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Numerical Methods Qualification Exam Problems and Solutions (University of Maryland)/Aug04 667

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Problem 5a

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State Newton's method for the approximate solution of



where is a real-valued function of the real variable

Solution 5a

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Problem 5b

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State and prove a convergence result for the method.

Solution 5b

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Problem 5c

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What is the typical order of convergence? Are there situations in which the order of convergence is higher? Explain your answers to these questions.

Solution 5c

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The typical order of local convergence is quadratic.


Consider the Newton's method as a fixed point iteration i.e.:



Then




Expanding around gives an expression for the error



Note that if , then we have better than quadratic convergence.

Problem 6

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Consider the boundary value problem


Problem 6a

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Derive a variational formulation for (1).

Solution 6a

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Find such that for all


Problem 6b

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What do we mean by Finite Element Approximation to

Solution 6b

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Let be a partition of . Choose a an appropriate discrete subspace and basis functions . Then



The coefficients can be found by solving the following system of equations:


For


Problem 6c

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State and prove an estimate for



Solution 6c

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Cea's Lemma:



In particular choose to be the linear interpolant of .


Then,



Alternative Solution 6c

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Let be a discrete mesh of with step size . Consider the following integral

.

For some , as is just a linear interpolation on this interval. Hence

.

Similarly, we can bound the norm of the error in the derivatives with . With such intervals we have

Problem 6d

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Prove the formula


Solution 6d

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Problem 7

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Consider the initial value problem



where satisfies the Lipschitz condition



for all . A numerical method called the midpoint rule for solving this problem is defined by



where is a time step and for . Here is given and is presumed to be computed by some other method.

Problem 7a

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Suppose the problem is posed on a finite interval where . Show directly,i.e., without citing any major results, that the midpoint rule is stable. That is show that if and satisfy



then there exists a constant independent of such that



for

Solution 7a

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Subtracting both equations, letting , and applying the Lipschitz property yields,



Therefore,


Problem 7b

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Suppose instead we are interested in the long term behavior of the midpoint rule applied to a particular example . That is, let be fixed and let so that the rule is applied over a long time interval. Show that in this case the midpoint rule does not produce an accurate approximation to the solution to the initial value problem.

Solution 7b

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Substituting into the midpoint rule we have,



or



The solution of this equation is given by



where or the roots of the quadratic



The quadratic formula yields



If is a small negative number, than one of the roots will be greater than 1. Hence, as instead of converging to zero since .