xref: /petsc/src/snes/tests/ex20.c (revision b122ec5aa1bd4469eb4e0673542fb7de3f411254)

static char help[] ="Nonlinear Radiative Transport PDE with multigrid in 3d.\n\
Uses 3-dimensional distributed arrays.\n\
A 3-dim simplified Radiative Transport test problem is used, with analytic Jacobian. \n\
\n\
  Solves the linear systems via multilevel methods \n\
\n\
The command line\n\
options are:\n\
  -tleft <tl>, where <tl> indicates the left Diriclet BC \n\
  -tright <tr>, where <tr> indicates the right Diriclet BC \n\
  -beta <beta>, where <beta> indicates the exponent in T \n\n";

/*T
   Concepts: SNES^solving a system of nonlinear equations
   Concepts: DM^using distributed arrays
   Concepts: multigrid;
   Processors: n
T*/

/*

    This example models the partial differential equation

         - Div(alpha* T^beta (GRAD T)) = 0.

    where beta = 2.5 and alpha = 1.0

    BC: T_left = 1.0, T_right = 0.1, dT/dn_top = dTdn_bottom = dT/dn_up = dT/dn_down = 0.

    in the unit square, which is uniformly discretized in each of x and
    y in this simple encoding.  The degrees of freedom are cell centered.

    A finite volume approximation with the usual 7-point stencil
    is used to discretize the boundary value problem to obtain a
    nonlinear system of equations.

    This code was contributed by Nickolas Jovanovic based on ex18.c

*/

#include <petscsnes.h>
#include <petscdm.h>
#include <petscdmda.h>

/* User-defined application context */

typedef struct {
  PetscReal tleft,tright;     /* Dirichlet boundary conditions */
  PetscReal beta,bm1,coef;    /* nonlinear diffusivity parameterizations */
} AppCtx;

#define POWFLOP 5 /* assume a pow() takes five flops */

extern PetscErrorCode FormInitialGuess(SNES,Vec,void*);
extern PetscErrorCode FormFunction(SNES,Vec,Vec,void*);
extern PetscErrorCode FormJacobian(SNES,Vec,Mat,Mat,void*);

int main(int argc,char **argv)
{
  SNES           snes;
  AppCtx         user;
  PetscInt       its,lits;
  PetscReal      litspit;
  DM             da;

  CHKERRQ(PetscInitialize(&argc,&argv,NULL,help));
  /* set problem parameters */
  user.tleft  = 1.0;
  user.tright = 0.1;
  user.beta   = 2.5;
  CHKERRQ(PetscOptionsGetReal(NULL,NULL,"-tleft",&user.tleft,NULL));
  CHKERRQ(PetscOptionsGetReal(NULL,NULL,"-tright",&user.tright,NULL));
  CHKERRQ(PetscOptionsGetReal(NULL,NULL,"-beta",&user.beta,NULL));
  user.bm1    = user.beta - 1.0;
  user.coef   = user.beta/2.0;

  /*
      Set the DMDA (grid structure) for the grids.
  */
  CHKERRQ(DMDACreate3d(PETSC_COMM_WORLD,DM_BOUNDARY_NONE,DM_BOUNDARY_NONE,DM_BOUNDARY_NONE,DMDA_STENCIL_STAR,5,5,5,PETSC_DECIDE,PETSC_DECIDE,PETSC_DECIDE,1,1,0,0,0,&da));
  CHKERRQ(DMSetFromOptions(da));
  CHKERRQ(DMSetUp(da));
  CHKERRQ(DMSetApplicationContext(da,&user));

  /*
     Create the nonlinear solver
  */
  CHKERRQ(SNESCreate(PETSC_COMM_WORLD,&snes));
  CHKERRQ(SNESSetDM(snes,da));
  CHKERRQ(SNESSetFunction(snes,NULL,FormFunction,&user));
  CHKERRQ(SNESSetJacobian(snes,NULL,NULL,FormJacobian,&user));
  CHKERRQ(SNESSetFromOptions(snes));
  CHKERRQ(SNESSetComputeInitialGuess(snes,FormInitialGuess,NULL));

  CHKERRQ(SNESSolve(snes,NULL,NULL));
  CHKERRQ(SNESGetIterationNumber(snes,&its));
  CHKERRQ(SNESGetLinearSolveIterations(snes,&lits));
  litspit = ((PetscReal)lits)/((PetscReal)its);
  CHKERRQ(PetscPrintf(PETSC_COMM_WORLD,"Number of SNES iterations = %D\n",its));
  CHKERRQ(PetscPrintf(PETSC_COMM_WORLD,"Number of Linear iterations = %D\n",lits));
  CHKERRQ(PetscPrintf(PETSC_COMM_WORLD,"Average Linear its / SNES = %e\n",(double)litspit));

  CHKERRQ(SNESDestroy(&snes));
  CHKERRQ(DMDestroy(&da));
  CHKERRQ(PetscFinalize());
  return 0;
}
/* --------------------  Form initial approximation ----------------- */
PetscErrorCode FormInitialGuess(SNES snes,Vec X,void *ctx)
{
  AppCtx         *user;
  PetscInt       i,j,k,xs,ys,xm,ym,zs,zm;
  PetscScalar    ***x;
  DM             da;

  PetscFunctionBeginUser;
  CHKERRQ(SNESGetDM(snes,&da));
  CHKERRQ(DMGetApplicationContext(da,&user));
  CHKERRQ(DMDAGetCorners(da,&xs,&ys,&zs,&xm,&ym,&zm));
  CHKERRQ(DMDAVecGetArray(da,X,&x));

  /* Compute initial guess */
  for (k=zs; k<zs+zm; k++) {
    for (j=ys; j<ys+ym; j++) {
      for (i=xs; i<xs+xm; i++) {
        x[k][j][i] = user->tleft;
      }
    }
  }
  CHKERRQ(DMDAVecRestoreArray(da,X,&x));
  PetscFunctionReturn(0);
}
/* --------------------  Evaluate Function F(x) --------------------- */
PetscErrorCode FormFunction(SNES snes,Vec X,Vec F,void *ptr)
{
  AppCtx         *user = (AppCtx*)ptr;
  PetscInt       i,j,k,mx,my,mz,xs,ys,zs,xm,ym,zm;
  PetscScalar    zero = 0.0,one = 1.0;
  PetscScalar    hx,hy,hz,hxhydhz,hyhzdhx,hzhxdhy;
  PetscScalar    t0,tn,ts,te,tw,an,as,ae,aw,dn,ds,de,dw,fn = 0.0,fs = 0.0,fe =0.0,fw = 0.0;
  PetscScalar    tleft,tright,beta,td,ad,dd,fd=0.0,tu,au,du=0.0,fu=0.0;
  PetscScalar    ***x,***f;
  Vec            localX;
  DM             da;

  PetscFunctionBeginUser;
  CHKERRQ(SNESGetDM(snes,&da));
  CHKERRQ(DMGetLocalVector(da,&localX));
  CHKERRQ(DMDAGetInfo(da,NULL,&mx,&my,&mz,0,0,0,0,0,0,0,0,0));
  hx      = one/(PetscReal)(mx-1);  hy    = one/(PetscReal)(my-1);  hz = one/(PetscReal)(mz-1);
  hxhydhz = hx*hy/hz;   hyhzdhx = hy*hz/hx;   hzhxdhy = hz*hx/hy;
  tleft   = user->tleft;         tright = user->tright;
  beta    = user->beta;

  /* Get ghost points */
  CHKERRQ(DMGlobalToLocalBegin(da,X,INSERT_VALUES,localX));
  CHKERRQ(DMGlobalToLocalEnd(da,X,INSERT_VALUES,localX));
  CHKERRQ(DMDAGetCorners(da,&xs,&ys,&zs,&xm,&ym,&zm));
  CHKERRQ(DMDAVecGetArray(da,localX,&x));
  CHKERRQ(DMDAVecGetArray(da,F,&f));

  /* Evaluate function */
  for (k=zs; k<zs+zm; k++) {
    for (j=ys; j<ys+ym; j++) {
      for (i=xs; i<xs+xm; i++) {
        t0 = x[k][j][i];

        if (i > 0 && i < mx-1 && j > 0 && j < my-1 && k > 0 && k < mz-1) {

          /* general interior volume */

          tw = x[k][j][i-1];
          aw = 0.5*(t0 + tw);
          dw = PetscPowScalar(aw,beta);
          fw = dw*(t0 - tw);

          te = x[k][j][i+1];
          ae = 0.5*(t0 + te);
          de = PetscPowScalar(ae,beta);
          fe = de*(te - t0);

          ts = x[k][j-1][i];
          as = 0.5*(t0 + ts);
          ds = PetscPowScalar(as,beta);
          fs = ds*(t0 - ts);

          tn = x[k][j+1][i];
          an = 0.5*(t0 + tn);
          dn = PetscPowScalar(an,beta);
          fn = dn*(tn - t0);

          td = x[k-1][j][i];
          ad = 0.5*(t0 + td);
          dd = PetscPowScalar(ad,beta);
          fd = dd*(t0 - td);

          tu = x[k+1][j][i];
          au = 0.5*(t0 + tu);
          du = PetscPowScalar(au,beta);
          fu = du*(tu - t0);

        } else if (i == 0) {

          /* left-hand (west) boundary */
          tw = tleft;
          aw = 0.5*(t0 + tw);
          dw = PetscPowScalar(aw,beta);
          fw = dw*(t0 - tw);

          te = x[k][j][i+1];
          ae = 0.5*(t0 + te);
          de = PetscPowScalar(ae,beta);
          fe = de*(te - t0);

          if (j > 0) {
            ts = x[k][j-1][i];
            as = 0.5*(t0 + ts);
            ds = PetscPowScalar(as,beta);
            fs = ds*(t0 - ts);
          } else {
            fs = zero;
          }

          if (j < my-1) {
            tn = x[k][j+1][i];
            an = 0.5*(t0 + tn);
            dn = PetscPowScalar(an,beta);
            fn = dn*(tn - t0);
          } else {
            fn = zero;
          }

          if (k > 0) {
            td = x[k-1][j][i];
            ad = 0.5*(t0 + td);
            dd = PetscPowScalar(ad,beta);
            fd = dd*(t0 - td);
          } else {
            fd = zero;
          }

          if (k < mz-1) {
            tu = x[k+1][j][i];
            au = 0.5*(t0 + tu);
            du = PetscPowScalar(au,beta);
            fu = du*(tu - t0);
          } else {
            fu = zero;
          }

        } else if (i == mx-1) {

          /* right-hand (east) boundary */
          tw = x[k][j][i-1];
          aw = 0.5*(t0 + tw);
          dw = PetscPowScalar(aw,beta);
          fw = dw*(t0 - tw);

          te = tright;
          ae = 0.5*(t0 + te);
          de = PetscPowScalar(ae,beta);
          fe = de*(te - t0);

          if (j > 0) {
            ts = x[k][j-1][i];
            as = 0.5*(t0 + ts);
            ds = PetscPowScalar(as,beta);
            fs = ds*(t0 - ts);
          } else {
            fs = zero;
          }

          if (j < my-1) {
            tn = x[k][j+1][i];
            an = 0.5*(t0 + tn);
            dn = PetscPowScalar(an,beta);
            fn = dn*(tn - t0);
          } else {
            fn = zero;
          }

          if (k > 0) {
            td = x[k-1][j][i];
            ad = 0.5*(t0 + td);
            dd = PetscPowScalar(ad,beta);
            fd = dd*(t0 - td);
          } else {
            fd = zero;
          }

          if (k < mz-1) {
            tu = x[k+1][j][i];
            au = 0.5*(t0 + tu);
            du = PetscPowScalar(au,beta);
            fu = du*(tu - t0);
          } else {
            fu = zero;
          }

        } else if (j == 0) {

          /* bottom (south) boundary, and i <> 0 or mx-1 */
          tw = x[k][j][i-1];
          aw = 0.5*(t0 + tw);
          dw = PetscPowScalar(aw,beta);
          fw = dw*(t0 - tw);

          te = x[k][j][i+1];
          ae = 0.5*(t0 + te);
          de = PetscPowScalar(ae,beta);
          fe = de*(te - t0);

          fs = zero;

          tn = x[k][j+1][i];
          an = 0.5*(t0 + tn);
          dn = PetscPowScalar(an,beta);
          fn = dn*(tn - t0);

          if (k > 0) {
            td = x[k-1][j][i];
            ad = 0.5*(t0 + td);
            dd = PetscPowScalar(ad,beta);
            fd = dd*(t0 - td);
          } else {
            fd = zero;
          }

          if (k < mz-1) {
            tu = x[k+1][j][i];
            au = 0.5*(t0 + tu);
            du = PetscPowScalar(au,beta);
            fu = du*(tu - t0);
          } else {
            fu = zero;
          }

        } else if (j == my-1) {

          /* top (north) boundary, and i <> 0 or mx-1 */
          tw = x[k][j][i-1];
          aw = 0.5*(t0 + tw);
          dw = PetscPowScalar(aw,beta);
          fw = dw*(t0 - tw);

          te = x[k][j][i+1];
          ae = 0.5*(t0 + te);
          de = PetscPowScalar(ae,beta);
          fe = de*(te - t0);

          ts = x[k][j-1][i];
          as = 0.5*(t0 + ts);
          ds = PetscPowScalar(as,beta);
          fs = ds*(t0 - ts);

          fn = zero;

          if (k > 0) {
            td = x[k-1][j][i];
            ad = 0.5*(t0 + td);
            dd = PetscPowScalar(ad,beta);
            fd = dd*(t0 - td);
          } else {
            fd = zero;
          }

          if (k < mz-1) {
            tu = x[k+1][j][i];
            au = 0.5*(t0 + tu);
            du = PetscPowScalar(au,beta);
            fu = du*(tu - t0);
          } else {
            fu = zero;
          }

        } else if (k == 0) {

          /* down boundary (interior only) */
          tw = x[k][j][i-1];
          aw = 0.5*(t0 + tw);
          dw = PetscPowScalar(aw,beta);
          fw = dw*(t0 - tw);

          te = x[k][j][i+1];
          ae = 0.5*(t0 + te);
          de = PetscPowScalar(ae,beta);
          fe = de*(te - t0);

          ts = x[k][j-1][i];
          as = 0.5*(t0 + ts);
          ds = PetscPowScalar(as,beta);
          fs = ds*(t0 - ts);

          tn = x[k][j+1][i];
          an = 0.5*(t0 + tn);
          dn = PetscPowScalar(an,beta);
          fn = dn*(tn - t0);

          fd = zero;

          tu = x[k+1][j][i];
          au = 0.5*(t0 + tu);
          du = PetscPowScalar(au,beta);
          fu = du*(tu - t0);

        } else if (k == mz-1) {

          /* up boundary (interior only) */
          tw = x[k][j][i-1];
          aw = 0.5*(t0 + tw);
          dw = PetscPowScalar(aw,beta);
          fw = dw*(t0 - tw);

          te = x[k][j][i+1];
          ae = 0.5*(t0 + te);
          de = PetscPowScalar(ae,beta);
          fe = de*(te - t0);

          ts = x[k][j-1][i];
          as = 0.5*(t0 + ts);
          ds = PetscPowScalar(as,beta);
          fs = ds*(t0 - ts);

          tn = x[k][j+1][i];
          an = 0.5*(t0 + tn);
          dn = PetscPowScalar(an,beta);
          fn = dn*(tn - t0);

          td = x[k-1][j][i];
          ad = 0.5*(t0 + td);
          dd = PetscPowScalar(ad,beta);
          fd = dd*(t0 - td);

          fu = zero;
        }

        f[k][j][i] = -hyhzdhx*(fe-fw) - hzhxdhy*(fn-fs) - hxhydhz*(fu-fd);
      }
    }
  }
  CHKERRQ(DMDAVecRestoreArray(da,localX,&x));
  CHKERRQ(DMDAVecRestoreArray(da,F,&f));
  CHKERRQ(DMRestoreLocalVector(da,&localX));
  CHKERRQ(PetscLogFlops((22.0 + 4.0*POWFLOP)*ym*xm));
  PetscFunctionReturn(0);
}
/* --------------------  Evaluate Jacobian F(x) --------------------- */
PetscErrorCode FormJacobian(SNES snes,Vec X,Mat J,Mat jac,void *ptr)
{
  AppCtx         *user = (AppCtx*)ptr;
  PetscInt       i,j,k,mx,my,mz,xs,ys,zs,xm,ym,zm;
  PetscScalar    one = 1.0;
  PetscScalar    hx,hy,hz,hxhydhz,hyhzdhx,hzhxdhy;
  PetscScalar    t0,tn,ts,te,tw,an,as,ae,aw,dn,ds,de,dw;
  PetscScalar    tleft,tright,beta,td,ad,dd,tu,au,du,v[7],bm1,coef;
  PetscScalar    ***x,bn,bs,be,bw,bu,bd,gn,gs,ge,gw,gu,gd;
  Vec            localX;
  MatStencil     c[7],row;
  DM             da;

  PetscFunctionBeginUser;
  CHKERRQ(SNESGetDM(snes,&da));
  CHKERRQ(DMGetLocalVector(da,&localX));
  CHKERRQ(DMDAGetInfo(da,NULL,&mx,&my,&mz,0,0,0,0,0,0,0,0,0));
  hx      = one/(PetscReal)(mx-1);  hy    = one/(PetscReal)(my-1);  hz = one/(PetscReal)(mz-1);
  hxhydhz = hx*hy/hz;   hyhzdhx = hy*hz/hx;   hzhxdhy = hz*hx/hy;
  tleft   = user->tleft;         tright = user->tright;
  beta    = user->beta;          bm1    = user->bm1;              coef = user->coef;

  /* Get ghost points */
  CHKERRQ(DMGlobalToLocalBegin(da,X,INSERT_VALUES,localX));
  CHKERRQ(DMGlobalToLocalEnd(da,X,INSERT_VALUES,localX));
  CHKERRQ(DMDAGetCorners(da,&xs,&ys,&zs,&xm,&ym,&zm));
  CHKERRQ(DMDAVecGetArray(da,localX,&x));

  /* Evaluate Jacobian of function */
  for (k=zs; k<zs+zm; k++) {
    for (j=ys; j<ys+ym; j++) {
      for (i=xs; i<xs+xm; i++) {
        t0    = x[k][j][i];
        row.k = k; row.j = j; row.i = i;
        if (i > 0 && i < mx-1 && j > 0 && j < my-1 && k > 0 && k < mz-1) {

          /* general interior volume */

          tw = x[k][j][i-1];
          aw = 0.5*(t0 + tw);
          bw = PetscPowScalar(aw,bm1);
          /* dw = bw * aw */
          dw = PetscPowScalar(aw,beta);
          gw = coef*bw*(t0 - tw);

          te = x[k][j][i+1];
          ae = 0.5*(t0 + te);
          be = PetscPowScalar(ae,bm1);
          /* de = be * ae; */
          de = PetscPowScalar(ae,beta);
          ge = coef*be*(te - t0);

          ts = x[k][j-1][i];
          as = 0.5*(t0 + ts);
          bs = PetscPowScalar(as,bm1);
          /* ds = bs * as; */
          ds = PetscPowScalar(as,beta);
          gs = coef*bs*(t0 - ts);

          tn = x[k][j+1][i];
          an = 0.5*(t0 + tn);
          bn = PetscPowScalar(an,bm1);
          /* dn = bn * an; */
          dn = PetscPowScalar(an,beta);
          gn = coef*bn*(tn - t0);

          td = x[k-1][j][i];
          ad = 0.5*(t0 + td);
          bd = PetscPowScalar(ad,bm1);
          /* dd = bd * ad; */
          dd = PetscPowScalar(ad,beta);
          gd = coef*bd*(t0 - td);

          tu = x[k+1][j][i];
          au = 0.5*(t0 + tu);
          bu = PetscPowScalar(au,bm1);
          /* du = bu * au; */
          du = PetscPowScalar(au,beta);
          gu = coef*bu*(tu - t0);

          c[0].k = k-1; c[0].j = j; c[0].i = i; v[0]   = -hxhydhz*(dd - gd);
          c[1].k = k; c[1].j = j-1; c[1].i = i;
          v[1]   = -hzhxdhy*(ds - gs);
          c[2].k = k; c[2].j = j; c[2].i = i-1;
          v[2]   = -hyhzdhx*(dw - gw);
          c[3].k = k; c[3].j = j; c[3].i = i;
          v[3]   =   hzhxdhy*(ds + dn + gs - gn) + hyhzdhx*(dw + de + gw - ge) + hxhydhz*(dd + du + gd - gu);
          c[4].k = k; c[4].j = j; c[4].i = i+1;
          v[4]   = -hyhzdhx*(de + ge);
          c[5].k = k; c[5].j = j+1; c[5].i = i;
          v[5]   = -hzhxdhy*(dn + gn);
          c[6].k = k+1; c[6].j = j; c[6].i = i;
          v[6]   = -hxhydhz*(du + gu);
          CHKERRQ(MatSetValuesStencil(jac,1,&row,7,c,v,INSERT_VALUES));

        } else if (i == 0) {

          /* left-hand plane boundary */
          tw = tleft;
          aw = 0.5*(t0 + tw);
          bw = PetscPowScalar(aw,bm1);
          /* dw = bw * aw */
          dw = PetscPowScalar(aw,beta);
          gw = coef*bw*(t0 - tw);

          te = x[k][j][i+1];
          ae = 0.5*(t0 + te);
          be = PetscPowScalar(ae,bm1);
          /* de = be * ae; */
          de = PetscPowScalar(ae,beta);
          ge = coef*be*(te - t0);

          /* left-hand bottom edge */
          if (j == 0) {

            tn = x[k][j+1][i];
            an = 0.5*(t0 + tn);
            bn = PetscPowScalar(an,bm1);
            /* dn = bn * an; */
            dn = PetscPowScalar(an,beta);
            gn = coef*bn*(tn - t0);

            /* left-hand bottom down corner */
            if (k == 0) {

              tu = x[k+1][j][i];
              au = 0.5*(t0 + tu);
              bu = PetscPowScalar(au,bm1);
              /* du = bu * au; */
              du = PetscPowScalar(au,beta);
              gu = coef*bu*(tu - t0);

              c[0].k = k; c[0].j = j; c[0].i = i;
              v[0]   =   hzhxdhy*(dn - gn) + hyhzdhx*(dw + de + gw - ge) + hxhydhz*(du - gu);
              c[1].k = k; c[1].j = j; c[1].i = i+1;
              v[1]   = -hyhzdhx*(de + ge);
              c[2].k = k; c[2].j = j+1; c[2].i = i;
              v[2]   = -hzhxdhy*(dn + gn);
              c[3].k = k+1; c[3].j = j; c[3].i = i;
              v[3]   = -hxhydhz*(du + gu);
              CHKERRQ(MatSetValuesStencil(jac,1,&row,4,c,v,INSERT_VALUES));

              /* left-hand bottom interior edge */
            } else if (k < mz-1) {

              tu = x[k+1][j][i];
              au = 0.5*(t0 + tu);
              bu = PetscPowScalar(au,bm1);
              /* du = bu * au; */
              du = PetscPowScalar(au,beta);
              gu = coef*bu*(tu - t0);

              td = x[k-1][j][i];
              ad = 0.5*(t0 + td);
              bd = PetscPowScalar(ad,bm1);
              /* dd = bd * ad; */
              dd = PetscPowScalar(ad,beta);
              gd = coef*bd*(td - t0);

              c[0].k = k-1; c[0].j = j; c[0].i = i;
              v[0]   = -hxhydhz*(dd - gd);
              c[1].k = k; c[1].j = j; c[1].i = i;
              v[1]   =   hzhxdhy*(dn - gn) + hyhzdhx*(dw + de + gw - ge) + hxhydhz*(dd + du + gd - gu);
              c[2].k = k; c[2].j = j; c[2].i = i+1;
              v[2]   = -hyhzdhx*(de + ge);
              c[3].k = k; c[3].j = j+1; c[3].i = i;
              v[3]   = -hzhxdhy*(dn + gn);
              c[4].k = k+1; c[4].j = j; c[4].i = i;
              v[4]   = -hxhydhz*(du + gu);
              CHKERRQ(MatSetValuesStencil(jac,1,&row,5,c,v,INSERT_VALUES));

              /* left-hand bottom up corner */
            } else {

              td = x[k-1][j][i];
              ad = 0.5*(t0 + td);
              bd = PetscPowScalar(ad,bm1);
              /* dd = bd * ad; */
              dd = PetscPowScalar(ad,beta);
              gd = coef*bd*(td - t0);

              c[0].k = k-1; c[0].j = j; c[0].i = i;
              v[0]   = -hxhydhz*(dd - gd);
              c[1].k = k; c[1].j = j; c[1].i = i;
              v[1]   =   hzhxdhy*(dn - gn) + hyhzdhx*(dw + de + gw - ge) + hxhydhz*(dd + gd);
              c[2].k = k; c[2].j = j; c[2].i = i+1;
              v[2]   = -hyhzdhx*(de + ge);
              c[3].k = k; c[3].j = j+1; c[3].i = i;
              v[3]   = -hzhxdhy*(dn + gn);
              CHKERRQ(MatSetValuesStencil(jac,1,&row,4,c,v,INSERT_VALUES));
            }

            /* left-hand top edge */
          } else if (j == my-1) {

            ts = x[k][j-1][i];
            as = 0.5*(t0 + ts);
            bs = PetscPowScalar(as,bm1);
            /* ds = bs * as; */
            ds = PetscPowScalar(as,beta);
            gs = coef*bs*(ts - t0);

            /* left-hand top down corner */
            if (k == 0) {

              tu = x[k+1][j][i];
              au = 0.5*(t0 + tu);
              bu = PetscPowScalar(au,bm1);
              /* du = bu * au; */
              du = PetscPowScalar(au,beta);
              gu = coef*bu*(tu - t0);

              c[0].k = k; c[0].j = j-1; c[0].i = i;
              v[0]   = -hzhxdhy*(ds - gs);
              c[1].k = k; c[1].j = j; c[1].i = i;
              v[1]   =   hzhxdhy*(ds + gs) + hyhzdhx*(dw + de + gw - ge) + hxhydhz*(du - gu);
              c[2].k = k; c[2].j = j; c[2].i = i+1;
              v[2]   = -hyhzdhx*(de + ge);
              c[3].k = k+1; c[3].j = j; c[3].i = i;
              v[3]   = -hxhydhz*(du + gu);
              CHKERRQ(MatSetValuesStencil(jac,1,&row,4,c,v,INSERT_VALUES));

              /* left-hand top interior edge */
            } else if (k < mz-1) {

              tu = x[k+1][j][i];
              au = 0.5*(t0 + tu);
              bu = PetscPowScalar(au,bm1);
              /* du = bu * au; */
              du = PetscPowScalar(au,beta);
              gu = coef*bu*(tu - t0);

              td = x[k-1][j][i];
              ad = 0.5*(t0 + td);
              bd = PetscPowScalar(ad,bm1);
              /* dd = bd * ad; */
              dd = PetscPowScalar(ad,beta);
              gd = coef*bd*(td - t0);

              c[0].k = k-1; c[0].j = j; c[0].i = i;
              v[0]   = -hxhydhz*(dd - gd);
              c[1].k = k; c[1].j = j-1; c[1].i = i;
              v[1]   = -hzhxdhy*(ds - gs);
              c[2].k = k; c[2].j = j; c[2].i = i;
              v[2]   =   hzhxdhy*(ds + gs) + hyhzdhx*(dw + de + gw - ge) + hxhydhz*(dd + du + gd - gu);
              c[3].k = k; c[3].j = j; c[3].i = i+1;
              v[3]   = -hyhzdhx*(de + ge);
              c[4].k = k+1; c[4].j = j; c[4].i = i;
              v[4]   = -hxhydhz*(du + gu);
              CHKERRQ(MatSetValuesStencil(jac,1,&row,5,c,v,INSERT_VALUES));

              /* left-hand top up corner */
            } else {

              td = x[k-1][j][i];
              ad = 0.5*(t0 + td);
              bd = PetscPowScalar(ad,bm1);
              /* dd = bd * ad; */
              dd = PetscPowScalar(ad,beta);
              gd = coef*bd*(td - t0);

              c[0].k = k-1; c[0].j = j; c[0].i = i;
              v[0]   = -hxhydhz*(dd - gd);
              c[1].k = k; c[1].j = j-1; c[1].i = i;
              v[1]   = -hzhxdhy*(ds - gs);
              c[2].k = k; c[2].j = j; c[2].i = i;
              v[2]   =   hzhxdhy*(ds + gs) + hyhzdhx*(dw + de + gw - ge) + hxhydhz*(dd + gd);
              c[3].k = k; c[3].j = j; c[3].i = i+1;
              v[3]   = -hyhzdhx*(de + ge);
              CHKERRQ(MatSetValuesStencil(jac,1,&row,4,c,v,INSERT_VALUES));
            }

          } else {

            ts = x[k][j-1][i];
            as = 0.5*(t0 + ts);
            bs = PetscPowScalar(as,bm1);
            /* ds = bs * as; */
            ds = PetscPowScalar(as,beta);
            gs = coef*bs*(t0 - ts);

            tn = x[k][j+1][i];
            an = 0.5*(t0 + tn);
            bn = PetscPowScalar(an,bm1);
            /* dn = bn * an; */
            dn = PetscPowScalar(an,beta);
            gn = coef*bn*(tn - t0);

            /* left-hand down interior edge */
            if (k == 0) {

              tu = x[k+1][j][i];
              au = 0.5*(t0 + tu);
              bu = PetscPowScalar(au,bm1);
              /* du = bu * au; */
              du = PetscPowScalar(au,beta);
              gu = coef*bu*(tu - t0);

              c[0].k = k; c[0].j = j-1; c[0].i = i;
              v[0]   = -hzhxdhy*(ds - gs);
              c[1].k = k; c[1].j = j; c[1].i = i;
              v[1]   =   hzhxdhy*(ds + dn + gs - gn) + hyhzdhx*(dw + de + gw - ge) + hxhydhz*(du - gu);
              c[2].k = k; c[2].j = j; c[2].i = i+1;
              v[2]   = -hyhzdhx*(de + ge);
              c[3].k = k; c[3].j = j+1; c[3].i = i;
              v[3]   = -hzhxdhy*(dn + gn);
              c[4].k = k+1; c[4].j = j; c[4].i = i;
              v[4]   = -hxhydhz*(du + gu);
              CHKERRQ(MatSetValuesStencil(jac,1,&row,5,c,v,INSERT_VALUES));

            } else if (k == mz-1) { /* left-hand up interior edge */

              td = x[k-1][j][i];
              ad = 0.5*(t0 + td);
              bd = PetscPowScalar(ad,bm1);
              /* dd = bd * ad; */
              dd = PetscPowScalar(ad,beta);
              gd = coef*bd*(t0 - td);

              c[0].k = k-1; c[0].j = j; c[0].i = i;
              v[0]   = -hxhydhz*(dd - gd);
              c[1].k = k; c[1].j = j-1; c[1].i = i;
              v[1]   = -hzhxdhy*(ds - gs);
              c[2].k = k; c[2].j = j; c[2].i = i;
              v[2]   =   hzhxdhy*(ds + dn + gs - gn) + hyhzdhx*(dw + de + gw - ge) + hxhydhz*(dd + gd);
              c[3].k = k; c[3].j = j; c[3].i = i+1;
              v[3]   = -hyhzdhx*(de + ge);
              c[4].k = k; c[4].j = j+1; c[4].i = i;
              v[4]   = -hzhxdhy*(dn + gn);
              CHKERRQ(MatSetValuesStencil(jac,1,&row,5,c,v,INSERT_VALUES));
            } else { /* left-hand interior plane */

              td = x[k-1][j][i];
              ad = 0.5*(t0 + td);
              bd = PetscPowScalar(ad,bm1);
              /* dd = bd * ad; */
              dd = PetscPowScalar(ad,beta);
              gd = coef*bd*(t0 - td);

              tu = x[k+1][j][i];
              au = 0.5*(t0 + tu);
              bu = PetscPowScalar(au,bm1);
              /* du = bu * au; */
              du = PetscPowScalar(au,beta);
              gu = coef*bu*(tu - t0);

              c[0].k = k-1; c[0].j = j; c[0].i = i;
              v[0]   = -hxhydhz*(dd - gd);
              c[1].k = k; c[1].j = j-1; c[1].i = i;
              v[1]   = -hzhxdhy*(ds - gs);
              c[2].k = k; c[2].j = j; c[2].i = i;
              v[2]   =   hzhxdhy*(ds + dn + gs - gn) + hyhzdhx*(dw + de + gw - ge) + hxhydhz*(dd + du + gd - gu);
              c[3].k = k; c[3].j = j; c[3].i = i+1;
              v[3]   = -hyhzdhx*(de + ge);
              c[4].k = k; c[4].j = j+1; c[4].i = i;
              v[4]   = -hzhxdhy*(dn + gn);
              c[5].k = k+1; c[5].j = j; c[5].i = i;
              v[5]   = -hxhydhz*(du + gu);
              CHKERRQ(MatSetValuesStencil(jac,1,&row,6,c,v,INSERT_VALUES));
            }
          }

        } else if (i == mx-1) {

          /* right-hand plane boundary */
          tw = x[k][j][i-1];
          aw = 0.5*(t0 + tw);
          bw = PetscPowScalar(aw,bm1);
          /* dw = bw * aw */
          dw = PetscPowScalar(aw,beta);
          gw = coef*bw*(t0 - tw);

          te = tright;
          ae = 0.5*(t0 + te);
          be = PetscPowScalar(ae,bm1);
          /* de = be * ae; */
          de = PetscPowScalar(ae,beta);
          ge = coef*be*(te - t0);

          /* right-hand bottom edge */
          if (j == 0) {

            tn = x[k][j+1][i];
            an = 0.5*(t0 + tn);
            bn = PetscPowScalar(an,bm1);
            /* dn = bn * an; */
            dn = PetscPowScalar(an,beta);
            gn = coef*bn*(tn - t0);

            /* right-hand bottom down corner */
            if (k == 0) {

              tu = x[k+1][j][i];
              au = 0.5*(t0 + tu);
              bu = PetscPowScalar(au,bm1);
              /* du = bu * au; */
              du = PetscPowScalar(au,beta);
              gu = coef*bu*(tu - t0);

              c[0].k = k; c[0].j = j; c[0].i = i-1;
              v[0]   = -hyhzdhx*(dw - gw);
              c[1].k = k; c[1].j = j; c[1].i = i;
              v[1]   =   hzhxdhy*(dn - gn) + hyhzdhx*(dw + de + gw - ge) + hxhydhz*(du - gu);
              c[2].k = k; c[2].j = j+1; c[2].i = i;
              v[2]   = -hzhxdhy*(dn + gn);
              c[3].k = k+1; c[3].j = j; c[3].i = i;
              v[3]   = -hxhydhz*(du + gu);
              CHKERRQ(MatSetValuesStencil(jac,1,&row,4,c,v,INSERT_VALUES));

              /* right-hand bottom interior edge */
            } else if (k < mz-1) {

              tu = x[k+1][j][i];
              au = 0.5*(t0 + tu);
              bu = PetscPowScalar(au,bm1);
              /* du = bu * au; */
              du = PetscPowScalar(au,beta);
              gu = coef*bu*(tu - t0);

              td = x[k-1][j][i];
              ad = 0.5*(t0 + td);
              bd = PetscPowScalar(ad,bm1);
              /* dd = bd * ad; */
              dd = PetscPowScalar(ad,beta);
              gd = coef*bd*(td - t0);

              c[0].k = k-1; c[0].j = j; c[0].i = i;
              v[0]   = -hxhydhz*(dd - gd);
              c[1].k = k; c[1].j = j; c[1].i = i-1;
              v[1]   = -hyhzdhx*(dw - gw);
              c[2].k = k; c[2].j = j; c[2].i = i;
              v[2]   =   hzhxdhy*(dn - gn) + hyhzdhx*(dw + de + gw - ge) + hxhydhz*(dd + du + gd - gu);
              c[3].k = k; c[3].j = j+1; c[3].i = i;
              v[3]   = -hzhxdhy*(dn + gn);
              c[4].k = k+1; c[4].j = j; c[4].i = i;
              v[4]   = -hxhydhz*(du + gu);
              CHKERRQ(MatSetValuesStencil(jac,1,&row,5,c,v,INSERT_VALUES));

              /* right-hand bottom up corner */
            } else {

              td = x[k-1][j][i];
              ad = 0.5*(t0 + td);
              bd = PetscPowScalar(ad,bm1);
              /* dd = bd * ad; */
              dd = PetscPowScalar(ad,beta);
              gd = coef*bd*(td - t0);

              c[0].k = k-1; c[0].j = j; c[0].i = i;
              v[0]   = -hxhydhz*(dd - gd);
              c[1].k = k; c[1].j = j; c[1].i = i-1;
              v[1]   = -hyhzdhx*(dw - gw);
              c[2].k = k; c[2].j = j; c[2].i = i;
              v[2]   =   hzhxdhy*(dn - gn) + hyhzdhx*(dw + de + gw - ge) + hxhydhz*(dd + gd);
              c[3].k = k; c[3].j = j+1; c[3].i = i;
              v[3]   = -hzhxdhy*(dn + gn);
              CHKERRQ(MatSetValuesStencil(jac,1,&row,4,c,v,INSERT_VALUES));
            }

            /* right-hand top edge */
          } else if (j == my-1) {

            ts = x[k][j-1][i];
            as = 0.5*(t0 + ts);
            bs = PetscPowScalar(as,bm1);
            /* ds = bs * as; */
            ds = PetscPowScalar(as,beta);
            gs = coef*bs*(ts - t0);

            /* right-hand top down corner */
            if (k == 0) {

              tu = x[k+1][j][i];
              au = 0.5*(t0 + tu);
              bu = PetscPowScalar(au,bm1);
              /* du = bu * au; */
              du = PetscPowScalar(au,beta);
              gu = coef*bu*(tu - t0);

              c[0].k = k; c[0].j = j-1; c[0].i = i;
              v[0]   = -hzhxdhy*(ds - gs);
              c[1].k = k; c[1].j = j; c[1].i = i-1;
              v[1]   = -hyhzdhx*(dw - gw);
              c[2].k = k; c[2].j = j; c[2].i = i;
              v[2]   =   hzhxdhy*(ds + gs) + hyhzdhx*(dw + de + gw - ge) + hxhydhz*(du - gu);
              c[3].k = k+1; c[3].j = j; c[3].i = i;
              v[3]   = -hxhydhz*(du + gu);
              CHKERRQ(MatSetValuesStencil(jac,1,&row,4,c,v,INSERT_VALUES));

              /* right-hand top interior edge */
            } else if (k < mz-1) {

              tu = x[k+1][j][i];
              au = 0.5*(t0 + tu);
              bu = PetscPowScalar(au,bm1);
              /* du = bu * au; */
              du = PetscPowScalar(au,beta);
              gu = coef*bu*(tu - t0);

              td = x[k-1][j][i];
              ad = 0.5*(t0 + td);
              bd = PetscPowScalar(ad,bm1);
              /* dd = bd * ad; */
              dd = PetscPowScalar(ad,beta);
              gd = coef*bd*(td - t0);

              c[0].k = k-1; c[0].j = j; c[0].i = i;
              v[0]   = -hxhydhz*(dd - gd);
              c[1].k = k; c[1].j = j-1; c[1].i = i;
              v[1]   = -hzhxdhy*(ds - gs);
              c[2].k = k; c[2].j = j; c[2].i = i-1;
              v[2]   = -hyhzdhx*(dw - gw);
              c[3].k = k; c[3].j = j; c[3].i = i;
              v[3]   =   hzhxdhy*(ds + gs) + hyhzdhx*(dw + de + gw - ge) + hxhydhz*(dd + du + gd - gu);
              c[4].k = k+1; c[4].j = j; c[4].i = i;
              v[4]   = -hxhydhz*(du + gu);
              CHKERRQ(MatSetValuesStencil(jac,1,&row,5,c,v,INSERT_VALUES));

              /* right-hand top up corner */
            } else {

              td = x[k-1][j][i];
              ad = 0.5*(t0 + td);
              bd = PetscPowScalar(ad,bm1);
              /* dd = bd * ad; */
              dd = PetscPowScalar(ad,beta);
              gd = coef*bd*(td - t0);

              c[0].k = k-1; c[0].j = j; c[0].i = i;
              v[0]   = -hxhydhz*(dd - gd);
              c[1].k = k; c[1].j = j-1; c[1].i = i;
              v[1]   = -hzhxdhy*(ds - gs);
              c[2].k = k; c[2].j = j; c[2].i = i-1;
              v[2]   = -hyhzdhx*(dw - gw);
              c[3].k = k; c[3].j = j; c[3].i = i;
              v[3]   =   hzhxdhy*(ds + gs) + hyhzdhx*(dw + de + gw - ge) + hxhydhz*(dd + gd);
              CHKERRQ(MatSetValuesStencil(jac,1,&row,4,c,v,INSERT_VALUES));
            }

          } else {

            ts = x[k][j-1][i];
            as = 0.5*(t0 + ts);
            bs = PetscPowScalar(as,bm1);
            /* ds = bs * as; */
            ds = PetscPowScalar(as,beta);
            gs = coef*bs*(t0 - ts);

            tn = x[k][j+1][i];
            an = 0.5*(t0 + tn);
            bn = PetscPowScalar(an,bm1);
            /* dn = bn * an; */
            dn = PetscPowScalar(an,beta);
            gn = coef*bn*(tn - t0);

            /* right-hand down interior edge */
            if (k == 0) {

              tu = x[k+1][j][i];
              au = 0.5*(t0 + tu);
              bu = PetscPowScalar(au,bm1);
              /* du = bu * au; */
              du = PetscPowScalar(au,beta);
              gu = coef*bu*(tu - t0);

              c[0].k = k; c[0].j = j-1; c[0].i = i;
              v[0]   = -hzhxdhy*(ds - gs);
              c[1].k = k; c[1].j = j; c[1].i = i-1;
              v[1]   = -hyhzdhx*(dw - gw);
              c[2].k = k; c[2].j = j; c[2].i = i;
              v[2]   =   hzhxdhy*(ds + dn + gs - gn) + hyhzdhx*(dw + de + gw - ge) + hxhydhz*(du - gu);
              c[3].k = k; c[3].j = j+1; c[3].i = i;
              v[3]   = -hzhxdhy*(dn + gn);
              c[4].k = k+1; c[4].j = j; c[4].i = i;
              v[4]   = -hxhydhz*(du + gu);
              CHKERRQ(MatSetValuesStencil(jac,1,&row,5,c,v,INSERT_VALUES));

            } else if (k == mz-1) { /* right-hand up interior edge */

              td = x[k-1][j][i];
              ad = 0.5*(t0 + td);
              bd = PetscPowScalar(ad,bm1);
              /* dd = bd * ad; */
              dd = PetscPowScalar(ad,beta);
              gd = coef*bd*(t0 - td);

              c[0].k = k-1; c[0].j = j; c[0].i = i;
              v[0]   = -hxhydhz*(dd - gd);
              c[1].k = k; c[1].j = j-1; c[1].i = i;
              v[1]   = -hzhxdhy*(ds - gs);
              c[2].k = k; c[2].j = j; c[2].i = i-1;
              v[2]   = -hyhzdhx*(dw - gw);
              c[3].k = k; c[3].j = j; c[3].i = i;
              v[3]   =   hzhxdhy*(ds + dn + gs - gn) + hyhzdhx*(dw + de + gw - ge) + hxhydhz*(dd + gd);
              c[4].k = k; c[4].j = j+1; c[4].i = i;
              v[4]   = -hzhxdhy*(dn + gn);
              CHKERRQ(MatSetValuesStencil(jac,1,&row,5,c,v,INSERT_VALUES));

            } else { /* right-hand interior plane */

              td = x[k-1][j][i];
              ad = 0.5*(t0 + td);
              bd = PetscPowScalar(ad,bm1);
              /* dd = bd * ad; */
              dd = PetscPowScalar(ad,beta);
              gd = coef*bd*(t0 - td);

              tu = x[k+1][j][i];
              au = 0.5*(t0 + tu);
              bu = PetscPowScalar(au,bm1);
              /* du = bu * au; */
              du = PetscPowScalar(au,beta);
              gu = coef*bu*(tu - t0);

              c[0].k = k-1; c[0].j = j; c[0].i = i;
              v[0]   = -hxhydhz*(dd - gd);
              c[1].k = k; c[1].j = j-1; c[1].i = i;
              v[1]   = -hzhxdhy*(ds - gs);
              c[2].k = k; c[2].j = j; c[2].i = i-1;
              v[2]   = -hyhzdhx*(dw - gw);
              c[3].k = k; c[3].j = j; c[3].i = i;
              v[3]   =   hzhxdhy*(ds + dn + gs - gn) + hyhzdhx*(dw + de + gw - ge) + hxhydhz*(dd + du + gd - gu);
              c[4].k = k; c[4].j = j+1; c[4].i = i;
              v[4]   = -hzhxdhy*(dn + gn);
              c[5].k = k+1; c[5].j = j; c[5].i = i;
              v[5]   = -hxhydhz*(du + gu);
              CHKERRQ(MatSetValuesStencil(jac,1,&row,6,c,v,INSERT_VALUES));
            }
          }

        } else if (j == 0) {

          tw = x[k][j][i-1];
          aw = 0.5*(t0 + tw);
          bw = PetscPowScalar(aw,bm1);
          /* dw = bw * aw */
          dw = PetscPowScalar(aw,beta);
          gw = coef*bw*(t0 - tw);

          te = x[k][j][i+1];
          ae = 0.5*(t0 + te);
          be = PetscPowScalar(ae,bm1);
          /* de = be * ae; */
          de = PetscPowScalar(ae,beta);
          ge = coef*be*(te - t0);

          tn = x[k][j+1][i];
          an = 0.5*(t0 + tn);
          bn = PetscPowScalar(an,bm1);
          /* dn = bn * an; */
          dn = PetscPowScalar(an,beta);
          gn = coef*bn*(tn - t0);

          /* bottom down interior edge */
          if (k == 0) {

            tu = x[k+1][j][i];
            au = 0.5*(t0 + tu);
            bu = PetscPowScalar(au,bm1);
            /* du = bu * au; */
            du = PetscPowScalar(au,beta);
            gu = coef*bu*(tu - t0);

            c[0].k = k; c[0].j = j; c[0].i = i-1;
            v[0]   = -hyhzdhx*(dw - gw);
            c[1].k = k; c[1].j = j; c[1].i = i;
            v[1]   =   hzhxdhy*(dn - gn) + hyhzdhx*(dw + de + gw - ge) + hxhydhz*(du - gu);
            c[2].k = k; c[2].j = j; c[2].i = i+1;
            v[2]   = -hyhzdhx*(de + ge);
            c[3].k = k; c[3].j = j+1; c[3].i = i;
            v[3]   = -hzhxdhy*(dn + gn);
            c[4].k = k+1; c[4].j = j; c[4].i = i;
            v[4]   = -hxhydhz*(du + gu);
            CHKERRQ(MatSetValuesStencil(jac,1,&row,5,c,v,INSERT_VALUES));

          } else if (k == mz-1) { /* bottom up interior edge */

            td = x[k-1][j][i];
            ad = 0.5*(t0 + td);
            bd = PetscPowScalar(ad,bm1);
            /* dd = bd * ad; */
            dd = PetscPowScalar(ad,beta);
            gd = coef*bd*(td - t0);

            c[0].k = k-1; c[0].j = j; c[0].i = i;
            v[0]   = -hxhydhz*(dd - gd);
            c[1].k = k; c[1].j = j; c[1].i = i-1;
            v[1]   = -hyhzdhx*(dw - gw);
            c[2].k = k; c[2].j = j; c[2].i = i;
            v[2]   =   hzhxdhy*(dn - gn) + hyhzdhx*(dw + de + gw - ge) + hxhydhz*(dd + gd);
            c[3].k = k; c[3].j = j; c[3].i = i+1;
            v[3]   = -hyhzdhx*(de + ge);
            c[4].k = k; c[4].j = j+1; c[4].i = i;
            v[4]   = -hzhxdhy*(dn + gn);
            CHKERRQ(MatSetValuesStencil(jac,1,&row,5,c,v,INSERT_VALUES));

          } else { /* bottom interior plane */

            tu = x[k+1][j][i];
            au = 0.5*(t0 + tu);
            bu = PetscPowScalar(au,bm1);
            /* du = bu * au; */
            du = PetscPowScalar(au,beta);
            gu = coef*bu*(tu - t0);

            td = x[k-1][j][i];
            ad = 0.5*(t0 + td);
            bd = PetscPowScalar(ad,bm1);
            /* dd = bd * ad; */
            dd = PetscPowScalar(ad,beta);
            gd = coef*bd*(td - t0);

            c[0].k = k-1; c[0].j = j; c[0].i = i;
            v[0]   = -hxhydhz*(dd - gd);
            c[1].k = k; c[1].j = j; c[1].i = i-1;
            v[1]   = -hyhzdhx*(dw - gw);
            c[2].k = k; c[2].j = j; c[2].i = i;
            v[2]   =   hzhxdhy*(dn - gn) + hyhzdhx*(dw + de + gw - ge) + hxhydhz*(dd + du + gd - gu);
            c[3].k = k; c[3].j = j; c[3].i = i+1;
            v[3]   = -hyhzdhx*(de + ge);
            c[4].k = k; c[4].j = j+1; c[4].i = i;
            v[4]   = -hzhxdhy*(dn + gn);
            c[5].k = k+1; c[5].j = j; c[5].i = i;
            v[5]   = -hxhydhz*(du + gu);
            CHKERRQ(MatSetValuesStencil(jac,1,&row,6,c,v,INSERT_VALUES));
          }

        } else if (j == my-1) {

          tw = x[k][j][i-1];
          aw = 0.5*(t0 + tw);
          bw = PetscPowScalar(aw,bm1);
          /* dw = bw * aw */
          dw = PetscPowScalar(aw,beta);
          gw = coef*bw*(t0 - tw);

          te = x[k][j][i+1];
          ae = 0.5*(t0 + te);
          be = PetscPowScalar(ae,bm1);
          /* de = be * ae; */
          de = PetscPowScalar(ae,beta);
          ge = coef*be*(te - t0);

          ts = x[k][j-1][i];
          as = 0.5*(t0 + ts);
          bs = PetscPowScalar(as,bm1);
          /* ds = bs * as; */
          ds = PetscPowScalar(as,beta);
          gs = coef*bs*(t0 - ts);

          /* top down interior edge */
          if (k == 0) {

            tu = x[k+1][j][i];
            au = 0.5*(t0 + tu);
            bu = PetscPowScalar(au,bm1);
            /* du = bu * au; */
            du = PetscPowScalar(au,beta);
            gu = coef*bu*(tu - t0);

            c[0].k = k; c[0].j = j-1; c[0].i = i;
            v[0]   = -hzhxdhy*(ds - gs);
            c[1].k = k; c[1].j = j; c[1].i = i-1;
            v[1]   = -hyhzdhx*(dw - gw);
            c[2].k = k; c[2].j = j; c[2].i = i;
            v[2]   =   hzhxdhy*(ds + gs) + hyhzdhx*(dw + de + gw - ge) + hxhydhz*(du - gu);
            c[3].k = k; c[3].j = j; c[3].i = i+1;
            v[3]   = -hyhzdhx*(de + ge);
            c[4].k = k+1; c[4].j = j; c[4].i = i;
            v[4]   = -hxhydhz*(du + gu);
            CHKERRQ(MatSetValuesStencil(jac,1,&row,5,c,v,INSERT_VALUES));

          } else if (k == mz-1) { /* top up interior edge */

            td = x[k-1][j][i];
            ad = 0.5*(t0 + td);
            bd = PetscPowScalar(ad,bm1);
            /* dd = bd * ad; */
            dd = PetscPowScalar(ad,beta);
            gd = coef*bd*(td - t0);

            c[0].k = k-1; c[0].j = j; c[0].i = i;
            v[0]   = -hxhydhz*(dd - gd);
            c[1].k = k; c[1].j = j-1; c[1].i = i;
            v[1]   = -hzhxdhy*(ds - gs);
            c[2].k = k; c[2].j = j; c[2].i = i-1;
            v[2]   = -hyhzdhx*(dw - gw);
            c[3].k = k; c[3].j = j; c[3].i = i;
            v[3]   =   hzhxdhy*(ds + gs) + hyhzdhx*(dw + de + gw - ge) + hxhydhz*(dd + gd);
            c[4].k = k; c[4].j = j; c[4].i = i+1;
            v[4]   = -hyhzdhx*(de + ge);
            CHKERRQ(MatSetValuesStencil(jac,1,&row,5,c,v,INSERT_VALUES));

          } else { /* top interior plane */

            tu = x[k+1][j][i];
            au = 0.5*(t0 + tu);
            bu = PetscPowScalar(au,bm1);
            /* du = bu * au; */
            du = PetscPowScalar(au,beta);
            gu = coef*bu*(tu - t0);

            td = x[k-1][j][i];
            ad = 0.5*(t0 + td);
            bd = PetscPowScalar(ad,bm1);
            /* dd = bd * ad; */
            dd = PetscPowScalar(ad,beta);
            gd = coef*bd*(td - t0);

            c[0].k = k-1; c[0].j = j; c[0].i = i;
            v[0]   = -hxhydhz*(dd - gd);
            c[1].k = k; c[1].j = j-1; c[1].i = i;
            v[1]   = -hzhxdhy*(ds - gs);
            c[2].k = k; c[2].j = j; c[2].i = i-1;
            v[2]   = -hyhzdhx*(dw - gw);
            c[3].k = k; c[3].j = j; c[3].i = i;
            v[3]   =   hzhxdhy*(ds + gs) + hyhzdhx*(dw + de + gw - ge) + hxhydhz*(dd + du + gd - gu);
            c[4].k = k; c[4].j = j; c[4].i = i+1;
            v[4]   = -hyhzdhx*(de + ge);
            c[5].k = k+1; c[5].j = j; c[5].i = i;
            v[5]   = -hxhydhz*(du + gu);
            CHKERRQ(MatSetValuesStencil(jac,1,&row,6,c,v,INSERT_VALUES));
          }

        } else if (k == 0) {

          /* down interior plane */

          tw = x[k][j][i-1];
          aw = 0.5*(t0 + tw);
          bw = PetscPowScalar(aw,bm1);
          /* dw = bw * aw */
          dw = PetscPowScalar(aw,beta);
          gw = coef*bw*(t0 - tw);

          te = x[k][j][i+1];
          ae = 0.5*(t0 + te);
          be = PetscPowScalar(ae,bm1);
          /* de = be * ae; */
          de = PetscPowScalar(ae,beta);
          ge = coef*be*(te - t0);

          ts = x[k][j-1][i];
          as = 0.5*(t0 + ts);
          bs = PetscPowScalar(as,bm1);
          /* ds = bs * as; */
          ds = PetscPowScalar(as,beta);
          gs = coef*bs*(t0 - ts);

          tn = x[k][j+1][i];
          an = 0.5*(t0 + tn);
          bn = PetscPowScalar(an,bm1);
          /* dn = bn * an; */
          dn = PetscPowScalar(an,beta);
          gn = coef*bn*(tn - t0);

          tu = x[k+1][j][i];
          au = 0.5*(t0 + tu);
          bu = PetscPowScalar(au,bm1);
          /* du = bu * au; */
          du = PetscPowScalar(au,beta);
          gu = coef*bu*(tu - t0);

          c[0].k = k; c[0].j = j-1; c[0].i = i;
          v[0]   = -hzhxdhy*(ds - gs);
          c[1].k = k; c[1].j = j; c[1].i = i-1;
          v[1]   = -hyhzdhx*(dw - gw);
          c[2].k = k; c[2].j = j; c[2].i = i;
          v[2]   =   hzhxdhy*(ds + dn + gs - gn) + hyhzdhx*(dw + de + gw - ge) + hxhydhz*(du - gu);
          c[3].k = k; c[3].j = j; c[3].i = i+1;
          v[3]   = -hyhzdhx*(de + ge);
          c[4].k = k; c[4].j = j+1; c[4].i = i;
          v[4]   = -hzhxdhy*(dn + gn);
          c[5].k = k+1; c[5].j = j; c[5].i = i;
          v[5]   = -hxhydhz*(du + gu);
          CHKERRQ(MatSetValuesStencil(jac,1,&row,6,c,v,INSERT_VALUES));

        } else if (k == mz-1) {

          /* up interior plane */

          tw = x[k][j][i-1];
          aw = 0.5*(t0 + tw);
          bw = PetscPowScalar(aw,bm1);
          /* dw = bw * aw */
          dw = PetscPowScalar(aw,beta);
          gw = coef*bw*(t0 - tw);

          te = x[k][j][i+1];
          ae = 0.5*(t0 + te);
          be = PetscPowScalar(ae,bm1);
          /* de = be * ae; */
          de = PetscPowScalar(ae,beta);
          ge = coef*be*(te - t0);

          ts = x[k][j-1][i];
          as = 0.5*(t0 + ts);
          bs = PetscPowScalar(as,bm1);
          /* ds = bs * as; */
          ds = PetscPowScalar(as,beta);
          gs = coef*bs*(t0 - ts);

          tn = x[k][j+1][i];
          an = 0.5*(t0 + tn);
          bn = PetscPowScalar(an,bm1);
          /* dn = bn * an; */
          dn = PetscPowScalar(an,beta);
          gn = coef*bn*(tn - t0);

          td = x[k-1][j][i];
          ad = 0.5*(t0 + td);
          bd = PetscPowScalar(ad,bm1);
          /* dd = bd * ad; */
          dd = PetscPowScalar(ad,beta);
          gd = coef*bd*(t0 - td);

          c[0].k = k-1; c[0].j = j; c[0].i = i;
          v[0]   = -hxhydhz*(dd - gd);
          c[1].k = k; c[1].j = j-1; c[1].i = i;
          v[1]   = -hzhxdhy*(ds - gs);
          c[2].k = k; c[2].j = j; c[2].i = i-1;
          v[2]   = -hyhzdhx*(dw - gw);
          c[3].k = k; c[3].j = j; c[3].i = i;
          v[3]   =   hzhxdhy*(ds + dn + gs - gn) + hyhzdhx*(dw + de + gw - ge) + hxhydhz*(dd + gd);
          c[4].k = k; c[4].j = j; c[4].i = i+1;
          v[4]   = -hyhzdhx*(de + ge);
          c[5].k = k; c[5].j = j+1; c[5].i = i;
          v[5]   = -hzhxdhy*(dn + gn);
          CHKERRQ(MatSetValuesStencil(jac,1,&row,6,c,v,INSERT_VALUES));
        }
      }
    }
  }
  CHKERRQ(MatAssemblyBegin(jac,MAT_FINAL_ASSEMBLY));
  CHKERRQ(MatAssemblyEnd(jac,MAT_FINAL_ASSEMBLY));
  if (jac != J) {
    CHKERRQ(MatAssemblyBegin(J,MAT_FINAL_ASSEMBLY));
    CHKERRQ(MatAssemblyEnd(J,MAT_FINAL_ASSEMBLY));
  }
  CHKERRQ(DMDAVecRestoreArray(da,localX,&x));
  CHKERRQ(DMRestoreLocalVector(da,&localX));

  CHKERRQ(PetscLogFlops((41.0 + 8.0*POWFLOP)*xm*ym));
  PetscFunctionReturn(0);
}

/*TEST

   test:
      nsize: 4
      args: -snes_monitor_short -pc_mg_type full -ksp_type fgmres -pc_type mg -snes_view -pc_mg_levels 2 -pc_mg_galerkin pmat
      requires: !single

TEST*/