Numerical Methods Qualification Exam Problems and Solutions (University of Maryland)/Aug06 667

Problem 4Edit

Suppose that   is smooth and that the boundary value problem


 


has unique solution.

Problem 4aEdit

For  , let  . Write down a system of   equations to obtain an approximation   for the solution   at   by replacing the second derivatives by a symmetric difference quotient.

Solution 4aEdit

The symmetric difference quotient is given by


 


Hence we have the following system equations that incorporates the initial conditions  .


 

Problem 4bEdit

Write the system of equations in the form  . Define domain and range of   and explain the meaning of the variable  .



Solution 4bEdit

 


Domain:  


Range:  


  is a vector containing   approximations   for the solution   at  


Problem 4cEdit

Formulate Newton's method for the solution of the system in (b) with  . Give explicit expressions for all objects involved (as far as this is reasonable). Determine a sufficient condition that ensures that the iterates   in the Newton scheme are defined. Without doing any further calculations, can you decide whether the sequence   converges. Why or why not?


Solution 4cEdit

Newton's MethodEdit

 


where   denotes the Jacobian of a matrix  .


Specifically,


 

Sufficient ConditionEdit

If   exists, then   iterates are defined.


Convergence of sequnceEdit

We cannot decide if the sequence converges since Newton's method only guarantees local convergence.


In general, for local convergence of Newton's method we need:


  •   differentriable


  •   invertible


  •   Lipschitz


  •   close to solution  

Problem 5Edit

Consider the boundary value problem


 


with boundary conditions   and  . Here   is a given positive number.


Problem 5aEdit

Describe a Galerkin method to solve this problem using piecewise linear functions with respect to a uniform mesh.

Weak FormulationEdit

Find   such that for all  


 


which after integrating by parts and plugging in initial conditions we have


 


Let   be the nodes of a uniform partition of   where   and  .


Let   be the standard "hat" functions defined as follows:


For  


 


 


Also   since  


Then   forms a basis for the discrete space  

Problem 5bEdit

Derive the matrix equations for this Galerkin method. Write out explicitly that equation of the linear system which involves  


Discrete Weak FormulationEdit

Find   such that for all  


 


Since   forms a basis, we have


 


Also for  


 


In matrix form

 

Problem 6Edit

Consider the linear multistep method


 


for the solution of the initial value problem  

Problem 6aEdit

Show that the truncation error is of order 2.


Problem 6bEdit

State the condition for consistency of a linear multistep method and verify it for the scheme in this problem.

Solution 6bEdit

 


Conditions:


(i)  


 


(ii) 


 

Problem 6cEdit

Does the scheme satisfy the root condition and or the strong root condition?


The scheme satisfies the root condition but not the strong root condition since the roots are given by


 


which implies   and