inline typename GI::Vector
    EulerUpstreamResidual<GI, RP, BC>::
    estimateCapPressureGradient(const FIt& f, const FIt& nbf, const std::vector<double>& saturation) const
    {
	typedef typename GI::CellIterator::FaceIterator Face;
	typedef typename Face::Cell Cell;
	typedef typename GI::Vector Vector;

	// At nonperiodic boundaries, we return a zero gradient.
	// That is (sort of) a trivial Neumann (noflow) condition for the capillary pressure.
	if (f->boundary() && !pboundary_->satCond(*f).isPeriodic()) {
	    return Vector(0.0);
	}
	// Find neighbouring cell and face: nbc and nbf.
	// If we are not on a periodic boundary, nbf is of course equal to f.
	Cell c = f->cell();
	Cell nb = f->boundary() ? (f == nbf ? c : nbf->cell()) : f->neighbourCell();

	// Estimate the gradient like a finite difference between
	// cell centers, except that in order to handle periodic
	// conditions we pass through the face centroid(s).
	Vector cell_c = c.centroid();
	Vector nb_c = nb.centroid();
	Vector f_c = f->centroid();
	Vector nbf_c = nbf->centroid();
	double d0 = (cell_c - f_c).two_norm();
	double d1 = (nb_c - nbf_c).two_norm();
	int cell = c.index();
	int nbcell = nb.index();
	double cp0 = cap_pressures_[cell];
	double cp1 = cap_pressures_[nbcell];
	double val = (cp1 - cp0)/(d0 + d1);
	Vector res = nb_c - nbf_c + f_c - cell_c;
	res /= res.two_norm();
	res *= val;
	return res;
    }
    inline void EulerUpstreamImplicit<GI, RP, BC>::initObj(const GI& g, const RP& r, const BC& b)
    {
        //residual_computer_.initObj(g, r, b);

        mygrid_.init(g.grid());
        porevol_.resize(mygrid_.numCells());
        for (int i = 0; i < mygrid_.numCells(); ++i){
            porevol_[i]= mygrid_.cellVolume(i)*r.porosity(i);
        }
        // int numf=mygrid_.numFaces();
        int num_cells = mygrid_.numCells();
        int ngconn  = mygrid_.c_grid()->cell_facepos[num_cells];
        //std::vector<double> htrans_(ngconn);
        htrans_.resize(ngconn);
        const double* perm = &(r.permeability(0)(0,0));
        tpfa_htrans_compute(mygrid_.c_grid(), perm, &htrans_[0]);
        // int count = 0;

        myrp_= r;

        typedef typename GI::CellIterator CIt;
        typedef typename CIt::FaceIterator FIt;
        std::vector<FIt> bid_to_face;
        int maxbid = 0;
        for (CIt c = g.cellbegin(); c != g.cellend(); ++c) {
            for (FIt f = c->facebegin(); f != c->faceend(); ++f) {
                int bid = f->boundaryId();
                maxbid = std::max(maxbid, bid);
            }
        }

        bid_to_face.resize(maxbid + 1);
        std::vector<int> egf_cf(mygrid_.numFaces());
        int cix=0;
        for (CIt c = g.cellbegin(); c != g.cellend(); ++c) {
            int loc_fix=0;
            for (FIt f = c->facebegin(); f != c->faceend(); ++f) {
                if (f->boundary() && b.satCond(*f).isPeriodic()) {
                    bid_to_face[f->boundaryId()] = f;
                }
                int egf=f->index();
                int cf=mygrid_.cellFace(cix,loc_fix);
                egf_cf[egf]=cf;
                loc_fix+=1;
            }
            cix+=1;
        }

#ifndef NDEBUG
        const UnstructuredGrid& c_grid=*mygrid_.c_grid();
#endif
        int hf_ind=0;
        int bf_ind=0;
        periodic_cells_.resize(0);
        periodic_faces_.resize(0);
        periodic_hfaces_.resize(0);
        periodic_nbfaces_.resize(0);
        //cell1 = cell0;
        direclet_cells_.resize(0);
        direclet_sat_.resize(0);
        direclet_sat_.resize(0);
        direclet_hfaces_.resize(0);

        assert(periodic_cells_.size()==0);
        for (CIt c = g.cellbegin(); c != g.cellend(); ++c) {
            int cell0 = c->index();
            for (FIt f = c->facebegin(); f != c->faceend(); ++f) {
                // Neighbour face, will be changed if on a periodic boundary.
                // Compute cell[1], cell_sat[1]
                FIt nbface = f;
                if (f->boundary()) {
                    bf_ind+=1;
                    if (b.satCond(*f).isPeriodic()) {
                        nbface = bid_to_face[b.getPeriodicPartner(f->boundaryId())];
                        assert(nbface != f);
                        int cell1 = nbface->cellIndex();
                        assert(cell0 != cell1);

                        int f_ind=f->index();

                        int fn_ind=nbface->index();
                        // mapping face indices
                        f_ind=egf_cf[f_ind];
                        fn_ind=egf_cf[fn_ind];
                        assert((c_grid.face_cells[2*f_ind]==-1) || (c_grid.face_cells[2*f_ind+1]==-1));
                        assert((c_grid.face_cells[2*fn_ind]==-1) || (c_grid.face_cells[2*fn_ind+1]==-1));
                        assert((c_grid.face_cells[2*f_ind]==cell0) || (c_grid.face_cells[2*f_ind+1]==cell0));
                        assert((c_grid.face_cells[2*fn_ind]==cell1) || (c_grid.face_cells[2*fn_ind+1]==cell1));
                        periodic_cells_.push_back(cell0);
                        periodic_cells_.push_back(cell1);
                        periodic_faces_.push_back(f_ind);
                        periodic_hfaces_.push_back(hf_ind);
                        periodic_nbfaces_.push_back(fn_ind);
                    } else if (!( b.flowCond(*f).isNeumann() && b.flowCond(*f).outflux() == 0.0)) {
                        //cell1 = cell0;
                        direclet_cells_.push_back(cell0);
                        direclet_sat_.push_back(b.satCond(*f).saturation());
                        direclet_sat_.push_back(1-b.satCond(*f).saturation());//only work for 2 phases
                        direclet_hfaces_.push_back(hf_ind);
                    }
                }
                hf_ind+=1;
            }
        }

        mygrid_.makeQPeriodic(periodic_hfaces_,periodic_cells_);
        // use fractional flow instead of saturation as src
        TwophaseFluid myfluid(myrp_);
        int num_b=direclet_cells_.size();
        for(int i=0; i <num_b; ++i){
            std::array<double,2> sat = {{direclet_sat_[2*i] ,direclet_sat_[2*i+1] }};
            std::array<double,2> mob;
            std::array<double,2*2> dmob;
            myfluid.mobility(direclet_cells_[i], sat, mob, dmob);
            double fl = mob[0]/(mob[0]+mob[1]);
            direclet_sat_[2*i] = fl;
            direclet_sat_[2*i+1] = 1-fl;
        }
    }
            void operator()(const CIt& c) const
            {
                // This is constant for the whole run.
                const double delta_rho = s.preservoir_properties_->densityDifference();
                int cell[2];
                double cell_sat[2];
                cell[0] = c->index();
                cell_sat[0] = saturation[cell[0]];

                // Loop over all cell faces.
                for (FIt f = c->facebegin(); f != c->faceend(); ++f) {
                    // Neighbour face, will be changed if on a periodic boundary.
                    FIt nbface = f;
                    double dS = 0.0;
                    // Compute cell[1], cell_sat[1]
                    if (f->boundary()) {
                        if (s.pboundary_->satCond(*f).isPeriodic()) {
                            nbface = s.bid_to_face_[s.pboundary_->getPeriodicPartner(f->boundaryId())];
                            assert(nbface != f);
                            cell[1] = nbface->cellIndex();
                            assert(cell[0] != cell[1]);
                            // Periodic faces will be visited twice, but only once
                            // should they contribute. We make sure that we skip the
                            // periodic faces half the time.
                            if (cell[0] > cell[1]) {
                                // We skip this face.
                                continue;
                            }
                            cell_sat[1] = saturation[cell[1]];
                        } else {
                            assert(s.pboundary_->satCond(*f).isDirichlet());
                            cell[1] = cell[0];
                            cell_sat[1] = s.pboundary_->satCond(*f).saturation();
                        }
                    } else {
                        cell[1] = f->neighbourCellIndex();
                        assert(cell[0] != cell[1]);
                        if (cell[0] > cell[1]) {
                            // We skip this face.
                            continue;
                        }
                        cell_sat[1] = saturation[cell[1]];
                    }

                    // Get some local properties.
                    const double loc_area = f->area();
                    const double loc_flux = pressure_sol.outflux(f);
                    const Vector loc_normal = f->normal();

                    // We will now try to establish the upstream directions for each
                    // phase. They may be the same, or different (due to gravity).
                    // Recall the equation for v_w (water phase velocity):
                    //   v_w  = lambda_w * (lambda_o + lambda_w)^{-1}
                    //          * (v + lambda_o * K * grad p_{cow} + lambda_o * K * (rho_w - rho_o) * g)
                    //             ^   ^^^^^^^^^^^^^^^^^^^^^^^^^^^   ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
                    //     viscous term       capillary term                    gravity term
                    //
                    // For the purpose of upstream weighting, we only consider the viscous and gravity terms.
                    // The question is, in which direction does v_w and v_o point? That is, what is the sign
                    // of v_w*loc_normal and v_o*loc_normal?
                    //
                    // For the case when the mobilities are scalar, the following analysis applies:
                    // The viscous contribution to v_w is loc_area*loc_normal*f_w*v == f_w*loc_flux.
                    // Then the phase fluxes become
                    //     flux_w = f_w*(loc_flux + loc_area*loc_normal*lambda_o*K*(rho_w - rho_o)*g)
                    //                              ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
                    //                                           := lambda_o*G (only scalar case)
                    //     flux_o = f_o*(loc_flux - lambda_w*G)
                    // In the above, we must decide where to evaluate K, and for this purpose (deciding
                    // upstream directions) we use a K averaged between the two cells.
                    // Since all mobilities and fractional flow functions are positive, the sign
                    // of one of these cases is trivial. If G >= 0, flux_w is in the same direction as
                    // loc_flux, if G <= 0, flux_o is in the same direction as loc_flux.
                    // The phase k for which flux_k and loc_flux are of same sign, is called the trivial
                    // phase in the code below.
                    //
                    // Assuming for the moment that G >=0, we know the direction of the water flux
                    // (same as loc_flux) and evaluate lambda_w in the upstream cell. Then we may use
                    // that lambda_w to evaluate flux_o using the above formula. Knowing flux_o, we know
                    // the direction of the oil flux, and can evaluate lambda_o in the corresponding
                    // upstream cell. Finally, we can use the equation for flux_w to compute that flux.
                    // The opposite case is similar.
                    //
                    // What about tensorial mobilities? In the following code, we make the assumption
                    // that the directions of vectors are not so changed by the multiplication with
                    // mobility tensors that upstream directions change. In other words, we let all
                    // the upstream logic stand as it is. This assumption may need to be revisited.
                    // A worse problem is that
                    // 1) we do not have v, just loc_area*loc_normal*v,
                    // 2) we cannot define G, since the lambdas do not commute with the dot product.

                    typedef typename UpstreamSolver::RP::Mobility Mob;
                    using Opm::utils::arithmeticAverage;
                    // Doing arithmetic averages. Should we consider harmonic or geometric instead?
                    const MutablePermTensor aver_perm
                        = arithAver(s.preservoir_properties_->permeability(cell[0]),
                                    s.preservoir_properties_->permeability(cell[1]));
                    // Computing the raw gravity influence vector = (rho_w - rho_o)Kg
                    Vector grav_influence = prod(aver_perm, gravity);
                    grav_influence *= delta_rho;
                    // Computing G. Note that we do not multiply with the mobility,
                    // so this G is wrong in case of anisotropic relperm.
                    const double G = s.method_gravity_ ?
                        loc_area*inner(loc_normal, grav_influence) 
                        : 0.0;
                    const int triv_phase = G >= 0.0 ? 0 : 1;
                    const int ups_cell = loc_flux >= 0.0 ? 0 : 1;
                    // Compute mobility of the trivial phase.
                    Mob m_ups[2];
                    s.preservoir_properties_->phaseMobility(triv_phase, cell[ups_cell],
                                                          cell_sat[ups_cell], m_ups[triv_phase].mob);
                    // Compute gravity flow of the nontrivial phase.
                    double sign_G[2] = { -1.0, 1.0 };
                    double grav_flux_nontriv = sign_G[triv_phase]*loc_area
                        *inner(loc_normal, m_ups[triv_phase].multiply(grav_influence));
                    // Find flow direction of nontrivial phase.
                    const int ups_cell_nontriv = (loc_flux + grav_flux_nontriv >= 0.0) ? 0 : 1;
                    const int nontriv_phase = (triv_phase + 1) % 2;
                    s.preservoir_properties_->phaseMobility(nontriv_phase, cell[ups_cell_nontriv],
                                                          cell_sat[ups_cell_nontriv], m_ups[nontriv_phase].mob);
                    // Now we have the upstream phase mobilities in m_ups[].
                    Mob m_tot;
                    m_tot.setToSum(m_ups[0], m_ups[1]);
                    Mob m_totinv;
                    m_totinv.setToInverse(m_tot);


                    const double aver_sat
                        = Opm::utils::arithmeticAverage<double, double>(cell_sat[0], cell_sat[1]);

                    Mob m1c0, m1c1, m2c0, m2c1;
                    s.preservoir_properties_->phaseMobility(0, cell[0], aver_sat, m1c0.mob);
                    s.preservoir_properties_->phaseMobility(0, cell[1], aver_sat, m1c1.mob);
                    s.preservoir_properties_->phaseMobility(1, cell[0], aver_sat, m2c0.mob);
                    s.preservoir_properties_->phaseMobility(1, cell[1], aver_sat, m2c1.mob);
                    Mob m_aver[2];
                    m_aver[0].setToAverage(m1c0, m1c1);
                    m_aver[1].setToAverage(m2c0, m2c1);
                    Mob m_aver_tot;
                    m_aver_tot.setToSum(m_aver[0], m_aver[1]);
                    Mob m_aver_totinv;
                    m_aver_totinv.setToInverse(m_aver_tot);

                    // Viscous (pressure driven) term.
                    if (s.method_viscous_) {
                        // v is not correct for anisotropic relperm.
                        Vector v(loc_normal);
                        v *= loc_flux;
                        const double visc_change = inner(loc_normal, m_ups[0].multiply(m_totinv.multiply(v)));
                        // 		    const double visc_change = (m_ups[0].mob/(m_ups[1].mob + m_ups[0].mob))*loc_flux;
                        // 		    std::cout << "New: " << visc_change_2 << "   old: " << visc_change << '\n';
                        dS += visc_change;
                    }

                    // Gravity term.
                    if (s.method_gravity_) {
                        if (cell[0] != cell[1]) {
                            // We only add gravity flux on internal or periodic faces.
                            const double grav_change = loc_area
                                *inner(loc_normal, m_ups[0].multiply(m_totinv.multiply(m_ups[1].multiply(grav_influence))));
                            // const double grav_change = (lambda_one*lambda_two/(lambda_two+lambda_one))*G;
                            // const double grav_change = (lambda_one*lambda_two/(lambda_two+lambda_one))*loc_gravity_flux;
                            dS += grav_change;
                        }
                    }

                    // Capillary term.
                    if (s.method_capillary_) {
                        // J(s_w) = \frac{p_c(s_w)\sqrt{k/\phi}}{\sigma \cos\theta}
                        // p_c = \frac{J \sigma \cos\theta}{\sqrt{k/\phi}}
                        Vector cap_influence = prod(aver_perm, s.estimateCapPressureGradient(f, nbface, saturation));
                        const double cap_change = loc_area
			    *inner(loc_normal, m_aver[0].multiply(m_aver_totinv.multiply(m_aver[1].multiply(cap_influence))));
                        // 		    const double cap_vel = inner(loc_normal, prod(aver_perm, estimateCapPressureGradient(f, nbface, saturation)));
                        // 		    const double loc_cap_flux = cap_vel*loc_area;
                        // //   		    const double cap_change = loc_cap_flux*(m_aver[1].mob*m_aver[0].mob
                        // //   							    /(m_aver[0].mob + m_aver[1].mob));
                        //  		    const double cap_change = loc_cap_flux*(aver_lambda_two*aver_lambda_one
                        //  							    /(aver_lambda_one + aver_lambda_two));
                        dS += cap_change;
                    }

                    // Modify saturation.
                    if (cell[0] != cell[1]){
                        residual[cell[0]] -= dS;
                        residual[cell[1]] += dS;
                    } else {
                        assert(cell[0] == cell[1]);
                        residual[cell[0]] -= dS;
                    }
                }
                // Source term.
                double rate = s.pinjection_rates_->element(cell[0]);
                if (rate < 0.0) {
                    // For anisotropic relperm, fractionalFlow does not really make sense
                    // as a scalar
                    rate *= s.preservoir_properties_->fractionalFlow(cell[0], cell_sat[0]);
                }
                residual[cell[0]] += rate;
            }