[petsc-users] Nested FieldSplit Implementation for FE using Fortran

[Lucia Barandiaran]

Dear PETSc community,

Our research group is employing the PETSc libraries within a 
Fortran-based finite element (FE) code to solve large-scale, coupled 
multi-physics problems. The systems of interest typically involve more 
than two primary fields, specifically displacements (*U*), water 
pressure (*W*), gas pressure (*G*), and temperature (*T*).

We have recently integrated the |FieldSplit|preconditioner into our 
solver pipeline and have achieved successful results on a 3D gas 
injection reservoir simulation model comprising approximately 130,000 
nodes. The indices for each field variable were defined using 
|ISCreateGeneral|, and the splits were established via 
|PCFieldSplitSetIS|(one split per variable). The following command-line 
options were utilized effectively for this case:

-ksp_type gmres -ksp_gmres_modifiedgramschmidt \
-pc_type fieldsplit -pc_fieldsplit_type schur \
-pc_fieldsplit_schur_fact_type lower \
-pc_fieldsplit_schur_precondition selfp \
-fieldsplit_U_ksp_type preonly \
-fieldsplit_U_pc_type hypre \
-fieldsplit_U_pc_hypre_type boomeramg \
-fieldsplit_U_pc_hypre_boomeramg_coarsen_type PMIS \
-fieldsplit_U_pc_hypre_boomeramg_strong_threshold 0.6 \
-fieldsplit_U_pc_hypre_boomeramg_max_levels 25
-fieldsplit_G_ksp_type preonly \
-fieldsplit_G_pc_type hypre \
-fieldsplit_G_pc_hypre_type boomeramg \
-fieldsplit_G_pc_hypre_boomeramg_coarsen_type PMIS \
-fieldsplit_G_pc_hypre_boomeramg_strong_threshold 0.6 \
-fieldsplit_G_pc_hypre_boomeramg_max_levels 25

Subsequently, we have also configured a case with three primary fields 
by combining *W*and *G*into a single monolithic *P*(pressure) variable, 
which was solved using a similar approach.

We are now interested in advancing our preconditioning strategy by 
implementing a *nested*(or recursive) Schur complement decomposition. 
Our objective is to define a hierarchical structure where, for instance, 
an outer Schur complement is first constructed, and then the primary 
split itself is solved using an inner Schur complement preconditioner.

We have encountered an example of this nested configuration in 
presentation slides for the PETSc example |ex31|(e.g., from [1]), which 
outlines a command-line structure similar to:

-ksp_type fgmres -pc_type fieldsplit -pc_fieldsplit_type schur
-pc_fieldsplit_0_fields 0,1 -pc_fieldsplit_1_fields 2
-pc_fieldsplit_schur_factorization_type upper
-fieldsplit_0_ksp_type fgmres -fieldsplit_0_pc_type fieldsplit
-fieldsplit_0_pc_fieldsplit_type schur
-fieldsplit_0_pc_fieldsplit_schur_factorization_type full
-fieldsplit_0_fieldsplit_velocity_ksp_type preonly
-fieldsplit_0_fieldsplit_velocity_pc_type lu
-fieldsplit_0_fieldsplit_pressure_ksp_rtol 1e-10
-fieldsplit_0_fieldsplit_pressure_pc_type jacobi
-fieldsplit_temperature_ksp_type gmres
-fieldsplit_temperature_pc_type lsc

We are currently seeking guidance on how to implement this nested 
|FieldSplit|functionality within our Fortran code. We were wondering if 
the source code for this specific |ex31|example is publicly available, 
or if you could direct us to any other Fortran examples that demonstrate 
the implementation of a recursive Schur complement preconditioner.

Any advice, references, or code examples you could provide would be 
immensely valuable to our research.

Thank you very much for your time and assistance.

Sincerely,

Lucía Barandiarán

Scientific software developer  - Dracsys

Collaborator at MECMAT group - Universitat Politècnica de Catalunya (UPC)

Reference:
[1] Knepley, M. G. (2016). /Advanced PETSc Preconditioning/. 
Presentation. Retrieved from 
https://urldefense.us/v3/__https://cse.buffalo.edu/*knepley/presentations/PresMIT2016.pdf__;fg!!G_uCfscf7eWS!cyBB84JdoI3gXz-X7IFeVHTUEuq5x6-N36gHKdnu1acRmIh47LSfDP3AqW10CDEpIUW2fWhiL5g7nsUDzIMbTGkWotRffgz__g$  
<https://urldefense.us/v3/__https://cse.buffalo.edu/*knepley/presentations/PresMIT2016.pdf__;fg!!G_uCfscf7eWS!cyBB84JdoI3gXz-X7IFeVHTUEuq5x6-N36gHKdnu1acRmIh47LSfDP3AqW10CDEpIUW2fWhiL5g7nsUDzIMbTGkWotRffgz__g$ >
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