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

  1. 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.

  2. 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.

  3. 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.

  4. 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.

  5. 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

  1. 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.

  2. 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.

  3. 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.

  4. 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.