//---------------------------------------------------------
DVec& NDG2D::PoissonIPDGbc2D
(DVec& ubc, //[in]
 DVec& qbc  //[in]
)
//---------------------------------------------------------
{
  // function [OP] = PoissonIPDGbc2D()
  // Purpose: Set up the discrete Poisson matrix directly
  //          using LDG. The operator is set up in the weak form

  // build DG derivative matrices
  int max_OP = (K*Np*Np*(1+Nfaces));

  // initialize parameters
  DVec faceR("faceR"), faceS("faceS");
  DMat V1D("V1D"), Dx("Dx"),Dy("Dy"), Dn1("Dn1"), mmE_Fm1("mmE(:,Fm1)");
  IVec Fm("Fm"), Fm1("Fm1"), fidM("fidM");
  double lnx=0.0,lny=0.0,lsJ=0.0,hinv=0.0,gtau=0.0;
  int i=0,k1=0,f1=0,id=0;
  IVec i1_Nfp = Range(1,Nfp);
  double N1N1 = double((N+1)*(N+1));

  // build local face matrices
  DMat massEdge[4]; // = zeros(Np,Np,Nfaces);
  for (i=1; i<=Nfaces; ++i) {
    massEdge[i].resize(Np,Np);
  }

  // face mass matrix 1
  Fm = Fmask(All,1); faceR = r(Fm); 
  V1D = Vandermonde1D(N, faceR);
  massEdge[1](Fm,Fm) = inv(V1D*trans(V1D));

  // face mass matrix 2
  Fm = Fmask(All,2); faceR = r(Fm); 
  V1D = Vandermonde1D(N, faceR);
  massEdge[2](Fm,Fm) = inv(V1D*trans(V1D));

  // face mass matrix 3
  Fm = Fmask(All,3); faceS = s(Fm); 
  V1D = Vandermonde1D(N, faceS); 
  massEdge[3](Fm,Fm) = inv(V1D*trans(V1D));
 
  // build DG right hand side
  DVec* pBC = new DVec(Np*K, "bc", OBJ_temp); 
  DVec& bc = (*pBC);  // reference, for syntax
  ////////////////////////////////////////////////////////////////

  umMSG(1, "\n ==> {OP} assembly [bc]: ");
  for (k1=1; k1<=K; ++k1)
  {
    if (! (k1%100)) { umMSG(1, "%d, ",k1); }

    // rows1 = outer(Range((k1-1)*Np+1,k1*Np), Ones(NGauss));

    // Build element-to-element parts of operator
    for (f1=1; f1<=Nfaces; ++f1)
    {
      if (BCType(k1,f1))
      {
        ////////////////////////added by Kevin ///////////////////////////////
        Fm1 = Fmask(All,f1); 
        fidM  = (k1-1)*Nfp*Nfaces + (f1-1)*Nfp + i1_Nfp;
        id = 1+(f1-1)*Nfp + (k1-1)*Nfp*Nfaces;

        lnx = nx(id); lny = ny(id); 
        lsJ = sJ(id); hinv = Fscale(id);

        Dx = rx(1,k1)*Dr + sx(1,k1)*Ds;  
        Dy = ry(1,k1)*Dr + sy(1,k1)*Ds;
        Dn1 = lnx*Dx + lny*Dy;

      //mmE = lsJ*massEdge(:,:,f1);
      //bc(All,k1) += (gtau*mmE(All,Fm1) - Dn1'*mmE(All,Fm1))*ubc(fidM);

        mmE_Fm1 = massEdge[f1](All,Fm1);  mmE_Fm1 *= lsJ;

        gtau = 10*N1N1*hinv; // set penalty scaling
        //bc(All,k1) += (gtau*mmE_Fm1 - trans(Dn1)*mmE_Fm1) * ubc(fidM);

        switch(BCType(k1,f1)){
	  case BC_Dirichlet: 
            bc(Np*(k1-1)+Range(1,Np)) += (gtau*mmE_Fm1 - trans(Dn1)*mmE_Fm1)*ubc(fidM);  
            break;
          case BC_Neuman:
            bc(Np*(k1-1)+Range(1,Np)) += mmE_Fm1*qbc(fidM);
            break;
	default:
	  std::cout<<"warning: boundary condition is incorrect"<<std::endl;
	}
      }
    }
  }
  return bc;
}
//---------------------------------------------------------
void NDG3D::PoissonIPDG3D(CSd& spOP, CSd& spMM)
//---------------------------------------------------------
{
  // function [OP,MM] = PoissonIPDG3D()
  //
  // Purpose: Set up the discrete Poisson matrix directly
  //          using LDG. The operator is set up in the weak form


  DVec faceR("faceR"), faceS("faceS"), faceT("faceT");
  DMat V2D;  IVec Fm("Fm");  IVec i1_Nfp = Range(1,Nfp);
  double opti1=0.0, opti2=0.0; int i=0; 

  umLOG(1, "\n ==> {OP,MM} assembly: ");
  opti1 = timer.read(); // time assembly

  // build local face matrices
  DMat massEdge[5]; // = zeros(Np,Np,Nfaces);
  for (i=1; i<=Nfaces; ++i) {
    massEdge[i].resize(Np,Np);
  }

  // face mass matrix 1
  Fm = Fmask(All,1); faceR=r(Fm); faceS=s(Fm); 
  V2D = Vandermonde2D(N, faceR, faceS);
  massEdge[1](Fm,Fm) = inv(V2D*trans(V2D));

  // face mass matrix 2
  Fm = Fmask(All,2); faceR = r(Fm); faceT = t(Fm);
  V2D = Vandermonde2D(N, faceR, faceT);
  massEdge[2](Fm,Fm) = inv(V2D*trans(V2D));

  // face mass matrix 3
  Fm = Fmask(All,3); faceS = s(Fm); faceT = t(Fm);
  V2D = Vandermonde2D(N, faceS, faceT); 
  massEdge[3](Fm,Fm) = inv(V2D*trans(V2D));

  // face mass matrix 4
  Fm = Fmask(All,4); faceS = s(Fm); faceT = t(Fm);
  V2D = Vandermonde2D(N, faceS, faceT); 
  massEdge[4](Fm,Fm) = inv(V2D*trans(V2D));

  // build local volume mass matrix
  MassMatrix = trans(invV)*invV;

  DMat Dx("Dx"),Dy("Dy"),Dz("Dz"), Dx2("Dx2"),Dy2("Dy2"),Dz2("Dz2");
  DMat Dn1("Dn1"),Dn2("Dn2"), mmE("mmE"), OP11("OP11"), OP12("OP12");
  DMat mmE_All_Fm1, mmE_Fm1_Fm1, Dn2_Fm2_All;
  IMat rows1,cols1,rows2,cols2;  int k1=0,f1=0,k2=0,f2=0,id=0;
  Index1D entries, entriesMM, idsM;  IVec fidM,vidM,Fm1,vidP,Fm2;
  double lnx=0.0,lny=0.0,lnz=0.0,lsJ=0.0,hinv=0.0,gtau=0.0;
  double N1N1 = double((N+1)*(N+1)); int NpNp = Np*Np;

  // build DG derivative matrices
  int max_OP = (K*Np*Np*(1+Nfaces));
  int max_MM = (K*Np*Np);

  // "OP" triplets (i,j,x), extracted to {Ai,Aj,Ax}
  IVec OPi(max_OP), OPj(max_OP), Ai,Aj; DVec OPx(max_OP), Ax;
  // "MM" triplets (i,j,x)
  IVec MMi(max_MM), MMj(max_MM); DVec MMx(max_MM);
  IVec OnesNp = Ones(Np);

  // global node numbering
  entries.reset(1,NpNp); entriesMM.reset(1,NpNp);

  OP12.resize(Np,Np);

  for (k1=1; k1<=K; ++k1)
  {
    if (! (k1%250)) { umLOG(1, "%d, ",k1); }

    rows1 = outer( Range((k1-1)*Np+1,k1*Np), OnesNp );
    cols1 = trans(rows1);

    // Build local operators  
    Dx = rx(1,k1)*Dr + sx(1,k1)*Ds + tx(1,k1)*Dt;   
    Dy = ry(1,k1)*Dr + sy(1,k1)*Ds + ty(1,k1)*Dt;
    Dz = rz(1,k1)*Dr + sz(1,k1)*Ds + tz(1,k1)*Dt;

    OP11 = J(1,k1)*(trans(Dx)*MassMatrix*Dx + 
                    trans(Dy)*MassMatrix*Dy + 
                    trans(Dz)*MassMatrix*Dz);

    // Build element-to-element parts of operator
    for (f1=1; f1<=Nfaces; ++f1) {
      k2 = EToE(k1,f1); f2 = EToF(k1,f1); 

      rows2 = outer( Range((k2-1)*Np+1, k2*Np), OnesNp );
      cols2 = trans(rows2);

      fidM  = (k1-1)*Nfp*Nfaces + (f1-1)*Nfp + i1_Nfp;
      vidM = vmapM(fidM); Fm1 = mod(vidM-1,Np)+1;
      vidP = vmapP(fidM); Fm2 = mod(vidP-1,Np)+1;

      id = 1+(f1-1)*Nfp + (k1-1)*Nfp*Nfaces;
      lnx = nx(id);  lny = ny(id);  lnz = nz(id); lsJ = sJ(id); 
      hinv = std::max(Fscale(id), Fscale(1+(f2-1)*Nfp, k2));    

      Dx2 = rx(1,k2)*Dr + sx(1,k2)*Ds + tx(1,k2)*Dt;   
      Dy2 = ry(1,k2)*Dr + sy(1,k2)*Ds + ty(1,k2)*Dt;
      Dz2 = rz(1,k2)*Dr + sz(1,k2)*Ds + tz(1,k2)*Dt;
      
      Dn1 = lnx*Dx  + lny*Dy  + lnz*Dz;
      Dn2 = lnx*Dx2 + lny*Dy2 + lnz*Dz2;

      mmE = lsJ*massEdge[f1];

      gtau = 2.0 * N1N1 * hinv; // set penalty scaling

      if (EToE(k1,f1)==k1) {
        OP11 += ( gtau*mmE - mmE*Dn1 - trans(Dn1)*mmE ); // ok
      }
      else 
      {
        // interior face variational terms
        OP11 += 0.5*( gtau*mmE - mmE*Dn1 - trans(Dn1)*mmE );

        // extract mapped regions:
        mmE_All_Fm1 = mmE(All,Fm1);
        mmE_Fm1_Fm1 = mmE(Fm1,Fm1);
        Dn2_Fm2_All = Dn2(Fm2,All);

        OP12 = 0.0;   // reset to zero
        OP12(All,Fm2)  = -0.5*(       gtau*mmE_All_Fm1 );
        OP12(Fm1,All) -=  0.5*(            mmE_Fm1_Fm1*Dn2_Fm2_All );
      //OP12(All,Fm2) -=  0.5*(-trans(Dn1)*mmE_All_Fm1 );
        OP12(All,Fm2) +=  0.5*( trans(Dn1)*mmE_All_Fm1 );

        // load this set of triplets
#if (1)
        OPi(entries)=rows1; OPj(entries)=cols2, OPx(entries)=OP12;
        entries += (NpNp);
#else
        //###########################################################
        // load only the lower triangle (after droptol test?)
        sk=0; start=entries(1);
        for (int i=1; i<=NpNp; ++i) {
          eid = start+i;
          id=entries(eid); rid=rows1(i); cid=cols2(i);
          if (rows1(rid) >= cid) {          // take lower triangle
            if ( fabs(OP12(id)) > 1e-15) {  // drop small entries
              ++sk; OPi(id)=rid; OPj(id)=cid, OPx(id)=OP12(id);
            }
          }
        }
        entries += sk;
        //###########################################################
#endif
      }
    }

    OPi(entries  )=rows1; OPj(entries  )=cols1, OPx(entries  )=OP11;
    MMi(entriesMM)=rows1; MMj(entriesMM)=cols1; MMx(entriesMM)=J(1,k1)*MassMatrix;
    entries += (NpNp); entriesMM += (NpNp);
  }
  umLOG(1, "\n ==> {OP,MM} to sparse\n");

  entries.reset(1, entries.hi()-Np*Np);

  // Extract triplets from the large buffers. Note: this 
  // requires copying each array, and since these arrays 
  // can be HUGE(!), we force immediate deallocation:

  Ai=OPi(entries);  OPi.Free();
  Aj=OPj(entries);  OPj.Free();
  Ax=OPx(entries);  OPx.Free();
  umLOG(1, " ==> triplets ready (OP) nnz = %10d\n", entries.hi());

  // adjust triplet indices for 0-based sparse operators
  Ai -= 1; Aj -= 1; MMi -= 1; MMj -= 1;  int npk=Np*K;

#if defined(NDG_USE_CHOLMOD) || defined(NDG_New_CHOLINC)
  // load only the lower triangle tril(OP)        free args?
  spOP.load(npk,npk, Ai,Aj,Ax, sp_LT, false,1e-15, true);  // {LT, false} -> TriL
#else
  // select {upper,lower,both} triangles
//spOP.load(npk,npk, Ai,Aj,Ax, sp_LT, true,1e-15,true);   // LT -> enforce symmetry
//spOP.load(npk,npk, Ai,Aj,Ax, sp_All,true,1e-15,true);   // All-> includes "noise"
//spOP.load(npk,npk, Ai,Aj,Ax, sp_UT, false,1e-15,true);  // UT -> triu(OP) only
#endif

  Ai.Free();  Aj.Free();  Ax.Free();

  umLOG(1, " ==> triplets ready (MM) nnz = %10d\n", entriesMM.hi());

  //-------------------------------------------------------
  // The mass matrix operator will NOT be factorised, 
  // Load ALL elements (both upper and lower triangles):
  //-------------------------------------------------------
  spMM.load(npk,npk, MMi,MMj,MMx, sp_All,false,1.00e-15,true);
  MMi.Free(); MMj.Free(); MMx.Free();

  opti2 = timer.read(); // time assembly
  umLOG(1, " ==> {OP,MM} converted to csc.  (%g secs)\n", opti2-opti1);
}
void NDG2D::PoissonIPDGbc2D(
  CSd& spOP //[out] sparse operator 
  )
{
  // function [OP] = PoissonIPDGbc2D()
  // Purpose: Set up the discrete Poisson matrix directly
  //          using LDG. The operator is set up in the weak form

  // build DG derivative matrices
  int max_OP = (K*Np*Np*(1+Nfaces));

  //initialize parameters
  DVec faceR("faceR"), faceS("faceS");
  IVec Fm("Fm"), Fm1("Fm1"), fidM("fidM");
  DMat V1D("V1D"); int i=0;

  // build local face matrices
  DMat massEdge[4]; // = zeros(Np,Np,Nfaces);
  for (i=1; i<=Nfaces; ++i) {
    massEdge[i].resize(Np,Np);
  }

  // face mass matrix 1
  Fm = Fmask(All,1); faceR = r(Fm); 
  V1D = Vandermonde1D(N, faceR);
  massEdge[1](Fm,Fm) = inv(V1D*trans(V1D));

  // face mass matrix 2
  Fm = Fmask(All,2); faceR = r(Fm); 
  V1D = Vandermonde1D(N, faceR);
  massEdge[2](Fm,Fm) = inv(V1D*trans(V1D));

  // face mass matrix 3
  Fm = Fmask(All,3); faceS = s(Fm); 
  V1D = Vandermonde1D(N, faceS); 
  massEdge[3](Fm,Fm) = inv(V1D*trans(V1D));

  //continue initialize parameters
  DMat Dx("Dx"),Dy("Dy"), Dn1("Dn1"), mmE_Fm1("mmE(:,Fm1)");
  double lnx=0.0,lny=0.0,lsJ=0.0,hinv=0.0,gtau=0.0;
  int k1=0,f1=0,id=0;
  IVec i1_Nfp = Range(1,Nfp);
  double N1N1 = double((N+1)*(N+1));
  
  // "OP" triplets (i,j,x), extracted to {Ai,Aj,Ax}
  IVec OPi(max_OP),OPj(max_OP), Ai,Aj; DVec OPx(max_OP), Ax;
  IMat rows1, cols1;  Index1D entries; DMat OP11(Np,Nfp, 0.0);

  // global node numbering
  entries.reset(1,Np*Nfp); 
  cols1 = outer(Ones(Np), Range(1,Nfp));

  umMSG(1, "\n ==> {OP} assembly [bc]: ");
  for (k1=1; k1<=K; ++k1)
  {
    if (! (k1%100)) { umMSG(1, "%d, ",k1); }
    rows1 = outer(Range((k1-1)*Np+1,k1*Np), Ones(Nfp));

    // Build element-to-element parts of operator
    for (f1=1; f1<=Nfaces; ++f1)
    {
      if (BCType(k1,f1))
      {   
        ////////////////////////added by Kevin ///////////////////////////////
        Fm1 = Fmask(All,f1); 
        fidM  = (k1-1)*Nfp*Nfaces + (f1-1)*Nfp + i1_Nfp;
        id = 1+(f1-1)*Nfp + (k1-1)*Nfp*Nfaces;

        lnx = nx(id); lny = ny(id); 
        lsJ = sJ(id); hinv = Fscale(id);

        Dx = rx(1,k1)*Dr + sx(1,k1)*Ds;  
        Dy = ry(1,k1)*Dr + sy(1,k1)*Ds;
        Dn1 = lnx*Dx + lny*Dy;

      //mmE = lsJ*massEdge(:,:,f1);
      //bc(All,k1) += (gtau*mmE(All,Fm1) - Dn1'*mmE(All,Fm1))*ubc(fidM);

        mmE_Fm1 = massEdge[f1](All,Fm1);  mmE_Fm1 *= lsJ;

        gtau = 10*N1N1*hinv; // set penalty scaling
        //bc(All,k1) += (gtau*mmE_Fm1 - trans(Dn1)*mmE_Fm1) * ubc(fidM);

        switch(BCType(k1,f1)){
	  case BC_Dirichlet: 
            OP11 = gtau*mmE_Fm1 - trans(Dn1)*mmE_Fm1;  
            break;
          case BC_Neuman:
            OP11 = mmE_Fm1;
            break;
	default:
	  std::cout<<"warning: boundary condition is incorrect"<<std::endl;
	}

        OPi(entries)=rows1; OPj(entries)=cols1; OPx(entries)=OP11; 
        entries += (Np*Nfp);
      }
      cols1 += Nfp;
    }
  }

  umMSG(1, "\n ==> {OPbc} to sparse\n");
  entries.reset(1, entries.hi()-(Np*Nfp));

  // extract triplets from large buffers
  Ai=OPi(entries); Aj=OPj(entries); Ax=OPx(entries);

  // These arrays can be HUGE, so force deallocation
  OPi.Free(); OPj.Free(); OPx.Free();

  // return 0-based sparse result
  Ai -= 1; Aj -= 1;

  //-------------------------------------------------------
  // This operator is not symmetric, and will NOT be 
  // factorised, only used to create reference RHS's:
  //
  //    refrhsbcPR = spOP1 * bcPR;
  //    refrhsbcUx = spOP2 * bcUx;
  //    refrhsbcUy = spOP2 * bcUy;
  //
  // Load ALL elements (both upper and lower triangles):
  //-------------------------------------------------------
  spOP.load(Np*K, Nfp*Nfaces*K, Ai,Aj,Ax, sp_All,false, 1e-15,true);

  Ai.Free();  Aj.Free();  Ax.Free();
  umMSG(1, " ==> {OPbc} ready.\n");

#if (1)
  // check on original estimates for nnx
  umMSG(1, " ==> max_OP: %12d\n", max_OP);
  umMSG(1, " ==> nnz_OP: %12d\n", entries.hi());
#endif
}