Math 621 - Numerical Analysis II
Numerical Methods for Partial Differential Equations
Fall 2007 - Matthias K. Gobbert
Class Presentations of the Final Projects

Friday, December 14, 2007, 02:30 p.m., MP 401

  1. 02:30-02:45
    The Matrix-Free Preconditioned Conjugate Gradient Method: Demonstrated on a 2D Poisson Equation Example
    Amanda Gassman
    The existing Preconditioned Conjugate Gradient method in Matlab does not efficiently compute the numerical solution and therefore an optimized method was developed. The algorithm was optimized in several stages and tested repeatedly at each stage on a two dimensional Poisson problem to ensure that it produces the same numerical solution as the original Matlab function. Some of the memory saving techniques include: creation of a function which reuses some vectors, development of a matrix-free method of computing matrix-vector multiplications, and derivation of a matrix-free method that implements preconditioners on the system matrix. All techniques were combined to create a matrix-free Preconditioned Conjugate Gradient method, which requires less time and less memory than the original. Ultimately, convergence results confirm the accuracy of the new method and demonstrate its superiority.

  2. 02:45-03:00
    Parallelism of the Load Vector Calculation in a 3D Upscaling Elastic Wave Propagation Code
    Sean Griffith
    Solving the elastic wave equation often requires dealing with data on multiple scales. However, computational limits often require the use of a grid that is coarse relative to finer features of the domain. The upscaling technique adequately deals with this issue by breaking a large domain into subdomains, and solving the problem on each sub-domain separately. Each sub-domain solution is then used to solve the problem on the coarse scale. Using fine grid information to solve the coarse problem imparts some of the fine scale accuracy to the coarse solution. Additionally, this technique has the advantage that subdomain problems can be solved in parallel. In this presentation, it is seen that this parallelizability extends to the coarse grid solution as well, in particular to the calculation of the load vector needed to solve the coarse problem via finite elements.

  3. 03:00-03:15
    A Convergence Study of Finite Element Solutions Using True Solutions in COMSOL Multiphysics
    Shiming Yang
    Finite element solutions' L2-norm errors based on true solutions are studied for Poisson equations. Smooth and non-smooth test problems are solved with Lagrange elements of all polynomial degrees available in COMSOL Multiphysics, with respect to two types of domain shapes. The results agree with the convegence theorem, but need more explanation for certain cases. Results in this work leads up to the development and testing of a procedure based on a reference solution in cases where a true solution is not available.

  4. 03:15-03:30
    Numerical Studies of the Asymptotic Behavior of a Reaction-Diffusion System with a Fast Reaction
    Yushu Yang
    A system of reaction-diffusion equations with a fast reaction is studied with an initial condition with three interfaces between regions of dominance by the two main reactants. The efficiency and accuracy of the numerical solver is studied for two cases of faster reaction rate values. We observe that the asymptotic limit of the problem is already approached for the values considered. The talk will also report ODE order k and Newton iteration steps for studying behavior of the numerical solution.

  5. 03:30-03:45
    Numerical Solution of the Second Order Wave Equation
    Noemi Zakarias
    The second-order two-dimensional wave equation is solved analytically using traditional techniques in Partial Differential Equations and numerically using both the finite difference method (implemented in MATLAB) and COMSOL Multiphysics solver package based on the finite element method. Stability theory for the finite difference method and numerical experiments are presented. Accuracy and stability are confirmed for the leapfrog finite difference scheme (centered second differences in time and space) and the convergence order of the finite element solution is estimated in the L2-norm using the COMSOL Script feature. The results show agreement between the true and numerical solutions.

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This page version 1.0, December 2007.