[petsc-dev] rename SNES methods ls, tr etc

[Anton Popov]

On 12/5/12 4:37 AM, Jed Brown wrote:
>
> So you start with the quasi-linear form
>
> F(u) := A(u) u - b = 0
>
> then we can rewrite the iteration
>
> w = A(u)^{-1} b
>
> in defect-correction form
>
> w = u - A(u)^{-1} F(u)
>
> because
>
> A^{-1} F(u) = A^{-1} (A u - b) = u - A^{-1} b
>
>
Jed,

Of course that is correct. The only advantage of using non 
defect-correction form is straightforward implementation of non-zero 
Dirichlet boundary conditions. As usual, one would just remove (or zero 
out) corresponding rows, multiply column coefficients with defined 
values and subtract from RHS. In defect correction form one at least 
needs to distinguish first iteration. Skipping correction of RHS in this 
case most likely will cause convergence problems.

Picard is equal to Newton only when following conditions hold:

1) F(u)  = R(u) which is true residual.
2) A(u) =  J(u) which is true Jacobian (does not necessarily follow from 
the first condition).

In more likely case Picard is equal to Newton with A(u) approximate 
Jacobian, or simply not applicable because residual cannot be expressed 
in a form of a linear operator (as was discussed before by you and Barry).

For general problem with zero initial guess, and non-zero Dirichlet BC I 
would do exactly one iteration of Picard with corrected RHS (no 
difference between defect/non defect-correction forms, no matter whether 
A(u) or J(u), if available), and then switch to Newton.

Anton


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