[petsc-users] fixed point interations

[Dominik Szczerba]

On Sun, Nov 6, 2011 at 6:59 PM, Matthew Knepley <knepley at gmail.com> wrote:
> On Sun, Nov 6, 2011 at 5:52 PM, Dominik Szczerba <dominik at itis.ethz.ch>
> wrote:
>>
>> >>> I want to start small by porting a very simple code using fixed point
>> >>> iterations as follows: A(x)x = b(x) is approximated as A(x0)x = b(x0),
>> >>> then solved by KSP for x, then x0 is updated to x, then repeat until
>> >>> convergence.
>> >
>> > Run the usual "Newton" methods with A(x) in place of the true Jacobian.
>>
>> When I substitute A(x) into eq. 5.2 I get:
>>
>> A(x) dx = -F(x) (1)
>> A(x) dx = -A(x) x + b(x) (2)
>> A(x) dx + A(x) x = b(x) (3)
>> A(x) (x+dx) = b(x) (4)
>>
>> My questions:
>>
>> * Will the procedure somehow optimally group the two A(x) terms into
>> one, as in 3-4? This requires knowledge, will this be efficiently
>> handled?
>
> There is no grouping. You solve for dx and do a vector addition.
>
>>
>> * I am solving for x+dx, while eq. 5.3 solves for dx. Is this, and
>> how, correctly handled? Should I somehow disable the update myself?
>
> Do not do any update yourself, just give the correct A at each iteration in
> your FormJacobian routine.
>    Matt

OK, no manual update, this is clear now. What is still not clear is
that by substituting A for F' I arrive at an equation in x+dx (my eq.
4), and not dx (Petsc eq. 5.3)...

Dominik