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06:20-06:30
Solving Linear Advection Equation with FEMLAB
Zorayr Manukyan
The advection equation describes how a conserved quantity such as
potential temperature or momentum is carried along with a flow of air or
water. The linear advection equation is a very good example of how
numerical models and reality do not always agree. In this talk we will
discuss solution to advection equation with FEMLAB and will present
several simulations of advection equation with different (in terms of
regularity) initial data.
-
06:35-06:45
Solving Partial Differential Equations with Spectral Methods
Jonathan Desi
The purpose of the presentation is to give other students insight into
another way to solve partial differential equations (PDEs), namely using
spectral methods. The project will be presented as an introduction where
I will motivate some concepts and show some simple one-dimensional
examples. Then I will show how spectral methods can solve a PDE, namely
the Poisson equation along with some error analysis. This analysis will
show that better accuracy can be obtained over finite difference methods
but with some trade-offs.
Program for Day 2
Wednesday, November 26, 2003, 05:30 p.m., MP 401
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05:35-05:45
Solving the Convection-Diffusion Equation by Finite Differences
with a Stability Condition Obtained by Fourier Analysis
Zhibin Sun
For the convection-diffusion equation, we can obtain the numerical
solution by using forward-time, backward-space (for the convection
term) and central-space (for the diffusion term) finite difference
scheme. To determine this scheme's theoretical stability
condition, Fourier analysis is used. We use a numerical example to validate
this condition.
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05:50-06:00
Fourier Techniques in the Stability Analysis of Semi-Discrete
Finite Difference Schemes for the Heat Equation
Dan Wang
The Fourier transform is an important tool in
the stability analysis of numerical schemes for partial differential
equations. I restrict myself here to semi-discrete schemes for the
Cauchy problem of the heat equation in one spatial dimension.
One nice theorem will be reached, and two practical examples will be
used to show how the theorem works.
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06:05-06:15
Solving a One-Dimensional Nonlinear System of Parabolic Equations
with FEMLAB
Ana Maria Soane
A chemical process involving three reactive species is modeled by a nonlinear
system of three reaction-diffusion equations in one spatial dimension.
The problem is challenging numerically, because one reaction is much
faster than the other one.
The goal of this project is to illustrate how this system can be solved
numerically using FEMLAB.
Results both for the steady-state problem (using adaptivity
in FEMLAB) and for the transient problem will be shown.
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06:20-06:30
Using Finite Elements to Model the Performance Gains Attained by
Supplementing Insulation Shielding with Conductive Sink Shielding
Michael Muscedere
FE modeling is used to solve the 2D problem for the temperature
distribution within a rectangular planter box near the presences of an
insulated hot water source pipe. The three options proposes to moderate
the temperature below a tolerable level inside the planter are: (1)
Insulator shielding (IS) along the inside walls of the planter or (2)
Conductive Sink Shielding (CSS) along the inside walls extended to the
outside boundary or (3) A combination of the two. The system is modeled
using the homogeneous heat equation with Neumann boundary conditions along
the entire outside boundary of the system and a Dirchelet boundary
condition describing the constant temperature of the hot water pipe. Since
the system model admits a FE weak formulation, FEMLAB is used to simulate
the model. The simulation results conclude that IS decreases the max
temperature with loss of planting area. CSS does decrease the temperature
but less effectively than IS alone. Finally, a combiniation of the IS and
CSS reduces temperature to an acceptable level with less loss of planting
area than experienced with IS alone.
Copyright © 2003 by Matthias K. Gobbert. All Rights Reserved.
This page version 1.8, December 2003.