Numerical Methods Qualification Exam Problems and Solutions (University of Maryland)/Aug08 667
- 1 Problem 4a
- 2 Solution 4a
- 3 Problem 4b
- 4 Solution 4b
- 5 Problem 5
- 6 Problem 5a
- 7 Solution 5a
- 8 Problem 5b
- 9 Solution 5b
- 10 Problem 5c
- 11 Solution 5c
- 12 Problem 5d
- 13 Solution 5d
- 14 Problem 5e
- 15 Solution 5e
- 16 Problem 5f
- 17 Problem 5g
- 18 Solution 5
- 19 Problem 6
- 20 Problem 6a
- 21 Solution 6a
- 22 Problem 6b
- 23 Solution 6b
- 24 Problem 6c
- 25 Solution 6c
Show that the two-step method
is of order 2 but does not satisfy the root condition.
Method of Order 2
Method finds an approximation for such that .
Let be the th point of evaluation where is the starting point and is the step size.
Taylor Expansion of y' about a_0
Substituting into the second term on the right hand side of and simplifying yields
Taylor Expansion of y about a_0
Since , we also take Taylor Expansion of about
Substituting and simplifying yields,
Take Difference of Taylor Expansions
Hence shows that (1) is a method of order 2.
Does Not Satisfy Root Condition
The Characteristic equation of (1) is
Giving the roots
clearly does not satisfy
Give an example to show that the method (1) need not converge when solving .
Let . Then . We have the difference equation
which has general solution (use the roots)
If , then as
Hence, . Therefore if , then .
Consider the boundary value problem
Prove that has at most one solution
Let and be solutions. Let .
By subtracting the two equations and their conditions we have
Multiplying by test function and integrating by parts from 0 to 1, we want to find such that for all
Let . Then, we have
Since , , and are all , . Hence .
Discretize the problem. Take a uniform partition of
The three point difference formula for is given by
Equations for i=2,...,n-2
Substituting into with our difference formula we have in matrix formulation
Equation for i=1
We can eliminate the variable by using the approximation
Using this relationship and the three difference formula, we have
Equations for i=n-1
Since , we can eliminate the variable by substituting into the n-1 equation.
Prove that the equation found in has a unique solution
Since the matrix is diagonally dominant, the system has unique solution.
Transform the problem into an equivalent problem with homogeneous boundary conditions.
Let ,a solution of the boundary value problem, be represented as the sum of solutions to two different boundary value problems i.e.
Suppose . Then
and which implies and hence
Substituting into , we then have
Since , we have
Therefore an equivalent boundary value problem with homegenous boundary conditions is given by
Obtain the variational formulation of the problem formulated in . Specify the Sobolev space involved. Prove that this problem has a unique solution, which we denote by .
Variational Formulation and Sobolev space
Using the problem's notation, we want to find such that for all , we have
The above comes from integrating by parts and applying the boundary conditions.
To show that is unique, we show that the hypothesis of the Lax-Milgram Theorem are met.
a(v,w) bounded and continuous
Consider the approximation of by piecewise linear finite elements. Define precisely the piecewise linear finite element subspace (use the partition (3)). Show that the finite element problem has a unique solution.
Show that and indicate how the constant depends on the derivatives of .
Let be a nonlinear function with zero :
Prove (4) is locally convergent.
is a fixed point iteration. By the contraction mapping theorem, if is a contraction in some neighborhood of then the iteration converges at least linearly.
We have to show there exists such that .
By the mean value theorem we have that , that is for some in our neighborhood of .
In particular, , implying that is a contraction and that the iterative method converges at least linearly.
Calculating the Jacobian
Show that the convergence is at least quadratic.
where satisfies when .
Then, we obtain .
Write the Newton iteration and compare it with (4)
The Newton iteration looks like this:
Where B is the inverse of the Jacobian of f.
That is, in the Newton Iteration gives (4).