Let , suppose and assume is nonsingular . Consider the following iteration
Derive the following error equation for
Note the following identity
The error is given by
Let be a fixed matrix. Find conditions on B that guarentee local convergence. What rate of convergence do you expect and why?
Assume is invertible, is bounded, and is Lipschitz.
This implies local superlinear convergence.
Find sufficient conditions on for the convergence to be superlinear. What choice of corresponds to the Newton method and what rate of convergence do you expect?
Super linear convergenceEdit
Find condition for super linear convergenceEdit
Since , if
as , we have super linear convergence i.e.
Let be uniformly Lipschitz with respect to . Let be the solution to the initial value problem : . Consider the trapezoid method
Find a condition on the stepsize that ensures (1) can be solved uniquely for .
The implicit method can be viewed as a fix point iteration:
Define a local truncation error and estimate it. Examine the additional regularity of needed for this estimate.
Re-writing (1) and replacing we have a formula for consistency of order p:
For uniform stepsize h
Expanding in Taylor Series about gives
Prove a global error estimate for (1)
Consider the 2-point boundary value problem
with constants and . Let be a uniform partition of [0,1] with meshsize .
Use centered finite differences to discretize (2). Write the system as
and identify . Prove that A is nonsingular.
Using Taylor Expansions, we can approximate the second derivative as follows
We can eliminate two equations from the n+2 equations by substituting the initial conditions into the equations for and respectively.
We then have the system
is nonsingular since it is diagonally dominant.
Define truncation error and derive a bound for this method in terms of . State without proof an error estimate of the form
and specify the order s.
Local Truncation ErrorEdit
Bound for Local Truncation ErrorEdit
Derive Bound for Max ErrorEdit
Let , , and is the local truncation error.
Subtracting the two last equations gives
, that is the error has order 2.
Prove the following discrete monotonicity property: If is the solution corresponding to a forcing , for , and then componentwise.
Note that is a matrix and hence the discrete maximum principle applies. (See January 05 667 test for definition of matrix)
Discrete Maximum PrincipleEdit
If , then .
Specifically let , then which implies