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# PCLSC
Preconditioning for Schur complements, based on Least Squares Commutators 
## Options Database Key

- ***-pc_lsc_scale_diag -*** Use the diagonal of A for scaling





## Notes
This preconditioner will normally be used with `PCFIELDSPLIT` to precondition the Schur complement, but
it can be used for any Schur complement system.  Consider the Schur complement

```none
   S = A11 - A10 inv(A00) A01
```


`PCLSC` currently doesn't do anything with A11, so let's assume it is 0.  The idea is that a good approximation to
inv(S) is given by

```none
   inv(A10 A01) A10 A00 A01 inv(A10 A01)
```


The product A10 A01 can be computed for you, but you can provide it (this is
usually more efficient anyway).  In the case of incompressible flow, A10 A01 is a Laplacian; call it L.  The current
interface is to hang L and a preconditioning matrix Lp on the preconditioning matrix.

If you had called `KSPSetOperators`(ksp,S,Sp), S should have type `MATSCHURCOMPLEMENT` and Sp can be any type you
like (`PCLSC` doesn't use it directly) but should have matrices composed with it, under the names "LSC_L" and "LSC_Lp".
For example, you might have setup code like this

```none
   PetscObjectCompose((PetscObject)Sp,"LSC_L",(PetscObject)L);
   PetscObjectCompose((PetscObject)Sp,"LSC_Lp",(PetscObject)Lp);
```


And then your Jacobian assembly would look like

```none
   PetscObjectQuery((PetscObject)Sp,"LSC_L",(PetscObject*)&L);
   PetscObjectQuery((PetscObject)Sp,"LSC_Lp",(PetscObject*)&Lp);
   if (L) { assembly L }
   if (Lp) { assemble Lp }
```


With this, you should be able to choose LSC preconditioning, using e.g. ML's algebraic multigrid to solve with L

```none
   -fieldsplit_1_pc_type lsc -fieldsplit_1_lsc_pc_type ml
```


Since we do not use the values in Sp, you can still put an assembled matrix there to use normal preconditioners.


## References

- **** -*** Elman, Howle, Shadid, Shuttleworth, and Tuminaro, Block preconditioners based on approximate commutators, 2006.
- **** -*** Silvester, Elman, Kay, Wathen, Efficient preconditioning of the linearized Navier Stokes equations for incompressible flow, 2001.



## See Also
 `PCCreate()`, `PCSetType()`, `PCType`, `PC`, `Block_Preconditioners`, `PCFIELDSPLIT`,
`PCFieldSplitGetSubKSP()`, `PCFieldSplitSetFields()`, `PCFieldSplitSetType()`, `PCFieldSplitSetIS()`, `PCFieldSplitSetSchurPre()`,
`MatCreateSchurComplement()`, `MatCreateSchurComplement()`, `MatSchurComplementSetSubMatrices()`, `MatSchurComplementUpdateSubMatrices()`,
`MatSchurComplementSetAinvType()`, `MatGetSchurComplement()`

## Level
intermediate

## Location
<A HREF="PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/ksp/pc/impls/lsc/lsc.c.html#PCLSC">src/ksp/pc/impls/lsc/lsc.c</A>


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[Edit on GitLab](https://gitlab.com/petsc/petsc/-/edit/release/src/ksp/pc/impls/lsc/lsc.c)


[Index of all PC routines](index.md)  
[Table of Contents for all manual pages](/manualpages/index.md)  
[Index of all manual pages](/manualpages/singleindex.md)