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

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Problem 4[edit]

Suppose that is smooth and that the boundary value problem



has unique solution.

Problem 4a[edit]

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 4a[edit]

The symmetric difference quotient is given by



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


Problem 4b[edit]

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



Solution 4b[edit]


Domain:


Range:


is a vector containing approximations for the solution at


Problem 4c[edit]

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 4c[edit]

Newton's Method[edit]


where denotes the Jacobian of a matrix .


Specifically,


Sufficient Condition[edit]

If exists, then iterates are defined.


Convergence of sequnce[edit]

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 5[edit]

Consider the boundary value problem



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


Problem 5a[edit]

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

Weak Formulation[edit]

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 5b[edit]

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


Discrete Weak Formulation[edit]

Find such that for all



Since forms a basis, we have



Also for



In matrix form

Problem 6[edit]

Consider the linear multistep method



for the solution of the initial value problem

Problem 6a[edit]

Show that the truncation error is of order 2.


Problem 6b[edit]

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

Solution 6b[edit]


Conditions:


(i)



(ii)


Problem 6c[edit]

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