Numerical Methods Qualification Exam Problems and Solutions (University of Maryland)/Jan04 667

Problem 4aEdit

Consider the boundary value problem


 


where  , and  . Formulate a difference method for the approximate solution of   on a uniform mesh of size  . Explain how   is approximated by a difference quotient

Solution 4aEdit

From Taylor expansion of   and   around  , we have


 


Let   be a uniform partition of   with step size  


Then for   we have


 


For  


 


For  


 

Problem 4bEdit

Suppose   and   in  . Formulate a finite element method for the approximate solution of   in this special case, again on a uniform mesh. Using the standard "hat functions" basis for the finite element space, write out the finite element equations explicitly. Show that if an appropriate quadrature formula is used on the right-hand side of the finiite element equations, they (the finite element equations) are the same as the finite difference equations.

Solution 4bEdit

 


Since we are integrating hat functions on the right hand side, an appropriate quadrature formula would be to take half of the midpoint rule. The regular midpoint rule would give double the actual integral value of a hat function.


Therefore  


Then the finite difference method and the finite element method yield the same matrix.

Problem 4cEdit

Show that the matrix in   is non singluar.

Solution 4cEdit

Since the matrix is diagonally dominant, it is non-singular.

To show that the matrix has a non-zero determinant, 2n elementary row operation can be used to show that

 

has the same determinant as


 

which is  .

Problem 5Edit

Consider the following dissipative initial value problem,


 


where   is smooth and satisfies  


Problem 5aEdit

Write the Backward Euler Method for (2). This gives rise to an algebraic equation. Explain how you would solve this equation.

Solution 5aEdit

Using Taylor Expansion we have


 


Thus we have Backwards Euler Method:


 


Let  

Problem 5bEdit

Derive an error estimate of the form


 


where  . Do this directly, not as an application of a standard theorem. (Note that there is no exponential on the right hand side.

Solution 5bEdit

Subtracting   and  , we have


 

Problem 6Edit


Solution 6Edit