Suppose that is smooth and that the boundary value problem
has unique solution.
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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.
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The symmetric difference quotient is given by
Hence we have the following system equations that incorporates the initial conditions
.
Write the system of equations in the form . Define domain and range of and explain the meaning of the variable .
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Domain:
Range:
is a vector containing
approximations
for the solution
at
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?
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where
denotes the Jacobian of a matrix
.
Specifically,
If
exists, then
iterates are defined.
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 ![{\displaystyle U^{*}\!\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f064ba19e43475132f8242c06114970349030a3)
Consider the boundary value problem
![{\displaystyle -u''=f,\quad 0<x<1\!\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1608c87943a632ce873128659c3957c9892d715d)
with boundary conditions and . Here is a given positive number.
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Describe a Galerkin method to solve this problem using piecewise linear functions with respect to a uniform mesh.
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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
Derive the matrix equations for this Galerkin method. Write out explicitly that equation of the linear system which involves
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Find
such that for all
Since
forms a basis, we have
Also for
In matrix form
Consider the linear multistep method
for the solution of the initial value problem
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Show that the truncation error is of order 2.
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State the condition for consistency of a linear multistep method and verify it for the scheme in this problem.
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Conditions:
(i)
(ii)
Does the scheme satisfy the root condition and or the strong root condition?
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The scheme satisfies the root condition but not the strong root condition since the roots are given by
which implies
and