dt/interface/dt.c

/* Discretization tools */

#include <petscdt.h>            /*I "petscdt.h" I*/
#include <petscblaslapack.h>
#include <petsc/private/petscimpl.h>
#include <petsc/private/dtimpl.h>
#include <petscviewer.h>
#include <petscdmplex.h>
#include <petscdmshell.h>

#if defined(PETSC_HAVE_MPFR)
#include <mpfr.h>
#endif

static PetscBool GaussCite       = PETSC_FALSE;
const char       GaussCitation[] = "@article{GolubWelsch1969,\n"
                                   "  author  = {Golub and Welsch},\n"
                                   "  title   = {Calculation of Quadrature Rules},\n"
                                   "  journal = {Math. Comp.},\n"
                                   "  volume  = {23},\n"
                                   "  number  = {106},\n"
                                   "  pages   = {221--230},\n"
                                   "  year    = {1969}\n}\n";

/*@
  PetscQuadratureCreate - Create a PetscQuadrature object

  Collective

  Input Parameter:
. comm - The communicator for the PetscQuadrature object

  Output Parameter:
. q  - The PetscQuadrature object

  Level: beginner

.seealso: PetscQuadratureDestroy(), PetscQuadratureGetData()
@*/
PetscErrorCode PetscQuadratureCreate(MPI_Comm comm, PetscQuadrature *q)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  PetscValidPointer(q, 2);
  ierr = PetscSysInitializePackage();CHKERRQ(ierr);
  ierr = PetscHeaderCreate(*q,PETSC_OBJECT_CLASSID,"PetscQuadrature","Quadrature","DT",comm,PetscQuadratureDestroy,PetscQuadratureView);CHKERRQ(ierr);
  (*q)->dim       = -1;
  (*q)->Nc        =  1;
  (*q)->order     = -1;
  (*q)->numPoints = 0;
  (*q)->points    = NULL;
  (*q)->weights   = NULL;
  PetscFunctionReturn(0);
}

/*@
  PetscQuadratureDuplicate - Create a deep copy of the PetscQuadrature object

  Collective on q

  Input Parameter:
. q  - The PetscQuadrature object

  Output Parameter:
. r  - The new PetscQuadrature object

  Level: beginner

.seealso: PetscQuadratureCreate(), PetscQuadratureDestroy(), PetscQuadratureGetData()
@*/
PetscErrorCode PetscQuadratureDuplicate(PetscQuadrature q, PetscQuadrature *r)
{
  PetscInt         order, dim, Nc, Nq;
  const PetscReal *points, *weights;
  PetscReal       *p, *w;
  PetscErrorCode   ierr;

  PetscFunctionBegin;
  PetscValidPointer(q, 2);
  ierr = PetscQuadratureCreate(PetscObjectComm((PetscObject) q), r);CHKERRQ(ierr);
  ierr = PetscQuadratureGetOrder(q, &order);CHKERRQ(ierr);
  ierr = PetscQuadratureSetOrder(*r, order);CHKERRQ(ierr);
  ierr = PetscQuadratureGetData(q, &dim, &Nc, &Nq, &points, &weights);CHKERRQ(ierr);
  ierr = PetscMalloc1(Nq*dim, &p);CHKERRQ(ierr);
  ierr = PetscMalloc1(Nq*Nc, &w);CHKERRQ(ierr);
  ierr = PetscArraycpy(p, points, Nq*dim);CHKERRQ(ierr);
  ierr = PetscArraycpy(w, weights, Nc * Nq);CHKERRQ(ierr);
  ierr = PetscQuadratureSetData(*r, dim, Nc, Nq, p, w);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*@
  PetscQuadratureDestroy - Destroys a PetscQuadrature object

  Collective on q

  Input Parameter:
. q  - The PetscQuadrature object

  Level: beginner

.seealso: PetscQuadratureCreate(), PetscQuadratureGetData()
@*/
PetscErrorCode PetscQuadratureDestroy(PetscQuadrature *q)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  if (!*q) PetscFunctionReturn(0);
  PetscValidHeaderSpecific((*q),PETSC_OBJECT_CLASSID,1);
  if (--((PetscObject)(*q))->refct > 0) {
    *q = NULL;
    PetscFunctionReturn(0);
  }
  ierr = PetscFree((*q)->points);CHKERRQ(ierr);
  ierr = PetscFree((*q)->weights);CHKERRQ(ierr);
  ierr = PetscHeaderDestroy(q);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*@
  PetscQuadratureGetOrder - Return the order of the method

  Not collective

  Input Parameter:
. q - The PetscQuadrature object

  Output Parameter:
. order - The order of the quadrature, i.e. the highest degree polynomial that is exactly integrated

  Level: intermediate

.seealso: PetscQuadratureSetOrder(), PetscQuadratureGetData(), PetscQuadratureSetData()
@*/
PetscErrorCode PetscQuadratureGetOrder(PetscQuadrature q, PetscInt *order)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(q, PETSC_OBJECT_CLASSID, 1);
  PetscValidPointer(order, 2);
  *order = q->order;
  PetscFunctionReturn(0);
}

/*@
  PetscQuadratureSetOrder - Return the order of the method

  Not collective

  Input Parameters:
+ q - The PetscQuadrature object
- order - The order of the quadrature, i.e. the highest degree polynomial that is exactly integrated

  Level: intermediate

.seealso: PetscQuadratureGetOrder(), PetscQuadratureGetData(), PetscQuadratureSetData()
@*/
PetscErrorCode PetscQuadratureSetOrder(PetscQuadrature q, PetscInt order)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(q, PETSC_OBJECT_CLASSID, 1);
  q->order = order;
  PetscFunctionReturn(0);
}

/*@
  PetscQuadratureGetNumComponents - Return the number of components for functions to be integrated

  Not collective

  Input Parameter:
. q - The PetscQuadrature object

  Output Parameter:
. Nc - The number of components

  Note: We are performing an integral int f(x) . w(x) dx, where both f and w (the weight) have Nc components.

  Level: intermediate

.seealso: PetscQuadratureSetNumComponents(), PetscQuadratureGetData(), PetscQuadratureSetData()
@*/
PetscErrorCode PetscQuadratureGetNumComponents(PetscQuadrature q, PetscInt *Nc)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(q, PETSC_OBJECT_CLASSID, 1);
  PetscValidPointer(Nc, 2);
  *Nc = q->Nc;
  PetscFunctionReturn(0);
}

/*@
  PetscQuadratureSetNumComponents - Return the number of components for functions to be integrated

  Not collective

  Input Parameters:
+ q  - The PetscQuadrature object
- Nc - The number of components

  Note: We are performing an integral int f(x) . w(x) dx, where both f and w (the weight) have Nc components.

  Level: intermediate

.seealso: PetscQuadratureGetNumComponents(), PetscQuadratureGetData(), PetscQuadratureSetData()
@*/
PetscErrorCode PetscQuadratureSetNumComponents(PetscQuadrature q, PetscInt Nc)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(q, PETSC_OBJECT_CLASSID, 1);
  q->Nc = Nc;
  PetscFunctionReturn(0);
}

/*@C
  PetscQuadratureGetData - Returns the data defining the quadrature

  Not collective

  Input Parameter:
. q  - The PetscQuadrature object

  Output Parameters:
+ dim - The spatial dimension
. Nc - The number of components
. npoints - The number of quadrature points
. points - The coordinates of each quadrature point
- weights - The weight of each quadrature point

  Level: intermediate

  Fortran Notes:
    From Fortran you must call PetscQuadratureRestoreData() when you are done with the data

.seealso: PetscQuadratureCreate(), PetscQuadratureSetData()
@*/
PetscErrorCode PetscQuadratureGetData(PetscQuadrature q, PetscInt *dim, PetscInt *Nc, PetscInt *npoints, const PetscReal *points[], const PetscReal *weights[])
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(q, PETSC_OBJECT_CLASSID, 1);
  if (dim) {
    PetscValidPointer(dim, 2);
    *dim = q->dim;
  }
  if (Nc) {
    PetscValidPointer(Nc, 3);
    *Nc = q->Nc;
  }
  if (npoints) {
    PetscValidPointer(npoints, 4);
    *npoints = q->numPoints;
  }
  if (points) {
    PetscValidPointer(points, 5);
    *points = q->points;
  }
  if (weights) {
    PetscValidPointer(weights, 6);
    *weights = q->weights;
  }
  PetscFunctionReturn(0);
}

/*@C
  PetscQuadratureSetData - Sets the data defining the quadrature

  Not collective

  Input Parameters:
+ q  - The PetscQuadrature object
. dim - The spatial dimension
. Nc - The number of components
. npoints - The number of quadrature points
. points - The coordinates of each quadrature point
- weights - The weight of each quadrature point

  Note: This routine owns the references to points and weights, so they must be allocated using PetscMalloc() and the user should not free them.

  Level: intermediate

.seealso: PetscQuadratureCreate(), PetscQuadratureGetData()
@*/
PetscErrorCode PetscQuadratureSetData(PetscQuadrature q, PetscInt dim, PetscInt Nc, PetscInt npoints, const PetscReal points[], const PetscReal weights[])
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(q, PETSC_OBJECT_CLASSID, 1);
  if (dim >= 0)     q->dim       = dim;
  if (Nc >= 0)      q->Nc        = Nc;
  if (npoints >= 0) q->numPoints = npoints;
  if (points) {
    PetscValidPointer(points, 4);
    q->points = points;
  }
  if (weights) {
    PetscValidPointer(weights, 5);
    q->weights = weights;
  }
  PetscFunctionReturn(0);
}

static PetscErrorCode PetscQuadratureView_Ascii(PetscQuadrature quad, PetscViewer v)
{
  PetscInt          q, d, c;
  PetscViewerFormat format;
  PetscErrorCode    ierr;

  PetscFunctionBegin;
  if (quad->Nc > 1) {ierr = PetscViewerASCIIPrintf(v, "Quadrature of order %D on %D points (dim %D) with %D components\n", quad->order, quad->numPoints, quad->dim, quad->Nc);CHKERRQ(ierr);}
  else              {ierr = PetscViewerASCIIPrintf(v, "Quadrature of order %D on %D points (dim %D)\n", quad->order, quad->numPoints, quad->dim);CHKERRQ(ierr);}
  ierr = PetscViewerGetFormat(v, &format);CHKERRQ(ierr);
  if (format != PETSC_VIEWER_ASCII_INFO_DETAIL) PetscFunctionReturn(0);
  for (q = 0; q < quad->numPoints; ++q) {
    ierr = PetscViewerASCIIPrintf(v, "p%D (", q);CHKERRQ(ierr);
    ierr = PetscViewerASCIIUseTabs(v, PETSC_FALSE);CHKERRQ(ierr);
    for (d = 0; d < quad->dim; ++d) {
      if (d) {ierr = PetscViewerASCIIPrintf(v, ", ");CHKERRQ(ierr);}
      ierr = PetscViewerASCIIPrintf(v, "%+g", (double)quad->points[q*quad->dim+d]);CHKERRQ(ierr);
    }
    ierr = PetscViewerASCIIPrintf(v, ") ");CHKERRQ(ierr);
    if (quad->Nc > 1) {ierr = PetscViewerASCIIPrintf(v, "w%D (", q);CHKERRQ(ierr);}
    for (c = 0; c < quad->Nc; ++c) {
      if (c) {ierr = PetscViewerASCIIPrintf(v, ", ");CHKERRQ(ierr);}
      ierr = PetscViewerASCIIPrintf(v, "%+g", (double)quad->weights[q*quad->Nc+c]);CHKERRQ(ierr);
    }
    if (quad->Nc > 1) {ierr = PetscViewerASCIIPrintf(v, ")");CHKERRQ(ierr);}
    ierr = PetscViewerASCIIPrintf(v, "\n");CHKERRQ(ierr);
    ierr = PetscViewerASCIIUseTabs(v, PETSC_TRUE);CHKERRQ(ierr);
  }
  PetscFunctionReturn(0);
}

/*@C
  PetscQuadratureView - Views a PetscQuadrature object

  Collective on quad

  Input Parameters:
+ quad  - The PetscQuadrature object
- viewer - The PetscViewer object

  Level: beginner

.seealso: PetscQuadratureCreate(), PetscQuadratureGetData()
@*/
PetscErrorCode PetscQuadratureView(PetscQuadrature quad, PetscViewer viewer)
{
  PetscBool      iascii;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  PetscValidHeader(quad, 1);
  if (viewer) PetscValidHeaderSpecific(viewer, PETSC_VIEWER_CLASSID, 2);
  if (!viewer) {ierr = PetscViewerASCIIGetStdout(PetscObjectComm((PetscObject) quad), &viewer);CHKERRQ(ierr);}
  ierr = PetscObjectTypeCompare((PetscObject) viewer, PETSCVIEWERASCII, &iascii);CHKERRQ(ierr);
  ierr = PetscViewerASCIIPushTab(viewer);CHKERRQ(ierr);
  if (iascii) {ierr = PetscQuadratureView_Ascii(quad, viewer);CHKERRQ(ierr);}
  ierr = PetscViewerASCIIPopTab(viewer);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*@C
  PetscQuadratureExpandComposite - Return a quadrature over the composite element, which has the original quadrature in each subelement

  Not collective

  Input Parameter:
+ q - The original PetscQuadrature
. numSubelements - The number of subelements the original element is divided into
. v0 - An array of the initial points for each subelement
- jac - An array of the Jacobian mappings from the reference to each subelement

  Output Parameters:
. dim - The dimension

  Note: Together v0 and jac define an affine mapping from the original reference element to each subelement

 Not available from Fortran

  Level: intermediate

.seealso: PetscFECreate(), PetscSpaceGetDimension(), PetscDualSpaceGetDimension()
@*/
PetscErrorCode PetscQuadratureExpandComposite(PetscQuadrature q, PetscInt numSubelements, const PetscReal v0[], const PetscReal jac[], PetscQuadrature *qref)
{
  const PetscReal *points,    *weights;
  PetscReal       *pointsRef, *weightsRef;
  PetscInt         dim, Nc, order, npoints, npointsRef, c, p, cp, d, e;
  PetscErrorCode   ierr;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(q, PETSC_OBJECT_CLASSID, 1);
  PetscValidPointer(v0, 3);
  PetscValidPointer(jac, 4);
  PetscValidPointer(qref, 5);
  ierr = PetscQuadratureCreate(PETSC_COMM_SELF, qref);CHKERRQ(ierr);
  ierr = PetscQuadratureGetOrder(q, &order);CHKERRQ(ierr);
  ierr = PetscQuadratureGetData(q, &dim, &Nc, &npoints, &points, &weights);CHKERRQ(ierr);
  npointsRef = npoints*numSubelements;
  ierr = PetscMalloc1(npointsRef*dim,&pointsRef);CHKERRQ(ierr);
  ierr = PetscMalloc1(npointsRef*Nc, &weightsRef);CHKERRQ(ierr);
  for (c = 0; c < numSubelements; ++c) {
    for (p = 0; p < npoints; ++p) {
      for (d = 0; d < dim; ++d) {
        pointsRef[(c*npoints + p)*dim+d] = v0[c*dim+d];
        for (e = 0; e < dim; ++e) {
          pointsRef[(c*npoints + p)*dim+d] += jac[(c*dim + d)*dim+e]*(points[p*dim+e] + 1.0);
        }
      }
      /* Could also use detJ here */
      for (cp = 0; cp < Nc; ++cp) weightsRef[(c*npoints+p)*Nc+cp] = weights[p*Nc+cp]/numSubelements;
    }
  }
  ierr = PetscQuadratureSetOrder(*qref, order);CHKERRQ(ierr);
  ierr = PetscQuadratureSetData(*qref, dim, Nc, npointsRef, pointsRef, weightsRef);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*@
   PetscDTLegendreEval - evaluate Legendre polynomial at points

   Not Collective

   Input Arguments:
+  npoints - number of spatial points to evaluate at
.  points - array of locations to evaluate at
.  ndegree - number of basis degrees to evaluate
-  degrees - sorted array of degrees to evaluate

   Output Arguments:
+  B - row-oriented basis evaluation matrix B[point*ndegree + degree] (dimension npoints*ndegrees, allocated by caller) (or NULL)
.  D - row-oriented derivative evaluation matrix (or NULL)
-  D2 - row-oriented second derivative evaluation matrix (or NULL)

   Level: intermediate

.seealso: PetscDTGaussQuadrature()
@*/
PetscErrorCode PetscDTLegendreEval(PetscInt npoints,const PetscReal *points,PetscInt ndegree,const PetscInt *degrees,PetscReal *B,PetscReal *D,PetscReal *D2)
{
  PetscInt i,maxdegree;

  PetscFunctionBegin;
  if (!npoints || !ndegree) PetscFunctionReturn(0);
  maxdegree = degrees[ndegree-1];
  for (i=0; i<npoints; i++) {
    PetscReal pm1,pm2,pd1,pd2,pdd1,pdd2,x;
    PetscInt  j,k;
    x    = points[i];
    pm2  = 0;
    pm1  = 1;
    pd2  = 0;
    pd1  = 0;
    pdd2 = 0;
    pdd1 = 0;
    k    = 0;
    if (degrees[k] == 0) {
      if (B) B[i*ndegree+k] = pm1;
      if (D) D[i*ndegree+k] = pd1;
      if (D2) D2[i*ndegree+k] = pdd1;
      k++;
    }
    for (j=1; j<=maxdegree; j++,k++) {
      PetscReal p,d,dd;
      p    = ((2*j-1)*x*pm1 - (j-1)*pm2)/j;
      d    = pd2 + (2*j-1)*pm1;
      dd   = pdd2 + (2*j-1)*pd1;
      pm2  = pm1;
      pm1  = p;
      pd2  = pd1;
      pd1  = d;
      pdd2 = pdd1;
      pdd1 = dd;
      if (degrees[k] == j) {
        if (B) B[i*ndegree+k] = p;
        if (D) D[i*ndegree+k] = d;
        if (D2) D2[i*ndegree+k] = dd;
      }
    }
  }
  PetscFunctionReturn(0);
}

/*@
   PetscDTGaussQuadrature - create Gauss quadrature

   Not Collective

   Input Arguments:
+  npoints - number of points
.  a - left end of interval (often-1)
-  b - right end of interval (often +1)

   Output Arguments:
+  x - quadrature points
-  w - quadrature weights

   Level: intermediate

   References:
.   1. - Golub and Welsch, Calculation of Quadrature Rules, Math. Comp. 23(106), 1969.

.seealso: PetscDTLegendreEval()
@*/
PetscErrorCode PetscDTGaussQuadrature(PetscInt npoints,PetscReal a,PetscReal b,PetscReal *x,PetscReal *w)
{
  PetscErrorCode ierr;
  PetscInt       i;
  PetscReal      *work;
  PetscScalar    *Z;
  PetscBLASInt   N,LDZ,info;

  PetscFunctionBegin;
  ierr = PetscCitationsRegister(GaussCitation, &GaussCite);CHKERRQ(ierr);
  /* Set up the Golub-Welsch system */
  for (i=0; i<npoints; i++) {
    x[i] = 0;                   /* diagonal is 0 */
    if (i) w[i-1] = 0.5 / PetscSqrtReal(1 - 1./PetscSqr(2*i));
  }
  ierr = PetscMalloc2(npoints*npoints,&Z,PetscMax(1,2*npoints-2),&work);CHKERRQ(ierr);
  ierr = PetscBLASIntCast(npoints,&N);CHKERRQ(ierr);
  LDZ  = N;
  ierr = PetscFPTrapPush(PETSC_FP_TRAP_OFF);CHKERRQ(ierr);
  PetscStackCallBLAS("LAPACKsteqr",LAPACKsteqr_("I",&N,x,w,Z,&LDZ,work,&info));
  ierr = PetscFPTrapPop();CHKERRQ(ierr);
  if (info) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"xSTEQR error");

  for (i=0; i<(npoints+1)/2; i++) {
    PetscReal y = 0.5 * (-x[i] + x[npoints-i-1]); /* enforces symmetry */
    x[i]           = (a+b)/2 - y*(b-a)/2;
    if (x[i] == -0.0) x[i] = 0.0;
    x[npoints-i-1] = (a+b)/2 + y*(b-a)/2;

    w[i] = w[npoints-1-i] = 0.5*(b-a)*(PetscSqr(PetscAbsScalar(Z[i*npoints])) + PetscSqr(PetscAbsScalar(Z[(npoints-i-1)*npoints])));
  }
  ierr = PetscFree2(Z,work);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

static void qAndLEvaluation(PetscInt n, PetscReal x, PetscReal *q, PetscReal *qp, PetscReal *Ln)
/*
  Compute the polynomial q(x) = L_{N+1}(x) - L_{n-1}(x) and its derivative in
  addition to L_N(x) as these are needed for computing the GLL points via Newton's method.
  Reference: "Implementing Spectral Methods for Partial Differential Equations: Algorithms
  for Scientists and Engineers" by David A. Kopriva.
*/
{
  PetscInt k;

  PetscReal Lnp;
  PetscReal Lnp1, Lnp1p;
  PetscReal Lnm1, Lnm1p;
  PetscReal Lnm2, Lnm2p;

  Lnm1  = 1.0;
  *Ln   = x;
  Lnm1p = 0.0;
  Lnp   = 1.0;

  for (k=2; k<=n; ++k) {
    Lnm2  = Lnm1;
    Lnm1  = *Ln;
    Lnm2p = Lnm1p;
    Lnm1p = Lnp;
    *Ln   = (2.*((PetscReal)k)-1.)/(1.0*((PetscReal)k))*x*Lnm1 - (((PetscReal)k)-1.)/((PetscReal)k)*Lnm2;
    Lnp   = Lnm2p + (2.0*((PetscReal)k)-1.)*Lnm1;
  }
  k     = n+1;
  Lnp1  = (2.*((PetscReal)k)-1.)/(((PetscReal)k))*x*(*Ln) - (((PetscReal)k)-1.)/((PetscReal)k)*Lnm1;
  Lnp1p = Lnm1p + (2.0*((PetscReal)k)-1.)*(*Ln);
  *q    = Lnp1 - Lnm1;
  *qp   = Lnp1p - Lnm1p;
}

/*@C
   PetscDTGaussLobattoLegendreQuadrature - creates a set of the locations and weights of the Gauss-Lobatto-Legendre
                      nodes of a given size on the domain [-1,1]

   Not Collective

   Input Parameter:
+  n - number of grid nodes
-  type - PETSCGAUSSLOBATTOLEGENDRE_VIA_LINEAR_ALGEBRA or PETSCGAUSSLOBATTOLEGENDRE_VIA_NEWTON

   Output Arguments:
+  x - quadrature points
-  w - quadrature weights

   Notes:
    For n > 30  the Newton approach computes duplicate (incorrect) values for some nodes because the initial guess is apparently not
          close enough to the desired solution

   These are useful for implementing spectral methods based on Gauss-Lobatto-Legendre (GLL) nodes

   See  https://epubs.siam.org/doi/abs/10.1137/110855442  https://epubs.siam.org/doi/abs/10.1137/120889873 for better ways to compute GLL nodes

   Level: intermediate

.seealso: PetscDTGaussQuadrature()

@*/
PetscErrorCode PetscDTGaussLobattoLegendreQuadrature(PetscInt npoints,PetscGaussLobattoLegendreCreateType type,PetscReal *x,PetscReal *w)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  if (npoints < 2) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Must provide at least 2 grid points per element");

  if (type == PETSCGAUSSLOBATTOLEGENDRE_VIA_LINEAR_ALGEBRA) {
    PetscReal      *M,si;
    PetscBLASInt   bn,lierr;
    PetscReal      x0,z0,z1,z2;
    PetscInt       i,p = npoints - 1,nn;

    x[0]   =-1.0;
    x[npoints-1] = 1.0;
    if (npoints-2 > 0){
      ierr = PetscMalloc1(npoints-1,&M);CHKERRQ(ierr);
      for (i=0; i<npoints-2; i++) {
        si  = ((PetscReal)i)+1.0;
        M[i]=0.5*PetscSqrtReal(si*(si+2.0)/((si+0.5)*(si+1.5)));
      }
      ierr = PetscBLASIntCast(npoints-2,&bn);CHKERRQ(ierr);
      ierr = PetscArrayzero(&x[1],bn);CHKERRQ(ierr);
      ierr = PetscFPTrapPush(PETSC_FP_TRAP_OFF);CHKERRQ(ierr);
      x0=0;
      PetscStackCallBLAS("LAPACKsteqr",LAPACKREALsteqr_("N",&bn,&x[1],M,&x0,&bn,M,&lierr));
      if (lierr) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_LIB,"Error in STERF Lapack routine %d",(int)lierr);
      ierr = PetscFPTrapPop();CHKERRQ(ierr);
      ierr = PetscFree(M);CHKERRQ(ierr);
    }
    if ((npoints-1)%2==0) {
      x[(npoints-1)/2]   = 0.0; /* hard wire to exactly 0.0 since linear algebra produces nonzero */
    }

    w[0] = w[p] = 2.0/(((PetscReal)(p))*(((PetscReal)p)+1.0));
    z2 = -1.;                      /* Dummy value to avoid -Wmaybe-initialized */
    for (i=1; i<p; i++) {
      x0  = x[i];
      z0 = 1.0;
      z1 = x0;
      for (nn=1; nn<p; nn++) {
        z2 = x0*z1*(2.0*((PetscReal)nn)+1.0)/(((PetscReal)nn)+1.0)-z0*(((PetscReal)nn)/(((PetscReal)nn)+1.0));
        z0 = z1;
        z1 = z2;
      }
      w[i]=2.0/(((PetscReal)p)*(((PetscReal)p)+1.0)*z2*z2);
    }
  } else {
    PetscInt  j,m;
    PetscReal z1,z,q,qp,Ln;
    PetscReal *pt;
    ierr = PetscMalloc1(npoints,&pt);CHKERRQ(ierr);

    if (npoints > 30) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"PETSCGAUSSLOBATTOLEGENDRE_VIA_NEWTON produces incorrect answers for n > 30");
    x[0]     = -1.0;
    x[npoints-1]   = 1.0;
    w[0]   = w[npoints-1] = 2./(((PetscReal)npoints)*(((PetscReal)npoints)-1.0));
    m  = (npoints-1)/2; /* The roots are symmetric, so we only find half of them. */
    for (j=1; j<=m; j++) { /* Loop over the desired roots. */
      z = -1.0*PetscCosReal((PETSC_PI*((PetscReal)j)+0.25)/(((PetscReal)npoints)-1.0))-(3.0/(8.0*(((PetscReal)npoints)-1.0)*PETSC_PI))*(1.0/(((PetscReal)j)+0.25));
      /* Starting with the above approximation to the ith root, we enter */
      /* the main loop of refinement by Newton's method.                 */
      do {
        qAndLEvaluation(npoints-1,z,&q,&qp,&Ln);
        z1 = z;
        z  = z1-q/qp; /* Newton's method. */
      } while (PetscAbs(z-z1) > 10.*PETSC_MACHINE_EPSILON);
      qAndLEvaluation(npoints-1,z,&q,&qp,&Ln);

      x[j]       = z;
      x[npoints-1-j]   = -z;      /* and put in its symmetric counterpart.   */
      w[j]     = 2.0/(((PetscReal)npoints)*(((PetscReal)npoints)-1.)*Ln*Ln);  /* Compute the weight */
      w[npoints-1-j] = w[j];                 /* and its symmetric counterpart. */
      pt[j]=qp;
    }

    if ((npoints-1)%2==0) {
      qAndLEvaluation(npoints-1,0.0,&q,&qp,&Ln);
      x[(npoints-1)/2]   = 0.0;
      w[(npoints-1)/2] = 2.0/(((PetscReal)npoints)*(((PetscReal)npoints)-1.)*Ln*Ln);
    }
    ierr = PetscFree(pt);CHKERRQ(ierr);
  }
  PetscFunctionReturn(0);
}

/*@
  PetscDTGaussTensorQuadrature - creates a tensor-product Gauss quadrature

  Not Collective

  Input Arguments:
+ dim     - The spatial dimension
. Nc      - The number of components
. npoints - number of points in one dimension
. a       - left end of interval (often-1)
- b       - right end of interval (often +1)

  Output Argument:
. q - A PetscQuadrature object

  Level: intermediate

.seealso: PetscDTGaussQuadrature(), PetscDTLegendreEval()
@*/
PetscErrorCode PetscDTGaussTensorQuadrature(PetscInt dim, PetscInt Nc, PetscInt npoints, PetscReal a, PetscReal b, PetscQuadrature *q)
{
  PetscInt       totpoints = dim > 1 ? dim > 2 ? npoints*PetscSqr(npoints) : PetscSqr(npoints) : npoints, i, j, k, c;
  PetscReal     *x, *w, *xw, *ww;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = PetscMalloc1(totpoints*dim,&x);CHKERRQ(ierr);
  ierr = PetscMalloc1(totpoints*Nc,&w);CHKERRQ(ierr);
  /* Set up the Golub-Welsch system */
  switch (dim) {
  case 0:
    ierr = PetscFree(x);CHKERRQ(ierr);
    ierr = PetscFree(w);CHKERRQ(ierr);
    ierr = PetscMalloc1(1, &x);CHKERRQ(ierr);
    ierr = PetscMalloc1(Nc, &w);CHKERRQ(ierr);
    x[0] = 0.0;
    for (c = 0; c < Nc; ++c) w[c] = 1.0;
    break;
  case 1:
    ierr = PetscMalloc1(npoints,&ww);CHKERRQ(ierr);
    ierr = PetscDTGaussQuadrature(npoints, a, b, x, ww);CHKERRQ(ierr);
    for (i = 0; i < npoints; ++i) for (c = 0; c < Nc; ++c) w[i*Nc+c] = ww[i];
    ierr = PetscFree(ww);CHKERRQ(ierr);
    break;
  case 2:
    ierr = PetscMalloc2(npoints,&xw,npoints,&ww);CHKERRQ(ierr);
    ierr = PetscDTGaussQuadrature(npoints, a, b, xw, ww);CHKERRQ(ierr);
    for (i = 0; i < npoints; ++i) {
      for (j = 0; j < npoints; ++j) {
        x[(i*npoints+j)*dim+0] = xw[i];
        x[(i*npoints+j)*dim+1] = xw[j];
        for (c = 0; c < Nc; ++c) w[(i*npoints+j)*Nc+c] = ww[i] * ww[j];
      }
    }
    ierr = PetscFree2(xw,ww);CHKERRQ(ierr);
    break;
  case 3:
    ierr = PetscMalloc2(npoints,&xw,npoints,&ww);CHKERRQ(ierr);
    ierr = PetscDTGaussQuadrature(npoints, a, b, xw, ww);CHKERRQ(ierr);
    for (i = 0; i < npoints; ++i) {
      for (j = 0; j < npoints; ++j) {
        for (k = 0; k < npoints; ++k) {
          x[((i*npoints+j)*npoints+k)*dim+0] = xw[i];
          x[((i*npoints+j)*npoints+k)*dim+1] = xw[j];
          x[((i*npoints+j)*npoints+k)*dim+2] = xw[k];
          for (c = 0; c < Nc; ++c) w[((i*npoints+j)*npoints+k)*Nc+c] = ww[i] * ww[j] * ww[k];
        }
      }
    }
    ierr = PetscFree2(xw,ww);CHKERRQ(ierr);
    break;
  default:
    SETERRQ1(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Cannot construct quadrature rule for dimension %d", dim);
  }
  ierr = PetscQuadratureCreate(PETSC_COMM_SELF, q);CHKERRQ(ierr);
  ierr = PetscQuadratureSetOrder(*q, 2*npoints-1);CHKERRQ(ierr);
  ierr = PetscQuadratureSetData(*q, dim, Nc, totpoints, x, w);CHKERRQ(ierr);
  ierr = PetscObjectChangeTypeName((PetscObject)*q,"GaussTensor");CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/* Evaluates the nth jacobi polynomial with weight parameters a,b at a point x.
   Recurrence relations implemented from the pseudocode given in Karniadakis and Sherwin, Appendix B */
PETSC_STATIC_INLINE PetscErrorCode PetscDTComputeJacobi(PetscReal a, PetscReal b, PetscInt n, PetscReal x, PetscReal *P)
{
  PetscReal apb, pn1, pn2;
  PetscInt  k;

  PetscFunctionBegin;
  if (!n) {*P = 1.0; PetscFunctionReturn(0);}
  if (n == 1) {*P = 0.5 * (a - b + (a + b + 2.0) * x); PetscFunctionReturn(0);}
  apb = a + b;
  pn2 = 1.0;
  pn1 = 0.5 * (a - b + (apb + 2.0) * x);
  *P  = 0.0;
  for (k = 2; k < n+1; ++k) {
    PetscReal a1 = 2.0 * k * (k + apb) * (2.0*k + apb - 2.0);
    PetscReal a2 = (2.0 * k + apb - 1.0) * (a*a - b*b);
    PetscReal a3 = (2.0 * k + apb - 2.0) * (2.0 * k + apb - 1.0) * (2.0 * k + apb);
    PetscReal a4 = 2.0 * (k + a - 1.0) * (k + b - 1.0) * (2.0 * k + apb);

    a2  = a2 / a1;
    a3  = a3 / a1;
    a4  = a4 / a1;
    *P  = (a2 + a3 * x) * pn1 - a4 * pn2;
    pn2 = pn1;
    pn1 = *P;
  }
  PetscFunctionReturn(0);
}

/* Evaluates the first derivative of P_{n}^{a,b} at a point x. */
PETSC_STATIC_INLINE PetscErrorCode PetscDTComputeJacobiDerivative(PetscReal a, PetscReal b, PetscInt n, PetscReal x, PetscReal *P)
{
  PetscReal      nP;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  if (!n) {*P = 0.0; PetscFunctionReturn(0);}
  ierr = PetscDTComputeJacobi(a+1, b+1, n-1, x, &nP);CHKERRQ(ierr);
  *P   = 0.5 * (a + b + n + 1) * nP;
  PetscFunctionReturn(0);
}

/* Maps from [-1,1]^2 to the (-1,1) reference triangle */
PETSC_STATIC_INLINE PetscErrorCode PetscDTMapSquareToTriangle_Internal(PetscReal x, PetscReal y, PetscReal *xi, PetscReal *eta)
{
  PetscFunctionBegin;
  *xi  = 0.5 * (1.0 + x) * (1.0 - y) - 1.0;
  *eta = y;
  PetscFunctionReturn(0);
}

/* Maps from [-1,1]^2 to the (-1,1) reference triangle */
PETSC_STATIC_INLINE PetscErrorCode PetscDTMapCubeToTetrahedron_Internal(PetscReal x, PetscReal y, PetscReal z, PetscReal *xi, PetscReal *eta, PetscReal *zeta)
{
  PetscFunctionBegin;
  *xi   = 0.25 * (1.0 + x) * (1.0 - y) * (1.0 - z) - 1.0;
  *eta  = 0.5  * (1.0 + y) * (1.0 - z) - 1.0;
  *zeta = z;
  PetscFunctionReturn(0);
}

static PetscErrorCode PetscDTGaussJacobiQuadrature1D_Internal(PetscInt npoints, PetscReal a, PetscReal b, PetscReal *x, PetscReal *w)
{
  PetscInt       maxIter = 100;
  PetscReal      eps     = 1.0e-8;
  PetscReal      a1, a2, a3, a4, a5, a6;
  PetscInt       k;
  PetscErrorCode ierr;

  PetscFunctionBegin;

  a1      = PetscPowReal(2.0, a+b+1);
#if defined(PETSC_HAVE_TGAMMA)
  a2      = PetscTGamma(a + npoints + 1);
  a3      = PetscTGamma(b + npoints + 1);
  a4      = PetscTGamma(a + b + npoints + 1);
#else
  {
    PetscInt ia, ib;

    ia = (PetscInt) a;
    ib = (PetscInt) b;
    if (ia == a && ib == b && ia + npoints + 1 > 0 && ib + npoints + 1 > 0 && ia + ib + npoints + 1 > 0) { /* All gamma(x) terms are (x-1)! terms */
      ierr = PetscDTFactorial(ia + npoints, &a2);CHKERRQ(ierr);
      ierr = PetscDTFactorial(ib + npoints, &a3);CHKERRQ(ierr);
      ierr = PetscDTFactorial(ia + ib + npoints, &a4);CHKERRQ(ierr);
    } else {
      SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"tgamma() - math routine is unavailable.");
    }
  }
#endif

  ierr = PetscDTFactorial(npoints, &a5);CHKERRQ(ierr);
  a6   = a1 * a2 * a3 / a4 / a5;
  /* Computes the m roots of P_{m}^{a,b} on [-1,1] by Newton's method with Chebyshev points as initial guesses.
   Algorithm implemented from the pseudocode given by Karniadakis and Sherwin and Python in FIAT */
  for (k = 0; k < npoints; ++k) {
    PetscReal r = -PetscCosReal((2.0*k + 1.0) * PETSC_PI / (2.0 * npoints)), dP;
    PetscInt  j;

    if (k > 0) r = 0.5 * (r + x[k-1]);
    for (j = 0; j < maxIter; ++j) {
      PetscReal s = 0.0, delta, f, fp;
      PetscInt  i;

      for (i = 0; i < k; ++i) s = s + 1.0 / (r - x[i]);
      ierr = PetscDTComputeJacobi(a, b, npoints, r, &f);CHKERRQ(ierr);
      ierr = PetscDTComputeJacobiDerivative(a, b, npoints, r, &fp);CHKERRQ(ierr);
      delta = f / (fp - f * s);
      r     = r - delta;
      if (PetscAbsReal(delta) < eps) break;
    }
    x[k] = r;
    ierr = PetscDTComputeJacobiDerivative(a, b, npoints, x[k], &dP);CHKERRQ(ierr);
    w[k] = a6 / (1.0 - PetscSqr(x[k])) / PetscSqr(dP);
  }
  PetscFunctionReturn(0);
}

/*@
  PetscDTGaussJacobiQuadrature - create Gauss-Jacobi quadrature for a simplex

  Not Collective

  Input Arguments:
+ dim     - The simplex dimension
. Nc      - The number of components
. npoints - The number of points in one dimension
. a       - left end of interval (often-1)
- b       - right end of interval (often +1)

  Output Argument:
. q - A PetscQuadrature object

  Level: intermediate

  References:
.  1. - Karniadakis and Sherwin.  FIAT

.seealso: PetscDTGaussTensorQuadrature(), PetscDTGaussQuadrature()
@*/
PetscErrorCode PetscDTGaussJacobiQuadrature(PetscInt dim, PetscInt Nc, PetscInt npoints, PetscReal a, PetscReal b, PetscQuadrature *q)
{
  PetscInt       totpoints = dim > 1 ? dim > 2 ? npoints*PetscSqr(npoints) : PetscSqr(npoints) : npoints;
  PetscReal     *px, *wx, *py, *wy, *pz, *wz, *x, *w;
  PetscInt       i, j, k, c;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  if ((a != -1.0) || (b != 1.0)) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must use default internal right now");
  ierr = PetscMalloc1(totpoints*dim, &x);CHKERRQ(ierr);
  ierr = PetscMalloc1(totpoints*Nc, &w);CHKERRQ(ierr);
  switch (dim) {
  case 0:
    ierr = PetscFree(x);CHKERRQ(ierr);
    ierr = PetscFree(w);CHKERRQ(ierr);
    ierr = PetscMalloc1(1, &x);CHKERRQ(ierr);
    ierr = PetscMalloc1(Nc, &w);CHKERRQ(ierr);
    x[0] = 0.0;
    for (c = 0; c < Nc; ++c) w[c] = 1.0;
    break;
  case 1:
    ierr = PetscMalloc1(npoints,&wx);CHKERRQ(ierr);
    ierr = PetscDTGaussJacobiQuadrature1D_Internal(npoints, 0.0, 0.0, x, wx);CHKERRQ(ierr);
    for (i = 0; i < npoints; ++i) for (c = 0; c < Nc; ++c) w[i*Nc+c] = wx[i];
    ierr = PetscFree(wx);CHKERRQ(ierr);
    break;
  case 2:
    ierr = PetscMalloc4(npoints,&px,npoints,&wx,npoints,&py,npoints,&wy);CHKERRQ(ierr);
    ierr = PetscDTGaussJacobiQuadrature1D_Internal(npoints, 0.0, 0.0, px, wx);CHKERRQ(ierr);
    ierr = PetscDTGaussJacobiQuadrature1D_Internal(npoints, 1.0, 0.0, py, wy);CHKERRQ(ierr);
    for (i = 0; i < npoints; ++i) {
      for (j = 0; j < npoints; ++j) {
        ierr = PetscDTMapSquareToTriangle_Internal(px[i], py[j], &x[(i*npoints+j)*2+0], &x[(i*npoints+j)*2+1]);CHKERRQ(ierr);
        for (c = 0; c < Nc; ++c) w[(i*npoints+j)*Nc+c] = 0.5 * wx[i] * wy[j];
      }
    }
    ierr = PetscFree4(px,wx,py,wy);CHKERRQ(ierr);
    break;
  case 3:
    ierr = PetscMalloc6(npoints,&px,npoints,&wx,npoints,&py,npoints,&wy,npoints,&pz,npoints,&wz);CHKERRQ(ierr);
    ierr = PetscDTGaussJacobiQuadrature1D_Internal(npoints, 0.0, 0.0, px, wx);CHKERRQ(ierr);
    ierr = PetscDTGaussJacobiQuadrature1D_Internal(npoints, 1.0, 0.0, py, wy);CHKERRQ(ierr);
    ierr = PetscDTGaussJacobiQuadrature1D_Internal(npoints, 2.0, 0.0, pz, wz);CHKERRQ(ierr);
    for (i = 0; i < npoints; ++i) {
      for (j = 0; j < npoints; ++j) {
        for (k = 0; k < npoints; ++k) {
          ierr = PetscDTMapCubeToTetrahedron_Internal(px[i], py[j], pz[k], &x[((i*npoints+j)*npoints+k)*3+0], &x[((i*npoints+j)*npoints+k)*3+1], &x[((i*npoints+j)*npoints+k)*3+2]);CHKERRQ(ierr);
          for (c = 0; c < Nc; ++c) w[((i*npoints+j)*npoints+k)*Nc+c] = 0.125 * wx[i] * wy[j] * wz[k];
        }
      }
    }
    ierr = PetscFree6(px,wx,py,wy,pz,wz);CHKERRQ(ierr);
    break;
  default:
    SETERRQ1(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Cannot construct quadrature rule for dimension %d", dim);
  }
  ierr = PetscQuadratureCreate(PETSC_COMM_SELF, q);CHKERRQ(ierr);
  ierr = PetscQuadratureSetOrder(*q, 2*npoints-1);CHKERRQ(ierr);
  ierr = PetscQuadratureSetData(*q, dim, Nc, totpoints, x, w);CHKERRQ(ierr);
  ierr = PetscObjectChangeTypeName((PetscObject)*q,"GaussJacobi");CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*@
  PetscDTTanhSinhTensorQuadrature - create tanh-sinh quadrature for a tensor product cell

  Not Collective

  Input Arguments:
+ dim   - The cell dimension
. level - The number of points in one dimension, 2^l
. a     - left end of interval (often-1)
- b     - right end of interval (often +1)

  Output Argument:
. q - A PetscQuadrature object

  Level: intermediate

.seealso: PetscDTGaussTensorQuadrature()
@*/
PetscErrorCode PetscDTTanhSinhTensorQuadrature(PetscInt dim, PetscInt level, PetscReal a, PetscReal b, PetscQuadrature *q)
{
  const PetscInt  p     = 16;                        /* Digits of precision in the evaluation */
  const PetscReal alpha = (b-a)/2.;                  /* Half-width of the integration interval */
  const PetscReal beta  = (b+a)/2.;                  /* Center of the integration interval */
  const PetscReal h     = PetscPowReal(2.0, -level); /* Step size, length between x_k */
  PetscReal       xk;                                /* Quadrature point x_k on reference domain [-1, 1] */
  PetscReal       wk    = 0.5*PETSC_PI;              /* Quadrature weight at x_k */
  PetscReal      *x, *w;
  PetscInt        K, k, npoints;
  PetscErrorCode  ierr;

  PetscFunctionBegin;
  if (dim > 1) SETERRQ1(PETSC_COMM_SELF, PETSC_ERR_SUP, "Dimension %d not yet implemented", dim);
  if (!level) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must give a number of significant digits");
  /* Find K such that the weights are < 32 digits of precision */
  for (K = 1; PetscAbsReal(PetscLog10Real(wk)) < 2*p; ++K) {
    wk = 0.5*h*PETSC_PI*PetscCoshReal(K*h)/PetscSqr(PetscCoshReal(0.5*PETSC_PI*PetscSinhReal(K*h)));
  }
  ierr = PetscQuadratureCreate(PETSC_COMM_SELF, q);CHKERRQ(ierr);
  ierr = PetscQuadratureSetOrder(*q, 2*K+1);CHKERRQ(ierr);
  npoints = 2*K-1;
  ierr = PetscMalloc1(npoints*dim, &x);CHKERRQ(ierr);
  ierr = PetscMalloc1(npoints, &w);CHKERRQ(ierr);
  /* Center term */
  x[0] = beta;
  w[0] = 0.5*alpha*PETSC_PI;
  for (k = 1; k < K; ++k) {
    wk = 0.5*alpha*h*PETSC_PI*PetscCoshReal(k*h)/PetscSqr(PetscCoshReal(0.5*PETSC_PI*PetscSinhReal(k*h)));
    xk = PetscTanhReal(0.5*PETSC_PI*PetscSinhReal(k*h));
    x[2*k-1] = -alpha*xk+beta;
    w[2*k-1] = wk;
    x[2*k+0] =  alpha*xk+beta;
    w[2*k+0] = wk;
  }
  ierr = PetscQuadratureSetData(*q, dim, 1, npoints, x, w);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode PetscDTTanhSinhIntegrate(void (*func)(PetscReal, PetscReal *), PetscReal a, PetscReal b, PetscInt digits, PetscReal *sol)
{
  const PetscInt  p     = 16;        /* Digits of precision in the evaluation */
  const PetscReal alpha = (b-a)/2.;  /* Half-width of the integration interval */
  const PetscReal beta  = (b+a)/2.;  /* Center of the integration interval */
  PetscReal       h     = 1.0;       /* Step size, length between x_k */
  PetscInt        l     = 0;         /* Level of refinement, h = 2^{-l} */
  PetscReal       osum  = 0.0;       /* Integral on last level */
  PetscReal       psum  = 0.0;       /* Integral on the level before the last level */
  PetscReal       sum;               /* Integral on current level */
  PetscReal       yk;                /* Quadrature point 1 - x_k on reference domain [-1, 1] */
  PetscReal       lx, rx;            /* Quadrature points to the left and right of 0 on the real domain [a, b] */
  PetscReal       wk;                /* Quadrature weight at x_k */
  PetscReal       lval, rval;        /* Terms in the quadature sum to the left and right of 0 */
  PetscInt        d;                 /* Digits of precision in the integral */

  PetscFunctionBegin;
  if (digits <= 0) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must give a positive number of significant digits");
  /* Center term */
  func(beta, &lval);
  sum = 0.5*alpha*PETSC_PI*lval;
  /* */
  do {
    PetscReal lterm, rterm, maxTerm = 0.0, d1, d2, d3, d4;
    PetscInt  k = 1;

    ++l;
    /* PetscPrintf(PETSC_COMM_SELF, "LEVEL %D sum: %15.15f\n", l, sum); */
    /* At each level of refinement, h --> h/2 and sum --> sum/2 */
    psum = osum;
    osum = sum;
    h   *= 0.5;
    sum *= 0.5;
    do {
      wk = 0.5*h*PETSC_PI*PetscCoshReal(k*h)/PetscSqr(PetscCoshReal(0.5*PETSC_PI*PetscSinhReal(k*h)));
      yk = 1.0/(PetscExpReal(0.5*PETSC_PI*PetscSinhReal(k*h)) * PetscCoshReal(0.5*PETSC_PI*PetscSinhReal(k*h)));
      lx = -alpha*(1.0 - yk)+beta;
      rx =  alpha*(1.0 - yk)+beta;
      func(lx, &lval);
      func(rx, &rval);
      lterm   = alpha*wk*lval;
      maxTerm = PetscMax(PetscAbsReal(lterm), maxTerm);
      sum    += lterm;
      rterm   = alpha*wk*rval;
      maxTerm = PetscMax(PetscAbsReal(rterm), maxTerm);
      sum    += rterm;
      ++k;
      /* Only need to evaluate every other point on refined levels */
      if (l != 1) ++k;
    } while (PetscAbsReal(PetscLog10Real(wk)) < p); /* Only need to evaluate sum until weights are < 32 digits of precision */

    d1 = PetscLog10Real(PetscAbsReal(sum - osum));
    d2 = PetscLog10Real(PetscAbsReal(sum - psum));
    d3 = PetscLog10Real(maxTerm) - p;
    if (PetscMax(PetscAbsReal(lterm), PetscAbsReal(rterm)) == 0.0) d4 = 0.0;
    else d4 = PetscLog10Real(PetscMax(PetscAbsReal(lterm), PetscAbsReal(rterm)));
    d  = PetscAbsInt(PetscMin(0, PetscMax(PetscMax(PetscMax(PetscSqr(d1)/d2, 2*d1), d3), d4)));
  } while (d < digits && l < 12);
  *sol = sum;

  PetscFunctionReturn(0);
}

#if defined(PETSC_HAVE_MPFR)
PetscErrorCode PetscDTTanhSinhIntegrateMPFR(void (*func)(PetscReal, PetscReal *), PetscReal a, PetscReal b, PetscInt digits, PetscReal *sol)
{
  const PetscInt  safetyFactor = 2;  /* Calculate abcissa until 2*p digits */
  PetscInt        l            = 0;  /* Level of refinement, h = 2^{-l} */
  mpfr_t          alpha;             /* Half-width of the integration interval */
  mpfr_t          beta;              /* Center of the integration interval */
  mpfr_t          h;                 /* Step size, length between x_k */
  mpfr_t          osum;              /* Integral on last level */
  mpfr_t          psum;              /* Integral on the level before the last level */
  mpfr_t          sum;               /* Integral on current level */
  mpfr_t          yk;                /* Quadrature point 1 - x_k on reference domain [-1, 1] */
  mpfr_t          lx, rx;            /* Quadrature points to the left and right of 0 on the real domain [a, b] */
  mpfr_t          wk;                /* Quadrature weight at x_k */
  PetscReal       lval, rval;        /* Terms in the quadature sum to the left and right of 0 */
  PetscInt        d;                 /* Digits of precision in the integral */
  mpfr_t          pi2, kh, msinh, mcosh, maxTerm, curTerm, tmp;

  PetscFunctionBegin;
  if (digits <= 0) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must give a positive number of significant digits");
  /* Create high precision storage */
  mpfr_inits2(PetscCeilReal(safetyFactor*digits*PetscLogReal(10.)/PetscLogReal(2.)), alpha, beta, h, sum, osum, psum, yk, wk, lx, rx, tmp, maxTerm, curTerm, pi2, kh, msinh, mcosh, NULL);
  /* Initialization */
  mpfr_set_d(alpha, 0.5*(b-a), MPFR_RNDN);
  mpfr_set_d(beta,  0.5*(b+a), MPFR_RNDN);
  mpfr_set_d(osum,  0.0,       MPFR_RNDN);
  mpfr_set_d(psum,  0.0,       MPFR_RNDN);
  mpfr_set_d(h,     1.0,       MPFR_RNDN);
  mpfr_const_pi(pi2, MPFR_RNDN);
  mpfr_mul_d(pi2, pi2, 0.5, MPFR_RNDN);
  /* Center term */
  func(0.5*(b+a), &lval);
  mpfr_set(sum, pi2, MPFR_RNDN);
  mpfr_mul(sum, sum, alpha, MPFR_RNDN);
  mpfr_mul_d(sum, sum, lval, MPFR_RNDN);
  /* */
  do {
    PetscReal d1, d2, d3, d4;
    PetscInt  k = 1;

    ++l;
    mpfr_set_d(maxTerm, 0.0, MPFR_RNDN);
    /* PetscPrintf(PETSC_COMM_SELF, "LEVEL %D sum: %15.15f\n", l, sum); */
    /* At each level of refinement, h --> h/2 and sum --> sum/2 */
    mpfr_set(psum, osum, MPFR_RNDN);
    mpfr_set(osum,  sum, MPFR_RNDN);
    mpfr_mul_d(h,   h,   0.5, MPFR_RNDN);
    mpfr_mul_d(sum, sum, 0.5, MPFR_RNDN);
    do {
      mpfr_set_si(kh, k, MPFR_RNDN);
      mpfr_mul(kh, kh, h, MPFR_RNDN);
      /* Weight */
      mpfr_set(wk, h, MPFR_RNDN);
      mpfr_sinh_cosh(msinh, mcosh, kh, MPFR_RNDN);
      mpfr_mul(msinh, msinh, pi2, MPFR_RNDN);
      mpfr_mul(mcosh, mcosh, pi2, MPFR_RNDN);
      mpfr_cosh(tmp, msinh, MPFR_RNDN);
      mpfr_sqr(tmp, tmp, MPFR_RNDN);
      mpfr_mul(wk, wk, mcosh, MPFR_RNDN);
      mpfr_div(wk, wk, tmp, MPFR_RNDN);
      /* Abscissa */
      mpfr_set_d(yk, 1.0, MPFR_RNDZ);
      mpfr_cosh(tmp, msinh, MPFR_RNDN);
      mpfr_div(yk, yk, tmp, MPFR_RNDZ);
      mpfr_exp(tmp, msinh, MPFR_RNDN);
      mpfr_div(yk, yk, tmp, MPFR_RNDZ);
      /* Quadrature points */
      mpfr_sub_d(lx, yk, 1.0, MPFR_RNDZ);
      mpfr_mul(lx, lx, alpha, MPFR_RNDU);
      mpfr_add(lx, lx, beta, MPFR_RNDU);
      mpfr_d_sub(rx, 1.0, yk, MPFR_RNDZ);
      mpfr_mul(rx, rx, alpha, MPFR_RNDD);
      mpfr_add(rx, rx, beta, MPFR_RNDD);
      /* Evaluation */
      func(mpfr_get_d(lx, MPFR_RNDU), &lval);
      func(mpfr_get_d(rx, MPFR_RNDD), &rval);
      /* Update */
      mpfr_mul(tmp, wk, alpha, MPFR_RNDN);
      mpfr_mul_d(tmp, tmp, lval, MPFR_RNDN);
      mpfr_add(sum, sum, tmp, MPFR_RNDN);
      mpfr_abs(tmp, tmp, MPFR_RNDN);
      mpfr_max(maxTerm, maxTerm, tmp, MPFR_RNDN);
      mpfr_set(curTerm, tmp, MPFR_RNDN);
      mpfr_mul(tmp, wk, alpha, MPFR_RNDN);
      mpfr_mul_d(tmp, tmp, rval, MPFR_RNDN);
      mpfr_add(sum, sum, tmp, MPFR_RNDN);
      mpfr_abs(tmp, tmp, MPFR_RNDN);
      mpfr_max(maxTerm, maxTerm, tmp, MPFR_RNDN);
      mpfr_max(curTerm, curTerm, tmp, MPFR_RNDN);
      ++k;
      /* Only need to evaluate every other point on refined levels */
      if (l != 1) ++k;
      mpfr_log10(tmp, wk, MPFR_RNDN);
      mpfr_abs(tmp, tmp, MPFR_RNDN);
    } while (mpfr_get_d(tmp, MPFR_RNDN) < safetyFactor*digits); /* Only need to evaluate sum until weights are < 32 digits of precision */
    mpfr_sub(tmp, sum, osum, MPFR_RNDN);
    mpfr_abs(tmp, tmp, MPFR_RNDN);
    mpfr_log10(tmp, tmp, MPFR_RNDN);
    d1 = mpfr_get_d(tmp, MPFR_RNDN);
    mpfr_sub(tmp, sum, psum, MPFR_RNDN);
    mpfr_abs(tmp, tmp, MPFR_RNDN);
    mpfr_log10(tmp, tmp, MPFR_RNDN);
    d2 = mpfr_get_d(tmp, MPFR_RNDN);
    mpfr_log10(tmp, maxTerm, MPFR_RNDN);
    d3 = mpfr_get_d(tmp, MPFR_RNDN) - digits;
    mpfr_log10(tmp, curTerm, MPFR_RNDN);
    d4 = mpfr_get_d(tmp, MPFR_RNDN);
    d  = PetscAbsInt(PetscMin(0, PetscMax(PetscMax(PetscMax(PetscSqr(d1)/d2, 2*d1), d3), d4)));
  } while (d < digits && l < 8);
  *sol = mpfr_get_d(sum, MPFR_RNDN);
  /* Cleanup */
  mpfr_clears(alpha, beta, h, sum, osum, psum, yk, wk, lx, rx, tmp, maxTerm, curTerm, pi2, kh, msinh, mcosh, NULL);
  PetscFunctionReturn(0);
}
#else

PetscErrorCode PetscDTTanhSinhIntegrateMPFR(void (*func)(PetscReal, PetscReal *), PetscReal a, PetscReal b, PetscInt digits, PetscReal *sol)
{
  SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "This method will not work without MPFR. Reconfigure using --download-mpfr --download-gmp");
}
#endif

/* Overwrites A. Can only handle full-rank problems with m>=n
 * A in column-major format
 * Ainv in row-major format
 * tau has length m
 * worksize must be >= max(1,n)
 */
static PetscErrorCode PetscDTPseudoInverseQR(PetscInt m,PetscInt mstride,PetscInt n,PetscReal *A_in,PetscReal *Ainv_out,PetscScalar *tau,PetscInt worksize,PetscScalar *work)
{
  PetscErrorCode ierr;
  PetscBLASInt   M,N,K,lda,ldb,ldwork,info;
  PetscScalar    *A,*Ainv,*R,*Q,Alpha;

  PetscFunctionBegin;
#if defined(PETSC_USE_COMPLEX)
  {
    PetscInt i,j;
    ierr = PetscMalloc2(m*n,&A,m*n,&Ainv);CHKERRQ(ierr);
    for (j=0; j<n; j++) {
      for (i=0; i<m; i++) A[i+m*j] = A_in[i+mstride*j];
    }
    mstride = m;
  }
#else
  A = A_in;
  Ainv = Ainv_out;
#endif

  ierr = PetscBLASIntCast(m,&M);CHKERRQ(ierr);
  ierr = PetscBLASIntCast(n,&N);CHKERRQ(ierr);
  ierr = PetscBLASIntCast(mstride,&lda);CHKERRQ(ierr);
  ierr = PetscBLASIntCast(worksize,&ldwork);CHKERRQ(ierr);
  ierr = PetscFPTrapPush(PETSC_FP_TRAP_OFF);CHKERRQ(ierr);
  PetscStackCallBLAS("LAPACKgeqrf",LAPACKgeqrf_(&M,&N,A,&lda,tau,work,&ldwork,&info));
  ierr = PetscFPTrapPop();CHKERRQ(ierr);
  if (info) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_LIB,"xGEQRF error");
  R = A; /* Upper triangular part of A now contains R, the rest contains the elementary reflectors */

  /* Extract an explicit representation of Q */
  Q = Ainv;
  ierr = PetscArraycpy(Q,A,mstride*n);CHKERRQ(ierr);
  K = N;                        /* full rank */
  PetscStackCallBLAS("LAPACKorgqr",LAPACKorgqr_(&M,&N,&K,Q,&lda,tau,work,&ldwork,&info));
  if (info) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_LIB,"xORGQR/xUNGQR error");

  /* Compute A^{-T} = (R^{-1} Q^T)^T = Q R^{-T} */
  Alpha = 1.0;
  ldb = lda;
  PetscStackCallBLAS("BLAStrsm",BLAStrsm_("Right","Upper","ConjugateTranspose","NotUnitTriangular",&M,&N,&Alpha,R,&lda,Q,&ldb));
  /* Ainv is Q, overwritten with inverse */

#if defined(PETSC_USE_COMPLEX)
  {
    PetscInt i;
    for (i=0; i<m*n; i++) Ainv_out[i] = PetscRealPart(Ainv[i]);
    ierr = PetscFree2(A,Ainv);CHKERRQ(ierr);
  }
#endif
  PetscFunctionReturn(0);
}

/* Computes integral of L_p' over intervals {(x0,x1),(x1,x2),...} */
static PetscErrorCode PetscDTLegendreIntegrate(PetscInt ninterval,const PetscReal *x,PetscInt ndegree,const PetscInt *degrees,PetscBool Transpose,PetscReal *B)
{
  PetscErrorCode ierr;
  PetscReal      *Bv;
  PetscInt       i,j;

  PetscFunctionBegin;
  ierr = PetscMalloc1((ninterval+1)*ndegree,&Bv);CHKERRQ(ierr);
  /* Point evaluation of L_p on all the source vertices */
  ierr = PetscDTLegendreEval(ninterval+1,x,ndegree,degrees,Bv,NULL,NULL);CHKERRQ(ierr);
  /* Integral over each interval: \int_a^b L_p' = L_p(b)-L_p(a) */
  for (i=0; i<ninterval; i++) {
    for (j=0; j<ndegree; j++) {
      if (Transpose) B[i+ninterval*j] = Bv[(i+1)*ndegree+j] - Bv[i*ndegree+j];
      else           B[i*ndegree+j]   = Bv[(i+1)*ndegree+j] - Bv[i*ndegree+j];
    }
  }
  ierr = PetscFree(Bv);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*@
   PetscDTReconstructPoly - create matrix representing polynomial reconstruction using cell intervals and evaluation at target intervals

   Not Collective

   Input Arguments:
+  degree - degree of reconstruction polynomial
.  nsource - number of source intervals
.  sourcex - sorted coordinates of source cell boundaries (length nsource+1)
.  ntarget - number of target intervals
-  targetx - sorted coordinates of target cell boundaries (length ntarget+1)

   Output Arguments:
.  R - reconstruction matrix, utarget = sum_s R[t*nsource+s] * usource[s]

   Level: advanced

.seealso: PetscDTLegendreEval()
@*/
PetscErrorCode PetscDTReconstructPoly(PetscInt degree,PetscInt nsource,const PetscReal *sourcex,PetscInt ntarget,const PetscReal *targetx,PetscReal *R)
{
  PetscErrorCode ierr;
  PetscInt       i,j,k,*bdegrees,worksize;
  PetscReal      xmin,xmax,center,hscale,*sourcey,*targety,*Bsource,*Bsinv,*Btarget;
  PetscScalar    *tau,*work;

  PetscFunctionBegin;
  PetscValidRealPointer(sourcex,3);
  PetscValidRealPointer(targetx,5);
  PetscValidRealPointer(R,6);
  if (degree >= nsource) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_INCOMP,"Reconstruction degree %D must be less than number of source intervals %D",degree,nsource);
#if defined(PETSC_USE_DEBUG)
  for (i=0; i<nsource; i++) {
    if (sourcex[i] >= sourcex[i+1]) SETERRQ3(PETSC_COMM_SELF,PETSC_ERR_ARG_CORRUPT,"Source interval %D has negative orientation (%g,%g)",i,(double)sourcex[i],(double)sourcex[i+1]);
  }
  for (i=0; i<ntarget; i++) {
    if (targetx[i] >= targetx[i+1]) SETERRQ3(PETSC_COMM_SELF,PETSC_ERR_ARG_CORRUPT,"Target interval %D has negative orientation (%g,%g)",i,(double)targetx[i],(double)targetx[i+1]);
  }
#endif
  xmin = PetscMin(sourcex[0],targetx[0]);
  xmax = PetscMax(sourcex[nsource],targetx[ntarget]);
  center = (xmin + xmax)/2;
  hscale = (xmax - xmin)/2;
  worksize = nsource;
  ierr = PetscMalloc4(degree+1,&bdegrees,nsource+1,&sourcey,nsource*(degree+1),&Bsource,worksize,&work);CHKERRQ(ierr);
  ierr = PetscMalloc4(nsource,&tau,nsource*(degree+1),&Bsinv,ntarget+1,&targety,ntarget*(degree+1),&Btarget);CHKERRQ(ierr);
  for (i=0; i<=nsource; i++) sourcey[i] = (sourcex[i]-center)/hscale;
  for (i=0; i<=degree; i++) bdegrees[i] = i+1;
  ierr = PetscDTLegendreIntegrate(nsource,sourcey,degree+1,bdegrees,PETSC_TRUE,Bsource);CHKERRQ(ierr);
  ierr = PetscDTPseudoInverseQR(nsource,nsource,degree+1,Bsource,Bsinv,tau,nsource,work);CHKERRQ(ierr);
  for (i=0; i<=ntarget; i++) targety[i] = (targetx[i]-center)/hscale;
  ierr = PetscDTLegendreIntegrate(ntarget,targety,degree+1,bdegrees,PETSC_FALSE,Btarget);CHKERRQ(ierr);
  for (i=0; i<ntarget; i++) {
    PetscReal rowsum = 0;
    for (j=0; j<nsource; j++) {
      PetscReal sum = 0;
      for (k=0; k<degree+1; k++) {
        sum += Btarget[i*(degree+1)+k] * Bsinv[k*nsource+j];
      }
      R[i*nsource+j] = sum;
      rowsum += sum;
    }
    for (j=0; j<nsource; j++) R[i*nsource+j] /= rowsum; /* normalize each row */
  }
  ierr = PetscFree4(bdegrees,sourcey,Bsource,work);CHKERRQ(ierr);
  ierr = PetscFree4(tau,Bsinv,targety,Btarget);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*@C
   PetscGaussLobattoLegendreIntegrate - Compute the L2 integral of a function on the GLL points

   Not Collective

   Input Parameter:
+  n - the number of GLL nodes
.  nodes - the GLL nodes
.  weights - the GLL weights
.  f - the function values at the nodes

   Output Parameter:
.  in - the value of the integral

   Level: beginner

.seealso: PetscDTGaussLobattoLegendreQuadrature()

@*/
PetscErrorCode PetscGaussLobattoLegendreIntegrate(PetscInt n,PetscReal *nodes,PetscReal *weights,const PetscReal *f,PetscReal *in)
{
  PetscInt          i;

  PetscFunctionBegin;
  *in = 0.;
  for (i=0; i<n; i++) {
    *in += f[i]*f[i]*weights[i];
  }
  PetscFunctionReturn(0);
}

/*@C
   PetscGaussLobattoLegendreElementLaplacianCreate - computes the Laplacian for a single 1d GLL element

   Not Collective

   Input Parameter:
+  n - the number of GLL nodes
.  nodes - the GLL nodes
.  weights - the GLL weights

   Output Parameter:
.  A - the stiffness element

   Level: beginner

   Notes:
    Destroy this with PetscGaussLobattoLegendreElementLaplacianDestroy()

   You can access entries in this array with AA[i][j] but in memory it is stored in contiguous memory, row oriented (the array is symmetric)

.seealso: PetscDTGaussLobattoLegendreQuadrature(), PetscGaussLobattoLegendreElementLaplacianDestroy()

@*/
PetscErrorCode PetscGaussLobattoLegendreElementLaplacianCreate(PetscInt n,PetscReal *nodes,PetscReal *weights,PetscReal ***AA)
{
  PetscReal        **A;
  PetscErrorCode  ierr;
  const PetscReal  *gllnodes = nodes;
  const PetscInt   p = n-1;
  PetscReal        z0,z1,z2 = -1,x,Lpj,Lpr;
  PetscInt         i,j,nn,r;

  PetscFunctionBegin;
  ierr = PetscMalloc1(n,&A);CHKERRQ(ierr);
  ierr = PetscMalloc1(n*n,&A[0]);CHKERRQ(ierr);
  for (i=1; i<n; i++) A[i] = A[i-1]+n;

  for (j=1; j<p; j++) {
    x  = gllnodes[j];
    z0 = 1.;
    z1 = x;
    for (nn=1; nn<p; nn++) {
      z2 = x*z1*(2.*((PetscReal)nn)+1.)/(((PetscReal)nn)+1.)-z0*(((PetscReal)nn)/(((PetscReal)nn)+1.));
      z0 = z1;
      z1 = z2;
    }
    Lpj=z2;
    for (r=1; r<p; r++) {
      if (r == j) {
        A[j][j]=2./(3.*(1.-gllnodes[j]*gllnodes[j])*Lpj*Lpj);
      } else {
        x  = gllnodes[r];
        z0 = 1.;
        z1 = x;
        for (nn=1; nn<p; nn++) {
          z2 = x*z1*(2.*((PetscReal)nn)+1.)/(((PetscReal)nn)+1.)-z0*(((PetscReal)nn)/(((PetscReal)nn)+1.));
          z0 = z1;
          z1 = z2;
        }
        Lpr     = z2;
        A[r][j] = 4./(((PetscReal)p)*(((PetscReal)p)+1.)*Lpj*Lpr*(gllnodes[j]-gllnodes[r])*(gllnodes[j]-gllnodes[r]));
      }
    }
  }
  for (j=1; j<p+1; j++) {
    x  = gllnodes[j];
    z0 = 1.;
    z1 = x;
    for (nn=1; nn<p; nn++) {
      z2 = x*z1*(2.*((PetscReal)nn)+1.)/(((PetscReal)nn)+1.)-z0*(((PetscReal)nn)/(((PetscReal)nn)+1.));
      z0 = z1;
      z1 = z2;
    }
    Lpj     = z2;
    A[j][0] = 4.*PetscPowRealInt(-1.,p)/(((PetscReal)p)*(((PetscReal)p)+1.)*Lpj*(1.+gllnodes[j])*(1.+gllnodes[j]));
    A[0][j] = A[j][0];
  }
  for (j=0; j<p; j++) {
    x  = gllnodes[j];
    z0 = 1.;
    z1 = x;
    for (nn=1; nn<p; nn++) {
      z2 = x*z1*(2.*((PetscReal)nn)+1.)/(((PetscReal)nn)+1.)-z0*(((PetscReal)nn)/(((PetscReal)nn)+1.));
      z0 = z1;
      z1 = z2;
    }
    Lpj=z2;

    A[p][j] = 4./(((PetscReal)p)*(((PetscReal)p)+1.)*Lpj*(1.-gllnodes[j])*(1.-gllnodes[j]));
    A[j][p] = A[p][j];
  }
  A[0][0]=0.5+(((PetscReal)p)*(((PetscReal)p)+1.)-2.)/6.;
  A[p][p]=A[0][0];
  *AA = A;
  PetscFunctionReturn(0);
}

/*@C
   PetscGaussLobattoLegendreElementLaplacianDestroy - frees the Laplacian for a single 1d GLL element

   Not Collective

   Input Parameter:
+  n - the number of GLL nodes
.  nodes - the GLL nodes
.  weights - the GLL weightss
-  A - the stiffness element

   Level: beginner

.seealso: PetscDTGaussLobattoLegendreQuadrature(), PetscGaussLobattoLegendreElementLaplacianCreate()

@*/
PetscErrorCode PetscGaussLobattoLegendreElementLaplacianDestroy(PetscInt n,PetscReal *nodes,PetscReal *weights,PetscReal ***AA)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = PetscFree((*AA)[0]);CHKERRQ(ierr);
  ierr = PetscFree(*AA);CHKERRQ(ierr);
  *AA  = NULL;
  PetscFunctionReturn(0);
}

/*@C
   PetscGaussLobattoLegendreElementGradientCreate - computes the gradient for a single 1d GLL element

   Not Collective

   Input Parameter:
+  n - the number of GLL nodes
.  nodes - the GLL nodes
.  weights - the GLL weights

   Output Parameter:
.  AA - the stiffness element
-  AAT - the transpose of AA (pass in NULL if you do not need this array)

   Level: beginner

   Notes:
    Destroy this with PetscGaussLobattoLegendreElementGradientDestroy()

   You can access entries in these arrays with AA[i][j] but in memory it is stored in contiguous memory, row oriented

.seealso: PetscDTGaussLobattoLegendreQuadrature(), PetscGaussLobattoLegendreElementLaplacianDestroy()

@*/
PetscErrorCode PetscGaussLobattoLegendreElementGradientCreate(PetscInt n,PetscReal *nodes,PetscReal *weights,PetscReal ***AA,PetscReal ***AAT)
{
  PetscReal        **A, **AT = NULL;
  PetscErrorCode  ierr;
  const PetscReal  *gllnodes = nodes;
  const PetscInt   p = n-1;
  PetscReal        q,qp,Li, Lj,d0;
  PetscInt         i,j;

  PetscFunctionBegin;
  ierr = PetscMalloc1(n,&A);CHKERRQ(ierr);
  ierr = PetscMalloc1(n*n,&A[0]);CHKERRQ(ierr);
  for (i=1; i<n; i++) A[i] = A[i-1]+n;

  if (AAT) {
    ierr = PetscMalloc1(n,&AT);CHKERRQ(ierr);
    ierr = PetscMalloc1(n*n,&AT[0]);CHKERRQ(ierr);
    for (i=1; i<n; i++) AT[i] = AT[i-1]+n;
  }

  if (n==1) {A[0][0] = 0.;}
  d0 = (PetscReal)p*((PetscReal)p+1.)/4.;
  for  (i=0; i<n; i++) {
    for  (j=0; j<n; j++) {
      A[i][j] = 0.;
      qAndLEvaluation(p,gllnodes[i],&q,&qp,&Li);
      qAndLEvaluation(p,gllnodes[j],&q,&qp,&Lj);
      if (i!=j)             A[i][j] = Li/(Lj*(gllnodes[i]-gllnodes[j]));
      if ((j==i) && (i==0)) A[i][j] = -d0;
      if (j==i && i==p)     A[i][j] = d0;
      if (AT) AT[j][i] = A[i][j];
    }
  }
  if (AAT) *AAT = AT;
  *AA  = A;
  PetscFunctionReturn(0);
}

/*@C
   PetscGaussLobattoLegendreElementGradientDestroy - frees the gradient for a single 1d GLL element obtained with PetscGaussLobattoLegendreElementGradientCreate()

   Not Collective

   Input Parameter:
+  n - the number of GLL nodes
.  nodes - the GLL nodes
.  weights - the GLL weights
.  AA - the stiffness element
-  AAT - the transpose of the element

   Level: beginner

.seealso: PetscDTGaussLobattoLegendreQuadrature(), PetscGaussLobattoLegendreElementLaplacianCreate(), PetscGaussLobattoLegendreElementAdvectionCreate()

@*/
PetscErrorCode PetscGaussLobattoLegendreElementGradientDestroy(PetscInt n,PetscReal *nodes,PetscReal *weights,PetscReal ***AA,PetscReal ***AAT)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = PetscFree((*AA)[0]);CHKERRQ(ierr);
  ierr = PetscFree(*AA);CHKERRQ(ierr);
  *AA  = NULL;
  if (*AAT) {
    ierr = PetscFree((*AAT)[0]);CHKERRQ(ierr);
    ierr = PetscFree(*AAT);CHKERRQ(ierr);
    *AAT  = NULL;
  }
  PetscFunctionReturn(0);
}

/*@C
   PetscGaussLobattoLegendreElementAdvectionCreate - computes the advection operator for a single 1d GLL element

   Not Collective

   Input Parameter:
+  n - the number of GLL nodes
.  nodes - the GLL nodes
.  weights - the GLL weightss

   Output Parameter:
.  AA - the stiffness element

   Level: beginner

   Notes:
    Destroy this with PetscGaussLobattoLegendreElementAdvectionDestroy()

   This is the same as the Gradient operator multiplied by the diagonal mass matrix

   You can access entries in this array with AA[i][j] but in memory it is stored in contiguous memory, row oriented

.seealso: PetscDTGaussLobattoLegendreQuadrature(), PetscGaussLobattoLegendreElementLaplacianCreate(), PetscGaussLobattoLegendreElementAdvectionDestroy()

@*/
PetscErrorCode PetscGaussLobattoLegendreElementAdvectionCreate(PetscInt n,PetscReal *nodes,PetscReal *weights,PetscReal ***AA)
{
  PetscReal       **D;
  PetscErrorCode  ierr;
  const PetscReal  *gllweights = weights;
  const PetscInt   glln = n;
  PetscInt         i,j;

  PetscFunctionBegin;
  ierr = PetscGaussLobattoLegendreElementGradientCreate(n,nodes,weights,&D,NULL);CHKERRQ(ierr);
  for (i=0; i<glln; i++){
    for (j=0; j<glln; j++) {
      D[i][j] = gllweights[i]*D[i][j];
    }
  }
  *AA = D;
  PetscFunctionReturn(0);
}

/*@C
   PetscGaussLobattoLegendreElementAdvectionDestroy - frees the advection stiffness for a single 1d GLL element

   Not Collective

   Input Parameter:
+  n - the number of GLL nodes
.  nodes - the GLL nodes
.  weights - the GLL weights
-  A - advection

   Level: beginner

.seealso: PetscDTGaussLobattoLegendreQuadrature(), PetscGaussLobattoLegendreElementAdvectionCreate()

@*/
PetscErrorCode PetscGaussLobattoLegendreElementAdvectionDestroy(PetscInt n,PetscReal *nodes,PetscReal *weights,PetscReal ***AA)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = PetscFree((*AA)[0]);CHKERRQ(ierr);
  ierr = PetscFree(*AA);CHKERRQ(ierr);
  *AA  = NULL;
  PetscFunctionReturn(0);
}

PetscErrorCode PetscGaussLobattoLegendreElementMassCreate(PetscInt n,PetscReal *nodes,PetscReal *weights,PetscReal ***AA)
{
  PetscReal        **A;
  PetscErrorCode  ierr;
  const PetscReal  *gllweights = weights;
  const PetscInt   glln = n;
  PetscInt         i,j;

  PetscFunctionBegin;
  ierr = PetscMalloc1(glln,&A);CHKERRQ(ierr);
  ierr = PetscMalloc1(glln*glln,&A[0]);CHKERRQ(ierr);
  for (i=1; i<glln; i++) A[i] = A[i-1]+glln;
  if (glln==1) {A[0][0] = 0.;}
  for  (i=0; i<glln; i++) {
    for  (j=0; j<glln; j++) {
      A[i][j] = 0.;
      if (j==i)     A[i][j] = gllweights[i];
    }
  }
  *AA  = A;
  PetscFunctionReturn(0);
}

PetscErrorCode PetscGaussLobattoLegendreElementMassDestroy(PetscInt n,PetscReal *nodes,PetscReal *weights,PetscReal ***AA)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = PetscFree((*AA)[0]);CHKERRQ(ierr);
  ierr = PetscFree(*AA);CHKERRQ(ierr);
  *AA  = NULL;
  PetscFunctionReturn(0);
}