Numerical Methods Qualification Exam Problems and Solutions (University of Maryland)/Jan08 667

Problem 4

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Consider the system

 .


Problem 4a

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Show that if the parameter   is chosen sufficiently small, then this system has a unique solution   within some rectangular region.

Solution 4a

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The system of equations may be expressed in matrix notation.

 


The Jacobian of  ,  , is computed using partial derivatives


 


If   is sufficiently small and   are restricted to a bounded region  ,


 


From the mean value theorem, for  , there exists   such that


 


Since   is bounded in the region   give sufficiently small  


 


Therefore,   is a contraction and from the contraction mapping theorem there exists an unique fixed point (solution) in a rectangular region.

Problem 4b

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Derive a fixed point iteration scheme for solving the system and show that it converges.  

Solution 4b

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Use Newton Method

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To solve this problem, we can use the Newton Method. In fact, we want to find the zeros of

 


The Jacobian of  ,  , is computed using partial derivatives


 


Then, the Newton method for solving this linear system of equations is given by


 

Show convergence by showing Newton hypothesis hold

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Jacobian of g is Lipschitz

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We want to show that   is a Lipschitz function. In fact,


 


and now, using that   is Lipschitz, we have


 

Jacobian of g is invertible

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Since   is a contraction, the spectral radius of the Jacobian of   is less than 1 i.e.


 .


On the other hand, we know that the eigenvalues of   are  .


Then, it follows that   or equivalently   is invertible.

inverse of Jacobian of g is bounded

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Since   exists,  . Given a bounded region (bounded  ), each entry of the above matrix is bounded. Therefore the norm is bounded.

norm of (Jacobian of g)^-1 (x_0) * g(x_0) bounded

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  since   and   is bounded.


Then, using a good enough approximation  , we have that the Newton's method is at least quadratically convergent, i.e,

 

Problem 5a

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Outline the derivation of the Adams-Bashforth methods for the numerical solution of the initial value problem  .

Solution 5a

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We want to solve the following initial value problem:  .


First, we integrate this expression over  , to obtain


 ,


To approximate the integral on the right hand side, we approximate its integrand   using an appropriate interpolation polynomial of degree   at  .


This idea generates the Adams-Bashforth methods.


 ,


where,   denotes the approximated solution,   and   denotes the associated Lagrange polynomial.

Problem 5b

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Derive the Adams-Bashforth formula

 

Solution 5b

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From (a) we have

  where  


Then if we let  , where h is the fixed step size we get

 


 


So we have the Adams-Bashforth method as desired

 

Problem 5c

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Analyze the method (1). To be specific, find the local truncation error, prove convergence and find the order of convergence.

Solution 5c

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Find Local Truncation Error Using Taylor Expansion

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Note that  . Also, denote the uniform step size   as h. Hence,


 


 


Therefore, the given equation may be written as


 

Expand Left Hand Side

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Expanding about  , we get


 

Expand Right Hand side

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Also expanding about   gives


 

Show Convergence by Showing Stability and Consistency

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A method is convergent if and only if it is both stable and consistent.

Stability

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It is easy to show that the method is zero stable as it satisfies the root condition. So stable.

Consistency

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Truncation error is of order 2. Truncation error tends to zero as h tends to zeros. So the method is consistent.

Order of Convergence

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Dahlquist principle: consistency + stability = convergent. Order of convergence will be 2.

Problem 6

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Consider the problem

 


Problem 6a

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Give a variational formulation of (2), i.e. express (2) as

 

Define the Space H, the bilinear form B and the linear functional F, and state the relation between (2) and (3).

Solution 6a

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Multiplying (2) by a test function and using integration by parts we obtain:


 


 


Thus, the weak form or variational form associated with the problem (2) reads as follows: Find   such that


  for all  


where  .

Problem 6b

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Let   be a mesh on   with  , and let

 .

Define the finite element approximation,   based on the approximation space  . What can be said about   the error on the Sobolev norm on  ?

Solution 6b

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Define piecewise linear basis

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For our basis of  , we use the set of hat functions  , i.e., for  


 

Define u_h

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Since   is a basis for  , and   we have


 .

Discrete Problem

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Now, we can write the discrete problem: Find   such that


  for all  


If we consider that   is a basis of   and the linearity of the bilinear form   and the functional  , we obtain the equivalent problem:


Find   such that


 


This last problem can be formulated as a matrix problem as follows:


Find   such that


 


where   and  .

Bound error

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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 6c

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Derive the estimate for  , the error in  . Hint: Let w solve

(#) : 

We characterize   variationally as

 .

Let   to get

 

Use formula (4) to estimate  .

Solution 6c

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Continuity of Bilinear Form

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Orthogonality of the Error

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 .

Bound for L2 norm of w

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Hence,


 

Bound for L2 norm of w

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From  , we have


 


Then,


 

Bound L2 Error

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Finally, from (#), we have that  . Then,


 ,


or equivalently,


 .

Problem 6d

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Suppose   is a basis for  . Show that

 

where   is the stiffness matrix.

Solution 6d

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We know that

 


where the substitution in the last lines come from the orthogonality of error.


Now,  


Then, we have obtained