xref: /petsc/src/snes/tutorials/ex5f90t.F90 (revision e7a95102f46630f317be643b805dc1c3f4655aeb)

!
!  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)
!
!
!  --------------------------------------------------------------------------
!
!  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 uniprocessor version of this code is snes/tutorials/ex4f.F
!
!  --------------------------------------------------------------------------
!  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/petscdmda.h>
#include <petsc/finclude/petscsnes.h>
#include <petsc/finclude/petscsys.h>
#include <petsc/finclude/petscmat.h>
module ex5f90tmodule
  use petscsnes
  use petscdmda
  implicit none
  type userctx
    type(tDM) da
    PetscInt xs, xe, xm, gxs, gxe, gxm
    PetscInt ys, ye, ym, gys, gye, gym
    PetscInt mx, my
    PetscMPIInt rank
    PetscReal lambda
  end type userctx

  interface
    subroutine SNESSetApplicationContext(snesIn, ctx, ierr)
      use petscsnes
      import userctx
      type(tSNES) snesIn
      type(userctx) ctx
      PetscErrorCode ierr
    end subroutine
    subroutine SNESGetApplicationContext(snesIn, ctx, ierr)
      use petscsnes
      import userctx
      type(tSNES) snesIn
      type(userctx), pointer :: ctx
      PetscErrorCode ierr
    end subroutine
  end interface

contains
! ---------------------------------------------------------------------
!
!  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
!
!  Notes:
!  This routine serves as a wrapper for the lower-level routine
!  "FormFunctionLocal", 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 FormFunction(snesIn, X, F, user, ierr)
!  Input/output variables:
    type(tSNES) snesIn
    type(tVec) X, F
    PetscErrorCode ierr
    type(userctx) user

!  Declarations for use with local arrays:
    PetscScalar, pointer :: lx_v(:), lf_v(:)
    type(tVec) localX

!  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.
    PetscCall(DMGetLocalVector(user%da, localX, ierr))
    PetscCall(DMGlobalToLocalBegin(user%da, X, INSERT_VALUES, localX, ierr))
    PetscCall(DMGlobalToLocalEnd(user%da, X, INSERT_VALUES, localX, ierr))

!  Get a pointer to vector data.
!    - VecGetArray90() returns a pointer to the data array.
!    - You MUST call VecRestoreArray() when you no longer need access to
!      the array.

    PetscCall(VecGetArray(localX, lx_v, ierr))
    PetscCall(VecGetArray(F, lf_v, ierr))

!  Compute function over the locally owned part of the grid
    PetscCall(FormFunctionLocal(lx_v, lf_v, user, ierr))

!  Restore vectors
    PetscCall(VecRestoreArray(localX, lx_v, ierr))
    PetscCall(VecRestoreArray(F, lf_v, ierr))

!  Insert values into global vector

    PetscCall(DMRestoreLocalVector(user%da, localX, ierr))
    PetscCall(PetscLogFlops(11.0d0*user%ym*user%xm, ierr))

!      PetscCall(VecView(X,PETSC_VIEWER_STDOUT_WORLD,ierr))
!      PetscCall(VecView(F,PETSC_VIEWER_STDOUT_WORLD,ierr))
  end subroutine formfunction

! ---------------------------------------------------------------------
!
!  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, X, ierr)
!  Input/output variables:
    type(tSNES) mysnes
    type(userctx), pointer:: puser
    type(tVec) X
    PetscErrorCode ierr

!  Declarations for use with local arrays:
    PetscScalar, pointer :: lx_v(:)

    ierr = 0
    PetscCallA(SNESGetApplicationContext(mysnes, puser, ierr))
!  Get a pointer to vector data.
!    - VecGetArray90() returns a pointer to the data array.
!    - You MUST call VecRestoreArray() when you no longer need access to
!      the array.

    PetscCallA(VecGetArray(X, lx_v, ierr))

!  Compute initial guess over the locally owned part of the grid
    PetscCallA(InitialGuessLocal(puser, lx_v, ierr))

!  Restore vector
    PetscCallA(VecRestoreArray(X, lx_v, ierr))

!  Insert values into global vector

  end

! ---------------------------------------------------------------------
!
!  InitialGuessLocal - Computes initial approximation, called by
!  the higher level routine FormInitialGuess().
!
!  Input Parameter:
!  x - 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(user, x, ierr)
!  Input/output variables:
    type(userctx) user
    PetscScalar x(user%xs:user%xe, user%ys:user%ye)
    PetscErrorCode ierr

!  Local variables:
    PetscInt i, j
    PetscScalar temp1, temp, hx, hy
    PetscScalar one

!  Set parameters

    ierr = 0
    one = 1.0
    hx = one/(PetscIntToReal(user%mx - 1))
    hy = one/(PetscIntToReal(user%my - 1))
    temp1 = user%lambda/(user%lambda + one)

    do j = user%ys, user%ye
      temp = PetscIntToReal(min(j - 1, user%my - j))*hy
      do i = user%xs, user%xe
        if (i == 1 .or. j == 1 .or. i == user%mx .or. j == user%my) then
          x(i, j) = 0.0
        else
          x(i, j) = temp1*sqrt(min(PetscIntToReal(min(i - 1, user%mx - i)*hx), PetscIntToReal(temp)))
        end if
      end do
    end do

  end

! ---------------------------------------------------------------------
!
!  FormFunctionLocal - 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 FormFunctionLocal(x, f, user, ierr)
!  Input/output variables:
    type(userctx) user
    PetscScalar x(user%gxs:user%gxe, user%gys:user%gye)
    PetscScalar f(user%xs:user%xe, user%ys:user%ye)
    PetscErrorCode ierr

!  Local variables:
    PetscScalar two, one, hx, hy, hxdhy, hydhx, sc
    PetscScalar u, uxx, uyy
    PetscInt i, j

    one = 1.0
    two = 2.0
    hx = one/PetscIntToReal(user%mx - 1)
    hy = one/PetscIntToReal(user%my - 1)
    sc = hx*hy*user%lambda
    hxdhy = hx/hy
    hydhx = hy/hx

!  Compute function over the locally owned part of the grid

    do j = user%ys, user%ye
      do i = user%xs, user%xe
        if (i == 1 .or. j == 1 .or. i == user%mx .or. j == user%my) then
          f(i, j) = x(i, j)
        else
          u = x(i, j)
          uxx = hydhx*(two*u - x(i - 1, j) - x(i + 1, j))
          uyy = hxdhy*(two*u - x(i, j - 1) - x(i, j + 1))
          f(i, j) = uxx + uyy - sc*exp(u)
        end if
      end do
    end do
    ierr = 0
  end

! ---------------------------------------------------------------------
!
!  FormJacobian - Evaluates Jacobian matrix.
!
!  Input Parameters:
!  snes     - the SNES context
!  x        - input vector
!  dummy    - optional user-defined context, as set by SNESSetJacobian()
!             (not used here)
!
!  Output Parameters:
!  jac      - Jacobian matrix
!  jac_prec - optionally different matrix used to construct the preconditioner (not used here)
!
!  Notes:
!  This routine serves as a wrapper for the lower-level routine
!  "FormJacobianLocal", 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 accesses the local vector data via
!  VecGetArray() and VecRestoreArray().
!
!  Notes:
!  Due to grid point reordering with DMDAs, we must always work
!  with the local grid points, and then transform them to the new
!  global numbering with the "ltog" mapping
!  We cannot work directly with the global numbers for the original
!  uniprocessor grid!
!
!  Two methods are available for imposing this transformation
!  when setting matrix entries:
!    (A) MatSetValuesLocal(), using the local ordering (including
!        ghost points!)
!        - Set matrix entries using the local ordering
!          by calling MatSetValuesLocal()
!    (B) MatSetValues(), using the global ordering
!        - Use DMGetLocalToGlobalMapping() then
!          ISLocalToGlobalMappingGetIndices() to extract the local-to-global map
!        - Then apply this map explicitly yourself
!        - Set matrix entries using the global ordering by calling
!          MatSetValues()
!  Option (A) seems cleaner/easier in many cases, and is the procedure
!  used in this example.
!
  subroutine FormJacobian(mysnes, X, jac, jac_prec, user, ierr)
!  Input/output variables:
    type(tSNES) mysnes
    type(tVec) X
    type(tMat) jac, jac_prec
    type(userctx) user
    PetscErrorCode ierr

!  Declarations for use with local arrays:
    PetscScalar, pointer :: lx_v(:)
    type(tVec) localX

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

    PetscCallA(DMGetLocalVector(user%da, localX, ierr))
    PetscCallA(DMGlobalToLocalBegin(user%da, X, INSERT_VALUES, localX, ierr))
    PetscCallA(DMGlobalToLocalEnd(user%da, X, INSERT_VALUES, localX, ierr))

!  Get a pointer to vector data
    PetscCallA(VecGetArray(localX, lx_v, ierr))

!  Compute entries for the locally owned part of the Jacobian preconditioner.
    PetscCallA(FormJacobianLocal(lx_v, jac_prec, user, ierr))

!  Assemble matrix, using the 2-step process:
!     MatAssemblyBegin(), MatAssemblyEnd()
!  Computations can be done while messages are in transition,
!  by placing code between these two statements.

    PetscCallA(MatAssemblyBegin(jac, MAT_FINAL_ASSEMBLY, ierr))
!      if (jac .ne. jac_prec) then
    PetscCallA(MatAssemblyBegin(jac_prec, MAT_FINAL_ASSEMBLY, ierr))
!      endif
    PetscCallA(VecRestoreArray(localX, lx_v, ierr))
    PetscCallA(DMRestoreLocalVector(user%da, localX, ierr))
    PetscCallA(MatAssemblyEnd(jac, MAT_FINAL_ASSEMBLY, ierr))
!      if (jac .ne. jac_prec) then
    PetscCallA(MatAssemblyEnd(jac_prec, MAT_FINAL_ASSEMBLY, ierr))
!      endif

!  Tell the matrix we will never add a new nonzero location to the
!  matrix. If we do it will generate an error.

    PetscCallA(MatSetOption(jac, MAT_NEW_NONZERO_LOCATION_ERR, PETSC_TRUE, ierr))

  end

! ---------------------------------------------------------------------
!
!  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_prec - Jacobian matrix used to compute the preconditioner
!  ierr     - error code
!
!  Notes:
!  This routine uses standard Fortran-style computations over a 2-dim array.
!
!  Notes:
!  Due to grid point reordering with DMDAs, we must always work
!  with the local grid points, and then transform them to the new
!  global numbering with the "ltog" mapping
!  We cannot work directly with the global numbers for the original
!  uniprocessor grid!
!
!  Two methods are available for imposing this transformation
!  when setting matrix entries:
!    (A) MatSetValuesLocal(), using the local ordering (including
!        ghost points!)
!        - Set matrix entries using the local ordering
!          by calling MatSetValuesLocal()
!    (B) MatSetValues(), using the global ordering
!        - Set matrix entries using the global ordering by calling
!          MatSetValues()
!  Option (A) seems cleaner/easier in many cases, and is the procedure
!  used in this example.
!
  subroutine FormJacobianLocal(x, jac_prec, user, ierr)
!  Input/output variables:
    type(userctx) user
    PetscScalar x(user%gxs:user%gxe, user%gys:user%gye)
    type(tMat) jac_prec
    PetscErrorCode ierr

!  Local variables:
    PetscInt row, col(5), i, j
    PetscInt ione, ifive
    PetscScalar two, one, hx, hy, hxdhy
    PetscScalar hydhx, sc, v(5)

!  Set parameters
    ione = 1
    ifive = 5
    one = 1.0
    two = 2.0
    hx = one/PetscIntToReal(user%mx - 1)
    hy = one/PetscIntToReal(user%my - 1)
    sc = hx*hy
    hxdhy = hx/hy
    hydhx = hy/hx

!  Compute entries for the locally owned part of the Jacobian.
!   - Currently, all PETSc parallel matrix formats are partitioned by
!     contiguous chunks of rows across the processors.
!   - Each processor needs to insert only elements that it owns
!     locally (but any non-local elements will be sent to the
!     appropriate processor during matrix assembly).
!   - Here, we set all entries for a particular row at once.
!   - We can set matrix entries either using either
!     MatSetValuesLocal() or MatSetValues(), as discussed above.
!   - Note that MatSetValues() uses 0-based row and column numbers
!     in Fortran as well as in C.

    do j = user%ys, user%ye
      row = (j - user%gys)*user%gxm + user%xs - user%gxs - 1
      do i = user%xs, user%xe
        row = row + 1
!           boundary points
        if (i == 1 .or. j == 1 .or. i == user%mx .or. j == user%my) then
          col(1) = row
          v(1) = one
          PetscCallA(MatSetValuesLocal(jac_prec, ione, [row], ione, col, v, INSERT_VALUES, ierr))
!           interior grid points
        else
          v(1) = -hxdhy
          v(2) = -hydhx
          v(3) = two*(hydhx + hxdhy) - sc*user%lambda*exp(x(i, j))
          v(4) = -hydhx
          v(5) = -hxdhy
          col(1) = row - user%gxm
          col(2) = row - 1
          col(3) = row
          col(4) = row + 1
          col(5) = row + user%gxm
          PetscCallA(MatSetValuesLocal(jac_prec, ione, [row], ifive, col, v, INSERT_VALUES, ierr))
        end if
      end do
    end do
  end

end module

program main
  use petscdmda
  use petscsnes
  use ex5f90tmodule
  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
!     matrix_free - flag - 1 indicates matrix-free version
!
  type(tSNES) mysnes
  type(tVec) x, r
  type(tMat) J
  PetscErrorCode ierr
  PetscInt its
  PetscBool flg, matrix_free, set
  PetscInt ione, nfour
  PetscReal lambda_max, lambda_min
  type(userctx) user
  type(userctx), pointer:: puser
  type(tPetscOptions) :: options

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

!  Initialize problem parameters
  options%v = 0
  lambda_max = 6.81
  lambda_min = 0.0
  user%lambda = 6.0
  ione = 1
  nfour = 4
  PetscCallA(PetscOptionsGetReal(options, PETSC_NULL_CHARACTER, '-par', user%lambda, flg, ierr))
  PetscCheckA(user%lambda < lambda_max .and. user%lambda > lambda_min, PETSC_COMM_SELF, PETSC_ERR_USER, 'Lambda provided with -par is out of range')

! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!  Create nonlinear solver context
! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  PetscCallA(SNESCreate(PETSC_COMM_WORLD, mysnes, ierr))

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

!  Create distributed array (DMDA) to manage parallel grid and vectors

! This really needs only the star-type stencil, but we use the box
! stencil temporarily.
  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, user%da, ierr))
  PetscCallA(DMSetFromOptions(user%da, ierr))
  PetscCallA(DMSetUp(user%da, ierr))
  PetscCallA(DMDAGetInfo(user%da, PETSC_NULL_INTEGER, user%mx, user%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))

!
!   Visualize the distribution of the array across the processors
!
!     PetscCallA(DMView(user%da,PETSC_VIEWER_DRAW_WORLD,ierr))

!  Extract global and local vectors from DMDA; then duplicate for remaining
!  vectors that are the same types
  PetscCallA(DMCreateGlobalVector(user%da, x, ierr))
  PetscCallA(VecDuplicate(x, r, ierr))

!  Get local grid boundaries (for 2-dimensional DMDA)
  PetscCallA(DMDAGetCorners(user%da, user%xs, user%ys, PETSC_NULL_INTEGER, user%xm, user%ym, PETSC_NULL_INTEGER, ierr))
  PetscCallA(DMDAGetGhostCorners(user%da, user%gxs, user%gys, PETSC_NULL_INTEGER, user%gxm, user%gym, PETSC_NULL_INTEGER, ierr))

!  Here we shift the starting indices up by one so that we can easily
!  use the Fortran convention of 1-based indices (rather 0-based indices).
  user%xs = user%xs + 1
  user%ys = user%ys + 1
  user%gxs = user%gxs + 1
  user%gys = user%gys + 1

  user%ye = user%ys + user%ym - 1
  user%xe = user%xs + user%xm - 1
  user%gye = user%gys + user%gym - 1
  user%gxe = user%gxs + user%gxm - 1

  PetscCallA(SNESSetApplicationContext(mysnes, user, ierr))

!  Set function evaluation routine and vector
  PetscCallA(SNESSetFunction(mysnes, r, FormFunction, user, ierr))

! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!  Create matrix data structure; set Jacobian evaluation routine
! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

!  Set Jacobian matrix data structure and default Jacobian evaluation
!  routine. User can override with:
!     -snes_fd : default finite differencing approximation of Jacobian
!     -snes_mf : matrix-free Newton-Krylov method with no preconditioning
!                (unless user explicitly sets preconditioner)
!     -snes_mf_operator : form matrix used to construct the preconditioner as set by the user,
!                         but use matrix-free approx for Jacobian-vector
!                         products within Newton-Krylov method
!
!  Note:  For the parallel case, vectors and matrices MUST be partitioned
!     accordingly.  When using distributed arrays (DMDAs) to create vectors,
!     the DMDAs determine the problem partitioning.  We must explicitly
!     specify the local matrix dimensions upon its creation for compatibility
!     with the vector distribution.  Thus, the generic MatCreate() routine
!     is NOT sufficient when working with distributed arrays.
!
!     Note: Here we only approximately preallocate storage space for the
!     Jacobian.  See the users manual for a discussion of better techniques
!     for preallocating matrix memory.

  PetscCallA(PetscOptionsHasName(options, PETSC_NULL_CHARACTER, '-snes_mf', matrix_free, ierr))
  if (.not. matrix_free) then
    PetscCallA(DMSetMatType(user%da, MATAIJ, ierr))
    PetscCallA(DMCreateMatrix(user%da, J, ierr))
    PetscCallA(SNESSetJacobian(mysnes, J, J, FormJacobian, user, ierr))
  end if

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

!     Test Fortran90 wrapper for SNESSet/Get ApplicationContext()
  PetscCallA(PetscOptionsGetBool(options, PETSC_NULL_CHARACTER, '-test_appctx', flg, set, ierr))
  if (flg) then
    PetscCallA(SNESGetApplicationContext(mysnes, puser, ierr))
  end if

! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!  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 (user%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.
! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  if (.not. matrix_free) PetscCallA(MatDestroy(J, ierr))
  PetscCallA(VecDestroy(x, ierr))
  PetscCallA(VecDestroy(r, ierr))
  PetscCallA(SNESDestroy(mysnes, ierr))
  PetscCallA(DMDestroy(user%da, ierr))

  PetscCallA(PetscFinalize(ierr))
end
!/*TEST
!
!   test:
!      nsize: 4
!      args: -snes_mf -pc_type none -da_processors_x 4 -da_processors_y 1 -snes_monitor_short -ksp_gmres_cgs_refinement_type refine_always
!
!TEST*/