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Math 228a, Fall 2012: Problem Set 01
Exercise 1 Convert the second-order equation
y + 2y + y = f (t)
into a 2 2 rst-order system y = Ay , where A is a 2 2 matrix. Evaluate
the matrix exponential etA analytically. (Note that A may not be diagonalizable;
Iterative methods for linear systems
Chris H. Rycroft
November 20th, 2007
Introduction
For many elliptic PDE problems, nite-dierence and nite-element methods are the techniques of choice. In a nite-dierence approach, we search for a solution uk on a set o
UCB Math 228A, Fall 2014: Homework Set 2
Due September 29, 2014
1. The matlab script poisson.m solves the Poisson problem on a square
mm grid with x = y = h, using the 5-point Laplacian. It is set up to
solve a test problem for which the exact solution is
UCB Math 228A, Fall 2014: Homework Set 1
Due September 15, 2014
1. Read the paper, Eror Analysis of the Bjrck-Pereyra Algorithms for Solvo
ing Vandermonde Systems, by N. J. Higham, in Numerische Mathematik
vol. 50, pp. 613632, 1987. Implement a Matlab pro
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Week 01: 28 August 2008
These notes will cover the numerical solution of initial value problems (IVPs)
and boundary value problems (BVPs) for ordinary dierential equations
(ODEs) and ordinary dierential-algebraic equations (DAEs). Well review
the basic
UCB Math 228A, Fall 2014: Homework Set 3
Due Oct. 20, 2014
1. The m-le iter bvp Asplit.m implements the Jacobi, Gauss-Seidel, and SOR matrix splitting methods on the linear system arising from the boundary value problem u (x) = f (x) in one space dimensio
UCB Math 228A, Fall 2014: Homework Set 4
Due Nov. 3, 2014
Code Submission: E-mail all requested and supporting MATLAB les to
Luming at lwang@berkeley.edu as a zip-le named lastname rstname 4.zip,
for example luming wang 4.zip.
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2
3
4
UCB Math 228A, Fall 2014: Homework Set 5
Due Nov. 24, 2014
Code Submission: E-mail all requested and supporting MATLAB les to
Luming at lwang@berkeley.edu as a zip-le named lastname rstname 5.zip,
for example luming wang 5.zip.
1
2
3
4
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Math 228a, Fall 2012: Problem Set 06
Exercise 1 Derive the order conditions for a 3-stage third-order explicit
Runge-Kutta method and show that they are satised by the Heun method
0000
1
1
00
3
3
2
2
030
3
1
3
04
4
and the Kutta method
0
0
1
2
1
2
00
00
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Math 228a, Fall 2012: Problem Set 05
Exercise 1 Imitate the proof of convergence for Eulers method to prove
second-order convergence of the midpoint rule
un+1 = un + hf
un+1 + un
.
2
Assume that any nonlinear equation can be solved exactly.
A function f
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Math 228a, Fall 2012: Problem Set 04
Exercise 1 Look up the nine-point Laplacian 9 in LeVeque. Show that 9
is a linear combination 9 = 5 + 5 of the ve-point Laplacian 5 and
5 , which denotes 5 rotated by 45 . Show that the local truncation error
of any
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Math 228a, Fall 2012: Problem Set 03 (Revised 9 September)
Exercise 1 Write an integral equation solver for two-point boundary value
problems, following the scheme of the handout. Use an equidistant mesh of
N intervals covering [a, b]. Use the 10-point
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Math 228a, Fall 2012: Problem Set 02
Exercise 1 Write a mstep deferred correction code. First solve the midpoint rule Lh u = f for the uncorrected secondorder solution u on an equidistant mesh of N points on the interval [a, b]. Save the matrix Lh in fa
The Finite Element Method Lecture Notes
Per-Olof Persson
persson@berkeley.edu
April 23, 2013
1
1.1
Introduction to FEM
A simple example
Consider the model problem
u (x) = 1, for x (0, 1)
u(0) = u(1) = 0
(1.1)
(1.2)
with exact solution u(x) = x(1 x)/2. Fin