Numerical Methods Qualification Exam Problems and Solutions (University of Maryland)/Aug05 667
Problem 4aEdit
Given a smooth function , state the secant method for the approximate solution of nonlinear equation in 
Solution 4aEdit
Problem 4bEdit
State the order of convergence for this method, and explain how to derive it. 
Solution 4bEdit
If is bounded and is close to , then the secant method has convergence order (the Golden ratio).
A partial proof of this can be found here
Problem 4cEdit
Are there situations in which the order of convergence is higher? Explain your answers. 
Solution 4cEdit
Problem 5Edit
Consider the initial value problem

Problem 5aEdit
Write the ODE in integral form and explain how to use the trapezoidal quadrature rule to derive the trapezoidal method with uniform time step :

Solution 5aEdit
Problem 5bEdit
Define the concept of absolute stability. That is, consider applying the method to the case with real . Show that the region of absolute stability contains the entire negative real axis of the complex plane. 
Solution 5bEdit
Letting , we have
If we let , and rearrange the equation we have
We require . This is true if is a negative real number.
Problem 5cEdit
Suppose that where is a symmetric matrix and . Examine the properties of which guarantee that the method is absolutely stable (Hint: study the eigenvalues of ). 
Solution 5Edit
We now want instead
i.e.
or (since is symmetric)
or (since multiplying by orthogonal matrices does not affect the norm)
or (by definition)
If is negative definite (all its eigenvalues are negative), the above inequality holds.
Problem 6Edit
Consider the following twopoint boundary value problem in

Problem 6aEdit
Give a variation formulation of (1), i.e, express it it as

Solution 6aEdit
Variational FormEdit
Derive the variational form by multiplying by test function and integrating from 0 to 1. Use integration by parts and substitute initial conditions to then have:
Find such that for all
Relationship between (1) and (2)Edit
(2) is an equivalent formulation of (1) but it does not involve second derivatives.
Existence of Unique SolutionEdit
By the LaxMilgram theorem, we have the existence of a unique solution.
 bilinear form continuous/bounded:
 bilinear form coercive:
 functional bounded:
Problem 6bEdit
Write the finite element method with piecewise linear elements over a uniform partition with meshsize . If is the vector of nodal values of the finite element solution, find the stiffness matrix and right hand side such that . Show that is symmetric and positive definite. Show that solution is unique 
Solution 6bEdit
Define hat functions as basis of the discrete space. Note that and have only half the support as the other basis functions. Using this basis we have
Observe that is symmetric. It is positive definite by Gergoshin's theorem. The solution is unique since is diagonally dominant.
Problem 6cEdit
Consider two partitions and of , with a refinement of . Let and be the corresponding piecewise linear finite element spaces. Show that is a subspace of 
Solution 6cEdit
If then it is also in since is a refinement of . In other words, since is piecewise linear over each intervals, it is also piecewise linear over a refinement of its interval.
Problem 6dEdit
Let and be the finite element solutions. Show the orthogonality equality.

Solution 6Edit
From orthogonality of error, we have
for all
Specifically,
Then