[petsc-users] Dear Sir or Madam

[Smith, Barry F.]

  In general the convergence will be different; though one hopes the resulting answer is the same. The reason is that the upper diagonal pattern determines the order the solution entries are solved for. In the first case it solves for x3, then x2, then x1, in the second case it solves for x3, x1, x2 in that order. When solving for x2 in the second case it has a different right hand size and hence, of course, a different solution. There is, I think, some trial and error in determining what ordering is best for one particular problem.



> On Jun 5, 2019, at 9:56 PM, Guoyi Ke via petsc-users <petsc-users at mcs.anl.gov> wrote:
> 
> Dear Sir or Madam,
> 
> Recently, our research group try to use the PCfieldsplit  method in Petsc to construct the preconditioner for the linear equation system. For example, if we have following system,
> K11, K12, K13      u1      f1
>  K21, K22, K23      u2  =  f2                                     (1)
>  K31, K32, K33      u3      f3
>  
> we construct the preconditioner as follows:
>  K11, K12, K13     
>  0,     K22, K23      
>  0,       0,    K33.      
> 
> We also can reorder the system (1):
>  K22, K21, K23      u2      f2
>  K12, K11, K13      u1  =   f1                                  (2)
>  K32, K31, K33      u3       f3
> 
> with the preconditioner 
> K22, K21, K23      
>  0,    K11, K13     
>  0,    0,     K33
> 
> We use the same solver for Eq.(1) and Eq. (2) with corresponding preconditioner. The question is: is solving the Eq. (1) equivalent to solving Eq. (2) in Petsc? From our results, it seems not. Thank you so much.
> 
> Best,
> Guoyi 
> 
> 
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