Suppose that is smooth and that the boundary value problem
has unique solution.
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.
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 .
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?
where denotes the Jacobian of a matrix .
If exists, then iterates are defined.
Convergence of sequnce
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:
- close to solution
Consider the boundary value problem
with boundary conditions and . Here is a given positive number.
Describe a Galerkin method to solve this problem using piecewise linear functions with respect to a uniform mesh.
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:
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
Discrete Weak Formulation
Find such that for all
Since forms a basis, we have
In matrix form
Consider the linear multistep method
for the solution of the initial value problem
Show that the truncation error is of order 2.
State the condition for consistency of a linear multistep method and verify it for the scheme in this problem.
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