xref: /petsc/src/snes/tutorials/ex73f90t.F90 (revision ceeae6b5899f2879f7a95602f98efecbe51ff614)

!
!  Description: Solves a nonlinear system in parallel with SNES.
!  We solve the  Bratu (SFI - solid fuel ignition) problem in a 2D rectangular
!  domain, using distributed arrays (DMDAs) to partition the parallel grid.
!  The command line options include:
!    -par <parameter>, where <parameter> indicates the nonlinearity of the problem
!       problem SFI:  <parameter> = Bratu parameter (0 <= par <= 6.81)
!
!  This system (A) is augmented with constraints:
!
!    A -B   *  phi  =  rho
!   -C  I      lam  = 0
!
!  where I is the identity, A is the "normal" Poisson equation, B is the "distributor" of the
!  total flux (the first block equation is the flux surface averaging equation).  The second
!  equation  lambda = C * x enforces the surface flux auxiliary equation.  B and C have all
!  positive entries, areas in C and fraction of area in B.
!

!
!  --------------------------------------------------------------------------
!
!  Solid Fuel Ignition (SFI) problem.  This problem is modeled by
!  the partial differential equation
!
!          -Laplacian u - lambda*exp(u) = 0,  0 < x,y < 1,
!
!  with boundary conditions
!
!           u = 0  for  x = 0, x = 1, y = 0, y = 1.
!
!  A finite difference approximation with the usual 5-point stencil
!  is used to discretize the boundary value problem to obtain a nonlinear
!  system of equations.
!
!  --------------------------------------------------------------------------
!  The following define must be used before including any PETSc include files
!  into a module or interface. This is because they can't handle declarations
!  in them
!
#include <petsc/finclude/petsc.h>
module ex73f90tmodule
  use petscdmda
  use petscdmcomposite
  use petscmat
  type ex73f90tmodule_type
    DM::da
!     temp A block stuff
    PetscInt mx, my
    PetscMPIInt rank
    PetscReal lambda
!     Mats
    Mat::Amat, AmatLin, Bmat, CMat, Dmat
    IS::isPhi, isLambda
  end type ex73f90tmodule_type

end module ex73f90tmodule

module ex73f90tmodule_interfaces
  use ex73f90tmodule

  Interface SNESSetApplicationContext
    Subroutine SNESSetApplicationContext(snesIn, ctx, ierr)
      use petscsnes
      use ex73f90tmodule
      SNES::    snesIn
      type(ex73f90tmodule_type) ctx
      PetscErrorCode ierr
    End Subroutine
  End Interface SNESSetApplicationContext

  Interface SNESGetApplicationContext
    Subroutine SNESGetApplicationContext(snesIn, ctx, ierr)
      use petscsnes
      use ex73f90tmodule
      SNES::     snesIn
      type(ex73f90tmodule_type), pointer :: ctx
      PetscErrorCode ierr
    End Subroutine
  End Interface SNESGetApplicationContext
end module ex73f90tmodule_interfaces

subroutine MyObjective(snes, x, result, ctx, ierr)
  use petsc
  implicit none
  PetscInt ctx
  Vec x, f
  SNES snes
  PetscErrorCode ierr
  PetscScalar result
  PetscReal fnorm

  PetscCall(VecDuplicate(x, f, ierr))
  PetscCall(SNESComputeFunction(snes, x, f, ierr))
  PetscCall(VecNorm(f, NORM_2, fnorm, ierr))
  result = .5*fnorm*fnorm
  PetscCall(VecDestroy(f, ierr))
end subroutine MyObjective

program main
  use petscsnes
  use ex73f90tmodule
  use ex73f90tmodule_interfaces
  implicit none
! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!                   Variable declarations
! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!
!  Variables:
!     mysnes      - nonlinear solver
!     x, r        - solution, residual vectors
!     J           - Jacobian matrix
!     its         - iterations for convergence
!     Nx, Ny      - number of preocessors in x- and y- directions
!
  SNES::       mysnes
  Vec::        x, r, x2, x1, x1loc, x2loc
  Mat::       Amat, Bmat, Cmat, Dmat, KKTMat, matArray(4)
!      Mat::       tmat
  DM::       daphi, dalam
  IS, pointer ::        isglobal(:)
  PetscErrorCode ierr
  PetscInt its, N1, N2, i, j, irow, row(1)
  PetscInt col(1), low, high, lamlow, lamhigh
  PetscBool flg
  PetscInt ione, nfour, itwo, nloc, nloclam
  PetscReal lambda_max, lambda_min
  type(ex73f90tmodule_type) solver
  PetscScalar bval(1), cval(1), one
  PetscBool useobjective

!  Note: Any user-defined Fortran routines (such as FormJacobian)
!  MUST be declared as external.
  external FormInitialGuess, FormJacobian, FormFunction, MyObjective

! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!  Initialize program
! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  PetscCallA(PetscInitialize(ierr))
  PetscCallMPIA(MPI_Comm_rank(PETSC_COMM_WORLD, solver%rank, ierr))

!  Initialize problem parameters
  lambda_max = 6.81_PETSC_REAL_KIND
  lambda_min = 0.0
  solver%lambda = 6.0
  ione = 1
  nfour = 4
  itwo = 2
  useobjective = PETSC_FALSE
  PetscCallA(PetscOptionsGetReal(PETSC_NULL_OPTIONS, PETSC_NULL_CHARACTER, '-par', solver%lambda, flg, ierr))
  PetscCheckA(solver%lambda <= lambda_max .and. solver%lambda >= lambda_min, PETSC_COMM_SELF, PETSC_ERR_USER, 'Lambda provided with -par is out of range')
  PetscCallA(PetscOptionsGetBool(PETSC_NULL_OPTIONS, PETSC_NULL_CHARACTER, '-objective', useobjective, flg, ierr))

! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!  Create vector data structures; set function evaluation routine
! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

!     just get size
  PetscCallA(DMDACreate2d(PETSC_COMM_WORLD, DM_BOUNDARY_NONE, DM_BOUNDARY_NONE, DMDA_STENCIL_BOX, nfour, nfour, PETSC_DECIDE, PETSC_DECIDE, ione, ione, PETSC_NULL_INTEGER_ARRAY, PETSC_NULL_INTEGER_ARRAY, daphi, ierr))
  PetscCallA(DMSetFromOptions(daphi, ierr))
  PetscCallA(DMSetUp(daphi, ierr))
  PetscCallA(DMDAGetInfo(daphi, PETSC_NULL_INTEGER, solver%mx, solver%my, PETSC_NULL_INTEGER, PETSC_NULL_INTEGER, PETSC_NULL_INTEGER, PETSC_NULL_INTEGER, PETSC_NULL_INTEGER, PETSC_NULL_INTEGER, PETSC_NULL_DMBOUNDARYTYPE, PETSC_NULL_DMBOUNDARYTYPE, PETSC_NULL_DMBOUNDARYTYPE, PETSC_NULL_DMDASTENCILTYPE, ierr))
  N1 = solver%my*solver%mx
  N2 = solver%my
  flg = .false.
  PetscCallA(PetscOptionsGetBool(PETSC_NULL_OPTIONS, PETSC_NULL_CHARACTER, '-no_constraints', flg, flg, ierr))
  if (flg) then
    N2 = 0
  end if

  PetscCallA(DMDestroy(daphi, ierr))

! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!  Create matrix data structure; set Jacobian evaluation routine
! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  PetscCallA(DMShellCreate(PETSC_COMM_WORLD, daphi, ierr))
  PetscCallA(DMSetOptionsPrefix(daphi, 'phi_', ierr))
  PetscCallA(DMSetFromOptions(daphi, ierr))

  PetscCallA(VecCreate(PETSC_COMM_WORLD, x1, ierr))
  PetscCallA(VecSetSizes(x1, PETSC_DECIDE, N1, ierr))
  PetscCallA(VecSetFromOptions(x1, ierr))

  PetscCallA(VecGetOwnershipRange(x1, low, high, ierr))
  nloc = high - low

  PetscCallA(MatCreate(PETSC_COMM_WORLD, Amat, ierr))
  PetscCallA(MatSetSizes(Amat, PETSC_DECIDE, PETSC_DECIDE, N1, N1, ierr))
  PetscCallA(MatSetUp(Amat, ierr))

  PetscCallA(MatCreate(PETSC_COMM_WORLD, solver%AmatLin, ierr))
  PetscCallA(MatSetSizes(solver%AmatLin, PETSC_DECIDE, PETSC_DECIDE, N1, N1, ierr))
  PetscCallA(MatSetUp(solver%AmatLin, ierr))

  PetscCallA(FormJacobianLocal(x1, solver%AmatLin, solver, .false., ierr))
  PetscCallA(MatAssemblyBegin(solver%AmatLin, MAT_FINAL_ASSEMBLY, ierr))
  PetscCallA(MatAssemblyEnd(solver%AmatLin, MAT_FINAL_ASSEMBLY, ierr))

  PetscCallA(DMShellSetGlobalVector(daphi, x1, ierr))
  PetscCallA(DMShellSetMatrix(daphi, Amat, ierr))

  PetscCallA(VecCreate(PETSC_COMM_SELF, x1loc, ierr))
  PetscCallA(VecSetSizes(x1loc, nloc, nloc, ierr))
  PetscCallA(VecSetFromOptions(x1loc, ierr))
  PetscCallA(DMShellSetLocalVector(daphi, x1loc, ierr))

! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!  Create B, C, & D matrices
! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  PetscCallA(MatCreate(PETSC_COMM_WORLD, Cmat, ierr))
  PetscCallA(MatSetSizes(Cmat, PETSC_DECIDE, PETSC_DECIDE, N2, N1, ierr))
  PetscCallA(MatSetUp(Cmat, ierr))
!      create data for C and B
  PetscCallA(MatCreate(PETSC_COMM_WORLD, Bmat, ierr))
  PetscCallA(MatSetSizes(Bmat, PETSC_DECIDE, PETSC_DECIDE, N1, N2, ierr))
  PetscCallA(MatSetUp(Bmat, ierr))
!     create data for D
  PetscCallA(MatCreate(PETSC_COMM_WORLD, Dmat, ierr))
  PetscCallA(MatSetSizes(Dmat, PETSC_DECIDE, PETSC_DECIDE, N2, N2, ierr))
  PetscCallA(MatSetUp(Dmat, ierr))

  PetscCallA(VecCreate(PETSC_COMM_WORLD, x2, ierr))
  PetscCallA(VecSetSizes(x2, PETSC_DECIDE, N2, ierr))
  PetscCallA(VecSetFromOptions(x2, ierr))

  PetscCallA(VecGetOwnershipRange(x2, lamlow, lamhigh, ierr))
  nloclam = lamhigh - lamlow

! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!  Set fake B and C
! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  one = 1.0
  if (N2 > 0) then
    bval(1) = -one/(solver%mx - 2)
!     cval = -one/(solver%my*solver%mx)
    cval(1) = -one
    do irow = low, high - 1
      j = irow/solver%mx   ! row in domain
      i = mod(irow, solver%mx)
      row(1) = irow
      col(1) = j
      if (i == 0 .or. j == 0 .or. i == solver%mx - 1 .or. j == solver%my - 1) then
        !     no op
      else
        PetscCallA(MatSetValues(Bmat, ione, row, ione, col, bval, INSERT_VALUES, ierr))
      end if
      row(1) = j
      PetscCallA(MatSetValues(Cmat, ione, row, ione, row, cval, INSERT_VALUES, ierr))
    end do
  end if
  PetscCallA(MatAssemblyBegin(Bmat, MAT_FINAL_ASSEMBLY, ierr))
  PetscCallA(MatAssemblyEnd(Bmat, MAT_FINAL_ASSEMBLY, ierr))
  PetscCallA(MatAssemblyBegin(Cmat, MAT_FINAL_ASSEMBLY, ierr))
  PetscCallA(MatAssemblyEnd(Cmat, MAT_FINAL_ASSEMBLY, ierr))

! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!  Set D (identity)
! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  do j = lamlow, lamhigh - 1
    row(1) = j
    cval(1) = one
    PetscCallA(MatSetValues(Dmat, ione, row, ione, row, cval, INSERT_VALUES, ierr))
  end do
  PetscCallA(MatAssemblyBegin(Dmat, MAT_FINAL_ASSEMBLY, ierr))
  PetscCallA(MatAssemblyEnd(Dmat, MAT_FINAL_ASSEMBLY, ierr))

! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!  DM for lambda (dalam) : temp driver for A block, setup A block solver data
! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  PetscCallA(DMShellCreate(PETSC_COMM_WORLD, dalam, ierr))
  PetscCallA(DMShellSetGlobalVector(dalam, x2, ierr))
  PetscCallA(DMShellSetMatrix(dalam, Dmat, ierr))

  PetscCallA(VecCreate(PETSC_COMM_SELF, x2loc, ierr))
  PetscCallA(VecSetSizes(x2loc, nloclam, nloclam, ierr))
  PetscCallA(VecSetFromOptions(x2loc, ierr))
  PetscCallA(DMShellSetLocalVector(dalam, x2loc, ierr))

  PetscCallA(DMSetOptionsPrefix(dalam, 'lambda_', ierr))
  PetscCallA(DMSetFromOptions(dalam, ierr))
! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!  Create field split DA
! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  PetscCallA(DMCompositeCreate(PETSC_COMM_WORLD, solver%da, ierr))
  PetscCallA(DMCompositeAddDM(solver%da, daphi, ierr))
  PetscCallA(DMCompositeAddDM(solver%da, dalam, ierr))
  PetscCallA(DMSetFromOptions(solver%da, ierr))
  PetscCallA(DMSetUp(solver%da, ierr))
  PetscCallA(DMCompositeGetGlobalISs(solver%da, isglobal, ierr))
  solver%isPhi = isglobal(1)
  PetscCallA(PetscObjectReference(solver%isPhi, ierr))
  solver%isLambda = isglobal(2)
  PetscCallA(PetscObjectReference(solver%isLambda, ierr))

!     cache matrices
  solver%Amat = Amat
  solver%Bmat = Bmat
  solver%Cmat = Cmat
  solver%Dmat = Dmat

  matArray(1) = Amat
  matArray(2) = Bmat
  matArray(3) = Cmat
  matArray(4) = Dmat

  PetscCallA(MatCreateNest(PETSC_COMM_WORLD, itwo, isglobal, itwo, isglobal, matArray, KKTmat, ierr))
  PetscCallA(DMCompositeRestoreGlobalISs(solver%da, isglobal, ierr))
  PetscCallA(MatSetFromOptions(KKTmat, ierr))

!  Extract global and local vectors from DMDA; then duplicate for remaining
!     vectors that are the same types
  PetscCallA(MatCreateVecs(KKTmat, x, PETSC_NULL_VEC, ierr))
  PetscCallA(VecDuplicate(x, r, ierr))

  PetscCallA(SNESCreate(PETSC_COMM_WORLD, mysnes, ierr))

  PetscCallA(SNESSetDM(mysnes, solver%da, ierr))

  PetscCallA(SNESSetApplicationContext(mysnes, solver, ierr))

  PetscCallA(SNESSetDM(mysnes, solver%da, ierr))

!  Set function evaluation routine and vector
  PetscCallA(SNESSetFunction(mysnes, r, FormFunction, solver, ierr))
  if (useobjective .eqv. PETSC_TRUE) then
    PetscCallA(SNESSetObjective(mysnes, MyObjective, solver, ierr))
  end if
  PetscCallA(SNESSetJacobian(mysnes, KKTmat, KKTmat, FormJacobian, solver, ierr))

! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!  Customize nonlinear solver; set runtime options
! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!  Set runtime options (e.g., -snes_monitor -snes_rtol <rtol> -ksp_type <type>)
  PetscCallA(SNESSetFromOptions(mysnes, ierr))

! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!  Evaluate initial guess; then solve nonlinear system.
! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!  Note: The user should initialize the vector, x, with the initial guess
!  for the nonlinear solver prior to calling SNESSolve().  In particular,
!  to employ an initial guess of zero, the user should explicitly set
!  this vector to zero by calling VecSet().

  PetscCallA(FormInitialGuess(mysnes, x, ierr))
  PetscCallA(SNESSolve(mysnes, PETSC_NULL_VEC, x, ierr))
  PetscCallA(SNESGetIterationNumber(mysnes, its, ierr))
  if (solver%rank == 0) then
    write (6, 100) its
  end if
100 format('Number of SNES iterations = ', i5)

! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!  Free work space.  All PETSc objects should be destroyed when they
!  are no longer needed.
! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  PetscCallA(MatDestroy(KKTmat, ierr))
  PetscCallA(MatDestroy(Amat, ierr))
  PetscCallA(MatDestroy(Dmat, ierr))
  PetscCallA(MatDestroy(Bmat, ierr))
  PetscCallA(MatDestroy(Cmat, ierr))
  PetscCallA(MatDestroy(solver%AmatLin, ierr))
  PetscCallA(ISDestroy(solver%isPhi, ierr))
  PetscCallA(ISDestroy(solver%isLambda, ierr))
  PetscCallA(VecDestroy(x, ierr))
  PetscCallA(VecDestroy(x2, ierr))
  PetscCallA(VecDestroy(x1, ierr))
  PetscCallA(VecDestroy(x1loc, ierr))
  PetscCallA(VecDestroy(x2loc, ierr))
  PetscCallA(VecDestroy(r, ierr))
  PetscCallA(SNESDestroy(mysnes, ierr))
  PetscCallA(DMDestroy(solver%da, ierr))
  PetscCallA(DMDestroy(daphi, ierr))
  PetscCallA(DMDestroy(dalam, ierr))

  PetscCallA(PetscFinalize(ierr))
end

! ---------------------------------------------------------------------
!
!  FormInitialGuess - Forms initial approximation.
!
!  Input Parameters:
!  X - vector
!
!  Output Parameter:
!  X - vector
!
!  Notes:
!  This routine serves as a wrapper for the lower-level routine
!  "InitialGuessLocal", where the actual computations are
!  done using the standard Fortran style of treating the local
!  vector data as a multidimensional array over the local mesh.
!  This routine merely handles ghost point scatters and accesses
!  the local vector data via VecGetArray() and VecRestoreArray().
!
subroutine FormInitialGuess(mysnes, Xnest, ierr)
  use petscsnes
  use ex73f90tmodule
  use ex73f90tmodule_interfaces
  implicit none
!  Input/output variables:
  SNES::     mysnes
  Vec::      Xnest
  PetscErrorCode ierr

!  Declarations for use with local arrays:
  type(ex73f90tmodule_type), pointer:: solver
  Vec::      Xsub(2)
  PetscInt::  izero, ione, itwo

  izero = 0
  ione = 1
  itwo = 2
  ierr = 0
  PetscCall(SNESGetApplicationContext(mysnes, solver, ierr))
  PetscCall(DMCompositeGetAccessArray(solver%da, Xnest, itwo, PETSC_NULL_INTEGER_ARRAY, Xsub, ierr))

  PetscCall(InitialGuessLocal(solver, Xsub(1), ierr))
  PetscCall(VecAssemblyBegin(Xsub(1), ierr))
  PetscCall(VecAssemblyEnd(Xsub(1), ierr))

!     zero out lambda
  PetscCall(VecZeroEntries(Xsub(2), ierr))
  PetscCall(DMCompositeRestoreAccessArray(solver%da, Xnest, itwo, PETSC_NULL_INTEGER_ARRAY, Xsub, ierr))

end subroutine FormInitialGuess

! ---------------------------------------------------------------------
!
!  InitialGuessLocal - Computes initial approximation, called by
!  the higher level routine FormInitialGuess().
!
!  Input Parameter:
!  X1 - local vector data
!
!  Output Parameters:
!  x - local vector data
!  ierr - error code
!
!  Notes:
!  This routine uses standard Fortran-style computations over a 2-dim array.
!
subroutine InitialGuessLocal(solver, X1, ierr)
  use petscsys
  use ex73f90tmodule
  implicit none
!  Input/output variables:
  type(ex73f90tmodule_type) solver
  Vec::      X1
  PetscErrorCode ierr

!  Local variables:
  PetscInt row, i, j, ione, low, high
  PetscReal temp1, temp, hx, hy, v
  PetscReal one

!  Set parameters
  ione = 1
  ierr = 0
  one = 1.0
  hx = one/(solver%mx - 1)
  hy = one/(solver%my - 1)
  temp1 = solver%lambda/(solver%lambda + one) + one

  PetscCall(VecGetOwnershipRange(X1, low, high, ierr))

  do row = low, high - 1
    j = row/solver%mx
    i = mod(row, solver%mx)
    temp = min(j, solver%my - j + 1)*hy
    if (i == 0 .or. j == 0 .or. i == solver%mx - 1 .or. j == solver%my - 1) then
      v = 0.0
    else
      v = temp1*sqrt(min(min(i, solver%mx - i + 1)*hx, temp))
    end if
    PetscCall(VecSetValues(X1, ione, [row], [v], INSERT_VALUES, ierr))
  end do

end subroutine InitialGuessLocal

! ---------------------------------------------------------------------
!
!  FormJacobian - Evaluates Jacobian matrix.
!
!  Input Parameters:
!  dummy     - the SNES context
!  x         - input vector
!  solver    - solver data
!
!  Output Parameters:
!  jac      - Jacobian matrix
!  jac_prec - optionally different matrix used to construct the preconditioner (not used here)
!
subroutine FormJacobian(dummy, X, jac, jac_prec, solver, ierr)
  use petscsnes
  use ex73f90tmodule
  implicit none
!  Input/output variables:
  SNES::     dummy
  Vec::      X
  Mat::     jac, jac_prec
  type(ex73f90tmodule_type) solver
  PetscErrorCode ierr

!  Declarations for use with local arrays:
  Vec::      Xsub(1)
  Mat::     Amat
  PetscInt ione

  ione = 1

  PetscCall(DMCompositeGetAccessArray(solver%da, X, ione, PETSC_NULL_INTEGER_ARRAY, Xsub, ierr))

!     Compute entries for the locally owned part of the Jacobian preconditioner.
  PetscCall(MatCreateSubMatrix(jac_prec, solver%isPhi, solver%isPhi, MAT_INITIAL_MATRIX, Amat, ierr))

  PetscCall(FormJacobianLocal(Xsub(1), Amat, solver, .true., ierr))
  PetscCall(MatDestroy(Amat, ierr)) ! discard our reference
  PetscCall(DMCompositeRestoreAccessArray(solver%da, X, ione, PETSC_NULL_INTEGER_ARRAY, Xsub, ierr))

  ! the rest of the matrix is not touched
  PetscCall(MatAssemblyBegin(jac_prec, MAT_FINAL_ASSEMBLY, ierr))
  PetscCall(MatAssemblyEnd(jac_prec, MAT_FINAL_ASSEMBLY, ierr))
  if (jac /= jac_prec) then
    PetscCall(MatAssemblyBegin(jac, MAT_FINAL_ASSEMBLY, ierr))
    PetscCall(MatAssemblyEnd(jac, MAT_FINAL_ASSEMBLY, ierr))
  end if

!     Tell the matrix we will never add a new nonzero location to the
!     matrix. If we do it will generate an error.
  PetscCall(MatSetOption(jac_prec, MAT_NEW_NONZERO_LOCATION_ERR, PETSC_TRUE, ierr))

end subroutine FormJacobian

! ---------------------------------------------------------------------
!
!  FormJacobianLocal - Computes Jacobian matrix used to compute the preconditioner,
!  called by the higher level routine FormJacobian().
!
!  Input Parameters:
!  x        - local vector data
!
!  Output Parameters:
!  jac - Jacobian matrix used to compute the preconditioner
!  ierr     - error code
!
!  Notes:
!  This routine uses standard Fortran-style computations over a 2-dim array.
!
subroutine FormJacobianLocal(X1, jac, solver, add_nl_term, ierr)
  use ex73f90tmodule
  implicit none
!  Input/output variables:
  type(ex73f90tmodule_type) solver
  Vec::      X1
  Mat::     jac
  logical add_nl_term
  PetscErrorCode ierr

!  Local variables:
  PetscInt irow, row(1), col(5), i, j
  PetscInt ione, ifive, low, high, ii
  PetscScalar two, one, hx, hy, hy2inv
  PetscScalar hx2inv, sc, v(5)
  PetscScalar, pointer :: lx_v(:)

!  Set parameters
  ione = 1
  ifive = 5
  one = 1.0
  two = 2.0
  hx = one/(solver%mx - 1)
  hy = one/(solver%my - 1)
  sc = solver%lambda
  hx2inv = one/(hx*hx)
  hy2inv = one/(hy*hy)

  PetscCall(VecGetOwnershipRange(X1, low, high, ierr))
  PetscCall(VecGetArrayRead(X1, lx_v, ierr))

  ii = 0
  do irow = low, high - 1
    j = irow/solver%mx
    i = mod(irow, solver%mx)
    ii = ii + 1            ! one based local index
!     boundary points
    if (i == 0 .or. j == 0 .or. i == solver%mx - 1 .or. j == solver%my - 1) then
      col(1) = irow
      row(1) = irow
      v(1) = one
      PetscCall(MatSetValues(jac, ione, row, ione, col, v, INSERT_VALUES, ierr))
!     interior grid points
    else
      v(1) = -hy2inv
      if (j - 1 == 0) v(1) = 0.0
      v(2) = -hx2inv
      if (i - 1 == 0) v(2) = 0.0
      v(3) = two*(hx2inv + hy2inv)
      if (add_nl_term) v(3) = v(3) - sc*exp(lx_v(ii))
      v(4) = -hx2inv
      if (i + 1 == solver%mx - 1) v(4) = 0.0
      v(5) = -hy2inv
      if (j + 1 == solver%my - 1) v(5) = 0.0
      col(1) = irow - solver%mx
      col(2) = irow - 1
      col(3) = irow
      col(4) = irow + 1
      col(5) = irow + solver%mx
      row(1) = irow
      PetscCall(MatSetValues(jac, ione, row, ifive, col, v, INSERT_VALUES, ierr))
    end if
  end do

  PetscCall(VecRestoreArrayRead(X1, lx_v, ierr))

end subroutine FormJacobianLocal

! ---------------------------------------------------------------------
!
!  FormFunction - Evaluates nonlinear function, F(x).
!
!  Input Parameters:
!  snes - the SNES context
!  X - input vector
!  dummy - optional user-defined context, as set by SNESSetFunction()
!          (not used here)
!
!  Output Parameter:
!  F - function vector
!
subroutine FormFunction(snesIn, X, F, solver, ierr)
  use petscsnes
  use ex73f90tmodule
  implicit none
!  Input/output variables:
  SNES::     snesIn
  Vec::      X, F
  PetscErrorCode ierr
  type(ex73f90tmodule_type) solver

!  Declarations for use with local arrays:
  Vec::              Xsub(2), Fsub(2)
  PetscInt itwo

!  Scatter ghost points to local vector, using the 2-step process
!     DMGlobalToLocalBegin(), DMGlobalToLocalEnd().
!  By placing code between these two statements, computations can
!  be done while messages are in transition.

  itwo = 2
  PetscCall(DMCompositeGetAccessArray(solver%da, X, itwo, PETSC_NULL_INTEGER_ARRAY, Xsub, ierr))
  PetscCall(DMCompositeGetAccessArray(solver%da, F, itwo, PETSC_NULL_INTEGER_ARRAY, Fsub, ierr))

  PetscCall(FormFunctionNLTerm(Xsub(1), Fsub(1), solver, ierr))
  PetscCall(MatMultAdd(solver%AmatLin, Xsub(1), Fsub(1), Fsub(1), ierr))

!     do rest of operator (linear)
  PetscCall(MatMult(solver%Cmat, Xsub(1), Fsub(2), ierr))
  PetscCall(MatMultAdd(solver%Bmat, Xsub(2), Fsub(1), Fsub(1), ierr))
  PetscCall(MatMultAdd(solver%Dmat, Xsub(2), Fsub(2), Fsub(2), ierr))

  PetscCall(DMCompositeRestoreAccessArray(solver%da, X, itwo, PETSC_NULL_INTEGER_ARRAY, Xsub, ierr))
  PetscCall(DMCompositeRestoreAccessArray(solver%da, F, itwo, PETSC_NULL_INTEGER_ARRAY, Fsub, ierr))
end subroutine formfunction

! ---------------------------------------------------------------------
!
!  FormFunctionNLTerm - Computes nonlinear function, called by
!  the higher level routine FormFunction().
!
!  Input Parameter:
!  x - local vector data
!
!  Output Parameters:
!  f - local vector data, f(x)
!  ierr - error code
!
!  Notes:
!  This routine uses standard Fortran-style computations over a 2-dim array.
!
subroutine FormFunctionNLTerm(X1, F1, solver, ierr)
  use ex73f90tmodule
  implicit none
!  Input/output variables:
  type(ex73f90tmodule_type) solver
  Vec::      X1, F1
  PetscErrorCode ierr
!  Local variables:
  PetscScalar sc
  PetscScalar u, v(1)
  PetscInt i, j, low, high, ii, ione, irow, row(1)
  PetscScalar, pointer :: lx_v(:)

  sc = solver%lambda
  ione = 1

  PetscCall(VecGetArrayRead(X1, lx_v, ierr))
  PetscCall(VecGetOwnershipRange(X1, low, high, ierr))

!     Compute function over the locally owned part of the grid
  ii = 0
  do irow = low, high - 1
    j = irow/solver%mx
    i = mod(irow, solver%mx)
    ii = ii + 1            ! one based local index
    row(1) = irow
    if (i == 0 .or. j == 0 .or. i == solver%mx - 1 .or. j == solver%my - 1) then
      v(1) = 0.0
    else
      u = lx_v(ii)
      v(1) = -sc*exp(u)
    end if
    PetscCall(VecSetValues(F1, ione, row, v, INSERT_VALUES, ierr))
  end do

  PetscCall(VecRestoreArrayRead(X1, lx_v, ierr))

  PetscCall(VecAssemblyBegin(F1, ierr))
  PetscCall(VecAssemblyEnd(F1, ierr))

  ierr = 0
end subroutine FormFunctionNLTerm

!/*TEST
!
!   build:
!      requires: !single !complex
!
!   test:
!      nsize: 4
!      args: -par 5.0 -da_grid_x 10 -da_grid_y 10 -snes_monitor_short -snes_linesearch_type basic -snes_converged_reason -ksp_type fgmres -ksp_norm_type unpreconditioned -pc_type fieldsplit -pc_fieldsplit_type schur -pc_fieldsplit_schur_fact_type upper -ksp_monitor_short -fieldsplit_lambda_ksp_type preonly -fieldsplit_lambda_pc_type jacobi -fieldsplit_phi_pc_type gamg -fieldsplit_phi_pc_gamg_esteig_ksp_type cg -fieldsplit_phi_pc_gamg_esteig_ksp_max_it 10 -fieldsplit_phi_pc_gamg_agg_nsmooths 1 -fieldsplit_phi_pc_gamg_threshold 0.
!
!   test:
!      args: -snes_linesearch_type {{secant cp}separate output} -objective {{false true}shared output}
!
!   test:
!      args: -snes_linesearch_type bt -objective {{false true}separate output}
!
!TEST*/