Numerical Methods Qualification Exam Problems and Solutions (University of Maryland)/Aug09 667

Problem 4Edit

Given the two-point boundary value problem

 

Problem 4aEdit

Set up the finite element approximation for this problem, based on piecewise linear elements in equidistant points. Determine the convergence rate in an appropriate norm

Solution 4aEdit

Let  

Find   such that for all  


 


or after integrating by parts and including initial conditions

 

Discrete Variational FormEdit

  piecewise linear  


  is basis for  ; 


For  


 

 

 

 

 


 


Find   such that for all  


 


Since   forms a basis


 


Therefore we have system of equations


For  


 


 


Convergence RateEdit

In general terms, we can use Cea's Lemma to obtain


 


In particular, we can consider   as the Lagrange interpolant, which we denote by  . Then,


 .


It's easy to prove that the finite element solution is nodally exact. Then it coincides with the Lagrange interpolant, and we have the following punctual estimation:


 

Problem 4bEdit

Explain whether   is necessary for the convergence in part (a).

Solution 4bEdit

If  , then the stiffness matrix is diagonally dominant and hence solvable.

Solution 4Edit

Problem 5Edit


Solution 5Edit

Problem 6Edit


Solution 6Edit