vector_poisson_boundary2.C

/*
 *  Method for vector Poisson equation inverting eqs. for V^r and eta as a block.
 *
 *    (see file vector.h for documentation).
 *
 */

/*
 *   Copyright (c) 2005  Jerome Novak
 *
 *   This file is part of LORENE.
 *
 *   LORENE is free software; you can redistribute it and/or modify
 *   it under the terms of the GNU General Public License version 2
 *   as published by the Free Software Foundation.
 *
 *   LORENE is distributed in the hope that it will be useful,
 *   but WITHOUT ANY WARRANTY; without even the implied warranty of
 *   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 *   GNU General Public License for more details.
 *
 *   You should have received a copy of the GNU General Public License
 *   along with LORENE; if not, write to the Free Software
 *   Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
 *
 */

char vector_poisson_boundary2_C[] = "$Header: /cvsroot/Lorene/C++/Source/Tensor/vector_poisson_boundary2.C,v 1.7 2008/08/20 15:07:36 n_vasset Exp $" ;

/*
 * $Id: vector_poisson_boundary2.C,v 1.7 2008/08/20 15:07:36 n_vasset Exp $
 * $Log: vector_poisson_boundary2.C,v $
 * Revision 1.7  2008/08/20 15:07:36  n_vasset
 * Cleaning up the code...
 *
 * Revision 1.5  2007/09/05 12:35:18  j_novak
 * Homogeneous solutions are no longer obtained through the analytic formula, but
 * by solving (again) the operator system with almost zero as r.h.s. This is to
 * lower the condition number of the matching system.
 *
 * Revision 1.4  2007/01/23 17:08:46  j_novak
 * New function pois_vect_r0.C to solve the l=0 part of the vector Poisson
 * equation, which involves only the r-component.
 *
 * Revision 1.3  2005/12/30 13:39:38  j_novak
 * Changed the Galerkin base in the nucleus to (hopefully) stabilise the solver
 * when used in an iteration. Similar changes in the CED too.
 *
 * Revision 1.2  2005/02/15 15:43:18  j_novak
 * First version of the block inversion for the vector Poisson equation (method 6).
 *
 *
 * $Header: /cvsroot/Lorene/C++/Source/Tensor/vector_poisson_boundary2.C,v 1.7 2008/08/20 15:07:36 n_vasset Exp $
 *
 */

// C headers
#include <assert.h>
#include <stdlib.h>
#include <math.h>

// Lorene headers
#include "metric.h"
#include "diff.h"
#include "param_elliptic.h"
#include "proto.h"
#include "graphique.h"
#include "map.h"
#include "utilitaires.h"



void Vector::poisson_boundary2(double lam, Vector& resu, Scalar boundvr, Scalar boundeta, Scalar boundmu, double dir_vr, double neum_vr, double dir_eta, double neum_eta, double dir_mu, double neum_mu) const {

    const Map_af* mpaff = dynamic_cast<const Map_af*>(mp) ;
#ifndef NDEBUG 
    for (int i=0; i<3; i++)
    assert(cmp[i]->check_dzpuis(4)) ;
    // All this has a meaning only for spherical components:
    const Base_vect_spher* bvs = dynamic_cast<const Base_vect_spher*>(triad) ;
    assert(bvs != 0x0) ; 
    //## ... and affine mapping, for the moment!
    assert( mpaff != 0x0) ;
#endif

     if (fabs(lam + 1.) < 1.e-8) {

       cout <<" lambda = -1 is not supported by this code!" << endl;
       abort(); 
     }



     // Extraction of Mtbl_cf objects from boundary informations; 
     Scalar bound_vr2 = boundvr; 
     bound_vr2.set_spectral_va().ylm(); 
     Scalar bound_eta2 = boundeta; 
     bound_eta2.set_spectral_va().ylm();
     Scalar bound_mu2 = boundmu; bound_mu2.set_spectral_va().ylm();
     
     const  Mtbl_cf *bound_vr = bound_vr2.get_spectral_va().c_cf; 
     const  Mtbl_cf *bound_mu = bound_mu2.get_spectral_va().c_cf;
     const Mtbl_cf *bound_eta = bound_eta2.get_spectral_va().c_cf;




    // Some working objects
    //---------------------
  const Mg3d& mg = *mpaff->get_mg() ;
  int nz = mg.get_nzone() ; int nzm1 = nz - 1;
  assert(mg.get_type_r(0) == RARE) ;
  assert(mg.get_type_r(nzm1) == UNSURR) ;
  Scalar S_r = *cmp[0] ;
  Scalar S_eta = eta() ;
  Scalar het(*mpaff) ; 
  Scalar vr(*mpaff) ; 
 
  if (S_r.get_etat() == ETATZERO) {
      if (S_eta.get_etat() == ETATZERO) {

      S_r.annule_hard() ;
      S_r.set_spectral_base(S_eta.get_spectral_base()) ;
      S_eta.annule_hard() ;
      S_eta.set_spectral_base(S_r.get_spectral_base()) ;
      }
      else {
      S_r.annule_hard() ;
      S_r.set_spectral_base(S_eta.get_spectral_base()) ;
      }
  }
  if (S_eta.get_etat() == ETATZERO) {
      S_eta.annule_hard() ;
      S_eta.set_spectral_base(S_r.get_spectral_base()) ;
  }

  S_r.set_spectral_va().ylm() ;
  S_eta.set_spectral_va().ylm() ;
  const Base_val& base = S_eta.get_spectral_va().base ;
  Mtbl_cf sol_part_eta(mg, base) ; sol_part_eta.annule_hard() ;
  Mtbl_cf sol_part_vr(mg, base) ; sol_part_vr.annule_hard() ;
  Mtbl_cf sol_hom_un_eta(mg, base) ; sol_hom_un_eta.annule_hard() ;
  Mtbl_cf sol_hom_deux_eta(mg, base) ; sol_hom_deux_eta.annule_hard() ;
  Mtbl_cf sol_hom_trois_eta(mg, base) ; sol_hom_trois_eta.annule_hard() ;
  Mtbl_cf sol_hom_quatre_eta(mg, base) ; sol_hom_quatre_eta.annule_hard() ;
  Mtbl_cf sol_hom_un_vr(mg, base) ; sol_hom_un_vr.annule_hard() ;
  Mtbl_cf sol_hom_deux_vr(mg, base) ; sol_hom_deux_vr.annule_hard() ;
  Mtbl_cf sol_hom_trois_vr(mg, base) ; sol_hom_trois_vr.annule_hard() ;
  Mtbl_cf sol_hom_quatre_vr(mg, base) ; sol_hom_quatre_vr.annule_hard() ;
 
  // Solution of the l=0 part for Vr
  //---------------------------------
  Scalar sou_l0 = (*cmp[0]) / (1. + lam) ;
  sou_l0.set_spectral_va().ylm() ;

 Param_elliptic param_l0(sou_l0) ;
  for (int l=0; l<nz; l++){
      param_l0.set_poisson_vect_r(l, true) ;
  }


   Scalar vrl0 = sou_l0.sol_elliptic_boundary(param_l0, *bound_vr, dir_vr, neum_vr) ;


  // Build-up & inversion of the system for (eta, V^r) in each domain (except for the nucleus!)
  //-----------------------------------------------------------------


  // Shells
  //-------

  int nr = mg.get_nr(1) ;
  int nt = mg.get_nt(1) ;
  int np = mg.get_np(1) ;
  double alpha = mpaff->get_alpha()[1] ; double alp2 = alpha*alpha ;
  double beta = mpaff->get_beta()[1] ;
 
  int l_q = 0 ; int m_q = 0; int base_r = 0 ;
  for (int zone=1 ; zone<nzm1 ; zone++) {
      nr = mg.get_nr(zone) ; 
      assert (nr > 5) ;
      assert(nt == mg.get_nt(zone)) ;
      assert(np == mg.get_np(zone)) ;
      alpha = mpaff->get_alpha()[zone] ;
      beta = mpaff->get_beta()[zone] ;
      double ech = beta / alpha ;

  // Loop on l and m
  //----------------
      for (int k=0 ; k<np+1 ; k++) {
      for (int j=0 ; j<nt ; j++) {
      base.give_quant_numbers(zone, k, j, m_q, l_q, base_r) ;
      if ( (nullite_plm(j, nt, k, np, base) == 1) && (l_q != 0) ) {
          int dege1 = 2 ; //degeneracy of eq.1
          int dege2 = 2 ;  //degeneracy of eq.2
          int nr_eq1 = nr - dege1 ; //Eq.1 is for eta
          int nr_eq2 = nr - dege2 ; //Eq.2 is for V^r
          int nrtot = 2*nr ;
          Matrice oper(nrtot, nrtot) ; oper.set_etat_qcq() ;
          Tbl sec_membre(nrtot) ; sec_membre.set_etat_qcq() ;
          Diff_x2dsdx2 x2d2(base_r, nr); const Matrice& m2d2 = x2d2.get_matrice() ;
          Diff_xdsdx2 xd2(base_r, nr) ; const Matrice& mxd2 = xd2.get_matrice() ;
          Diff_dsdx2 d2(base_r, nr) ; const Matrice& md2 = d2.get_matrice() ;
          Diff_xdsdx xd(base_r, nr) ; const Matrice& mxd = xd.get_matrice() ;
          Diff_dsdx d1(base_r, nr) ; const Matrice& md = d1.get_matrice() ;
          Diff_id id(base_r, nr) ; const Matrice& mid = id.get_matrice() ;

          // Building the operator
          //----------------------
          for (int lin=0; lin<nr_eq1; lin++) { 
          for (int col=0; col<nr; col++) 
              oper.set(lin,col) 
              = m2d2(lin,col) + 2*ech*mxd2(lin,col) + ech*ech*md2(lin,col) 
              + 2*(mxd(lin,col) + ech*md(lin,col)) 
              - (lam+1)*l_q*(l_q+1)*mid(lin,col) ;
          for (int col=0; col<nr; col++) 
              oper.set(lin,col+nr) 
              = lam*(mxd(lin,col) + ech*md(lin,col)) 
              + 2*(1+lam)*mid(lin,col) ; 
          }
          oper.set(nr_eq1, 0) = 1 ;
          for (int col=1; col<2*nr; col++) 
          oper.set(nr_eq1, col) = 0 ;
          oper.set(nr_eq1+1, 0) = 0 ;
          oper.set(nr_eq1+1, 1) = 1 ;
          for (int col=2; col<2*nr; col++) 
          oper.set(nr_eq1+1, col) = 0 ;
          for (int lin=0; lin<nr_eq2; lin++) {
          for (int col=0; col<nr; col++)
              oper.set(lin+nr,col) 
              = -l_q*(l_q+1)*(lam*(mxd(lin,col) + ech*md(lin,col))
                      - (lam+2)*mid(lin,col)) ;
          for (int col=0; col<nr; col++)
              oper.set(lin+nr, col+nr) 
              = (lam+1)*(m2d2(lin,col) + 2*ech*mxd2(lin,col) 
                     + ech*ech*md2(lin,col) 
                     + 2*(mxd(lin,col) + ech*md(lin,col)))
              -(2*(lam+1)+l_q*(l_q+1))*mid(lin,col) ;         
          }
          for (int col=0; col<nr; col++) 
          oper.set(nr+nr_eq2, col) = 0 ;
          oper.set(nr+nr_eq2, nr) = 1 ;
          for (int col=nr+1; col<2*nr; col++) 
          oper.set(nr+nr_eq2, col) = 0 ;
          for (int col=0; col<nr+1; col++) 
          oper.set(nr+nr_eq2+1, col) = 0 ;
          oper.set(nr+nr_eq2+1, nr+1) = 1 ;
          for (int col=nr+2; col<2*nr; col++) 
          oper.set(nr+nr_eq2+1, col) = 0 ;

          oper.set_lu() ;
          
          // Filling the r.h.s
          //------------------
          Tbl sr(nr) ; sr.set_etat_qcq() ; 
          Tbl seta(nr) ; seta.set_etat_qcq() ;
          for (int i=0; i<nr; i++) {
          sr.set(i) = (*S_r.get_spectral_va().c_cf)(zone, k, j, i);
          seta.set(i) = (*S_eta.get_spectral_va().c_cf)(zone, k, j, i) ;
          }
          Tbl xsr= sr ;  Tbl x2sr= sr ;
          Tbl xseta= seta ; Tbl x2seta = seta ;
          multx2_1d(nr, &x2sr.t, base_r) ; multx_1d(nr, &xsr.t, base_r) ;
          multx2_1d(nr, &x2seta.t, base_r) ; multx_1d(nr, &xseta.t, base_r) ;
          
          for (int i=0; i<nr_eq1; i++) 
          sec_membre.set(i) = alpha*(alpha*x2seta(i) + 2*beta*xseta(i))
              + beta*beta*seta(i);
          sec_membre.set(nr_eq1) = 0 ;
          sec_membre.set(nr_eq1+1) = 0 ;
          for (int i=0; i<nr_eq2; i++)
          sec_membre.set(i+nr) = beta*beta*sr(i) 
              + alpha*(alpha*x2sr(i) + 2*beta*xsr(i)) ;
          sec_membre.set(nr+nr_eq2) = 0 ;
          sec_membre.set(nr+nr_eq2+1) = 0 ;

          // Inversion of the "big" operator
          //--------------------------------
          Tbl big_res = oper.inverse(sec_membre) ;
          for (int i=0; i<nr; i++) {
          sol_part_eta.set(zone, k, j, i) = big_res(i) ;
          sol_part_vr.set(zone, k, j, i) = big_res(nr+i) ;        
          }
          
          // Getting the homogeneous solutions
          //----------------------------------
          sec_membre.annule_hard() ;
          sec_membre.set(nr_eq1) = 1 ;
          big_res = oper.inverse(sec_membre) ;
          for (int i=0 ; i<nr ; i++) {
          sol_hom_un_eta.set(zone, k, j, i) = big_res(i) ;
          sol_hom_un_vr.set(zone, k, j, i) = big_res(nr+i) ;
          }
          sec_membre.set(nr_eq1) = 0 ;
          sec_membre.set(nr_eq1+1) = 1 ;
          big_res = oper.inverse(sec_membre) ;
          for (int i=0 ; i<nr ; i++) {
          sol_hom_deux_eta.set(zone, k, j, i) = big_res(i) ;
          sol_hom_deux_vr.set(zone, k, j, i) = big_res(nr+i) ;
          }
          sec_membre.set(nr_eq1+1) = 0 ;
          sec_membre.set(nr+nr_eq2) = 1 ;
          big_res = oper.inverse(sec_membre) ;
          for (int i=0 ; i<nr ; i++) {
          sol_hom_trois_eta.set(zone, k, j, i) = big_res(i) ;
          sol_hom_trois_vr.set(zone, k, j, i) = big_res(nr+i) ;
          }
          sec_membre.set(nr+nr_eq2) = 0 ;
          sec_membre.set(nr+nr_eq2+1) = 1 ;
          big_res = oper.inverse(sec_membre) ;
          for (int i=0 ; i<nr ; i++) {
          sol_hom_quatre_eta.set(zone, k, j, i) = big_res(i) ;
          sol_hom_quatre_vr.set(zone, k, j, i) = big_res(nr+i) ;
          }

      } 
      }
      }
  }

  // Compactified external domain
  //-----------------------------
  nr = mg.get_nr(nzm1) ;
  assert(nt == mg.get_nt(nzm1)) ;
  assert(np == mg.get_np(nzm1)) ;
  alpha = mpaff->get_alpha()[nzm1] ; alp2 = alpha*alpha ;
  assert (nr > 4) ;
  
  // Loop on l and m
  //----------------
  for (int k=0 ; k<np+1 ; k++) {
      for (int j=0 ; j<nt ; j++) { 
      base.give_quant_numbers(nzm1, k, j, m_q, l_q, base_r) ;
      if ( (nullite_plm(j, nt, k, np, base) == 1) && (l_q != 0) ) {
          int dege1 = 3; //degeneracy of eq.1
          int dege2 = (l_q == 1 ? 2 : 3); //degeneracy of eq.2
          int nr_eq1 = nr - dege1 ; //Eq.1 is for eta
          int nr_eq2 = nr - dege2 ; //Eq.2 is the div-free condition
          int nrtot = 2*nr ;
          Matrice oper(nrtot, nrtot) ; oper.set_etat_qcq() ;
          Tbl sec_membre(nrtot) ; sec_membre.set_etat_qcq() ;
          Diff_dsdx2 d2(base_r, nr) ; const Matrice& md2 = d2.get_matrice() ;
          Diff_sxdsdx sxd(base_r, nr) ; const Matrice& mxd = sxd.get_matrice() ;
          Diff_sx2 sx2(base_r, nr) ; const Matrice& ms2 = sx2.get_matrice() ;

          // Building the operator
          //----------------------
          for (int lin=0; lin<nr_eq1; lin++) { 
          for (int col=0; col<nr; col++) 
              oper.set(lin,col) 
              = (md2(lin,col) - (lam+1)*l_q*(l_q+1)*ms2(lin,col))/alp2 ;
              for (int col=0; col<nr; col++) 
                  oper.set(lin,col+nr) = 
                  (-lam*mxd(lin,col) + 2*(1+lam)*ms2(lin,col)) / alp2 ;
          }
          for (int col=0; col<nr; col++)
          oper.set(nr_eq1, col) = 1 ;
          for (int col=nr; col<nrtot; col++)
          oper.set(nr_eq1, col) = 0 ;
          int pari = -1 ;
          for (int col=0; col<nr; col++) {
          oper.set(nr_eq1+1, col) = pari*col*col ;
          pari = pari ;
          }
          for (int col=nr; col<nrtot; col++)
          oper.set(nr_eq1+1, col) = 0 ;
          oper.set(nr_eq1+2, 0) = 1 ;
          for (int col=1; col<nrtot; col++)
          oper.set(nr_eq1+2, col) = 0 ;
          for (int lin=0; lin<nr_eq2; lin++) {
          for (int col=0; col<nr; col++)
              oper.set(lin+nr,col) = (l_q*(l_q+1)*(lam*mxd(lin,col) 
                   + (lam+2)*ms2(lin,col))) / alp2 ;
          for (int col=0; col<nr; col++)
              oper.set(lin+nr, col+nr) = ((lam+1)*md2(lin,col) 
                 - (2*(lam+1) + l_q*(l_q+1))*ms2(lin,col)) / alp2 ;
          }
          for (int col=0; col<nr; col++)
          oper.set(nr+nr_eq2, col) = 0 ;
          for (int col=nr; col<nrtot; col++)
          oper.set(nr+nr_eq2, col) = 1 ;
          for (int col=0; col<nr; col++)
          oper.set(nr+nr_eq2+1, col) = 0 ;
          pari = -1 ;
          for (int col=0; col<nr; col++) {
          oper.set(nr+nr_eq2+1, nr+col) = pari*col*col ;
          pari = pari ;
          }
          if (l_q > 1) {
          for (int col=0; col<nr; col++)
              oper.set(nr+nr_eq2+2, col) = 0 ;
          oper.set(nr+nr_eq2+2, nr) = 1 ;
          for (int col=nr+1; col<nrtot; col++)
              oper.set(nr+nr_eq2+2, col) = 0 ;
          }
          oper.set_lu() ;

          // Filling the r.h.s
          //------------------
          for (int i=0; i<nr_eq1; i++) 
          sec_membre.set(i) = (*S_eta.get_spectral_va().c_cf)(nzm1, k, j, i) ;
          for (int i=nr_eq1; i<nr; i++)
          sec_membre.set(i) = 0 ;
          for (int i=0; i<nr_eq2; i++)
          sec_membre.set(i+nr) =(*S_r.get_spectral_va().c_cf)(nzm1, k, j, i);
          for (int i=nr_eq2; i<nr; i++)
          sec_membre.set(nr+i) = 0 ;
          Tbl big_res = oper.inverse(sec_membre) ;
          
          // Putting coefficients of h and v to individual arrays
          //-----------------------------------------------------
          for (int i=0; i<nr; i++) {
          sol_part_eta.set(nzm1, k, j, i) = big_res(i) ;
          sol_part_vr.set(nzm1, k, j, i) = big_res(i+nr) ;
          }

          // Homogeneous solutions (only 1/r^(l+2) and 1/r^l in the CED)
          //------------------------------------------------------------
          if (l_q == 1) {         
          Tbl sol_hom1 = solh(nr, 0, 0., base_r) ;
          Tbl sol_hom2 = solh(nr, 2, 0., base_r) ;
          double fac_eta = lam + 2. ;
          double fac_vr = 2*lam + 2. ;
          for (int i=0 ; i<nr ; i++) {
              sol_hom_deux_eta.set(nzm1, k, j, i) = sol_hom2(i) ;
              sol_hom_quatre_eta.set(nzm1, k, j, i) = fac_eta*sol_hom1(i) ;
              sol_hom_deux_vr.set(nzm1, k, j, i) = -2*sol_hom2(i) ;
              sol_hom_quatre_vr.set(nzm1, k, j, i) = fac_vr*sol_hom1(i) ;
          }
          }
          else {
          sec_membre.annule_hard() ;
          sec_membre.set(nr-1) = 1 ;
          big_res = oper.inverse(sec_membre) ;

          for (int i=0; i<nr; i++) {
              sol_hom_deux_eta.set(nzm1, k, j, i) = big_res(i) ;
              sol_hom_deux_vr.set(nzm1, k, j, i) = big_res(nr+i) ;
          }
          sec_membre.set(nr-1) = 0 ;
          sec_membre.set(2*nr-1) = 1 ;
          big_res = oper.inverse(sec_membre) ;
          for (int i=0; i<nr; i++) {
              sol_hom_quatre_eta.set(nzm1, k, j, i) = big_res(i) ;
              sol_hom_quatre_vr.set(nzm1, k, j, i) = big_res(nr+i) ;
          }
          }
      }
      }
  }

  // Now let's match everything ...
  //-------------------------------

  // Resulting V^r & eta
  vr.set_etat_qcq() ;
  vr.set_spectral_base(base) ;
  vr.set_spectral_va().set_etat_cf_qcq() ;
  Mtbl_cf& cf_vr = *vr.set_spectral_va().c_cf ;
  cf_vr.annule_hard() ;
  het.set_etat_qcq() ;
  het.set_spectral_base(base) ;
  het.set_spectral_va().set_etat_cf_qcq() ;
  Mtbl_cf& cf_eta = *het.set_spectral_va().c_cf ;
  cf_eta.annule_hard() ;
  int taille = 4*nzm1 -2 ; // Two less than R3 case 
  Tbl sec_membre(taille) ; 
  Matrice systeme(taille, taille) ; 
  systeme.set_etat_qcq() ;
  int ligne ;  int colonne ;
  
  // Loop on l and m
  //----------------
  for (int k=0 ; k<np+1 ; k++)
      for (int j=0 ; j<nt ; j++) {
      base.give_quant_numbers(0, k, j, m_q, l_q, base_r) ;
      if ((nullite_plm(j, nt, k, np, base) == 1)&&(l_q != 0)) {
        
          ligne = 0 ;
          colonne = 0 ;
          systeme.annule_hard() ;
          sec_membre.annule_hard() ;


              //shell 1 (boundary condition)
          int zone1 = 1;
          nr = mg.get_nr(zone1) ;
          alpha = mpaff->get_alpha()[zone1] ;

          // Here we prepare for a Robyn-type boundary condition on eta. 
          // Parameters are dir_eta and neum_eta
          systeme.set(ligne, colonne) 
              = -dir_eta*sol_hom_un_eta.val_in_bound_jk(zone1, j, k)  ;
          systeme.set(ligne, colonne+1) 
              = -dir_eta * sol_hom_deux_eta.val_in_bound_jk(zone1, j, k)  ;
          systeme.set(ligne, colonne+2) 
              = -dir_eta*sol_hom_trois_eta.val_in_bound_jk(zone1, j, k)  ;
          systeme.set(ligne, colonne+3) 
              = -dir_eta*sol_hom_quatre_eta.val_in_bound_jk(zone1, j, k)  ;
 
 


          sec_membre.set(ligne) += dir_eta* sol_part_eta.val_in_bound_jk(zone1, j, k) ;



          int pari = -1 ;
          for (int i=0; i<nr; i++) {
              systeme.set(ligne, colonne) 
              -= neum_eta*pari*i*i*sol_hom_un_eta(zone1, k, j, i)/alpha ;
              systeme.set(ligne, colonne+1) 
              -= neum_eta*pari*i*i*sol_hom_deux_eta(zone1, k, j, i)/alpha ;
              systeme.set(ligne, colonne+2) 
              -= neum_eta*pari*i*i*sol_hom_trois_eta(zone1, k, j, i)/alpha ;
              systeme.set(ligne, colonne+3) 
              -= neum_eta*pari*i*i*sol_hom_quatre_eta(zone1, k, j, i)/alpha ;
              sec_membre.set(ligne) 
              += neum_eta*pari*i*i* sol_part_eta(zone1, k, j, i)/alpha ;
              pari = -pari ;
          }

          sec_membre.set(ligne) -= (*bound_eta).val_in_bound_jk(1,j,k);
          ligne++ ;
 
                 

          // ... and their couterparts for V^r
          systeme.set(ligne, colonne) 
              = - dir_vr*sol_hom_un_vr.val_in_bound_jk(zone1, j, k)  ;
          systeme.set(ligne, colonne+1) 
              = - dir_vr*sol_hom_deux_vr.val_in_bound_jk(zone1, j, k)  ;
          systeme.set(ligne, colonne+2) 
              = -dir_vr*sol_hom_trois_vr.val_in_bound_jk(zone1, j, k)  ;
          systeme.set(ligne, colonne+3) 
              = -dir_vr*sol_hom_quatre_vr.val_in_bound_jk(zone1, j, k)  ;
          sec_membre.set(ligne) += dir_vr*sol_part_vr.val_in_bound_jk(zone1, j, k) ;
    

    
    
          pari = -1 ;
          for (int i=0; i<nr; i++) {
              systeme.set(ligne, colonne) 
              -= neum_vr*pari*i*i*sol_hom_un_vr(zone1, k, j, i)/alpha ;
              systeme.set(ligne, colonne+1) 
              -= neum_vr*pari*i*i*sol_hom_deux_vr(zone1, k, j, i)/alpha ;
              systeme.set(ligne, colonne+2) 
              -= neum_vr*pari*i*i*sol_hom_trois_vr(zone1, k, j, i)/alpha ;
              systeme.set(ligne, colonne+3) 
              -= neum_vr*pari*i*i*sol_hom_quatre_vr(zone1, k, j, i)/alpha ;
              sec_membre.set(ligne) 
              += neum_vr*pari*i*i* sol_part_vr(zone1, k, j, i)/alpha ;
              pari = -pari ;
          }

                  sec_membre.set(ligne) -= (*bound_vr).val_in_bound_jk(1,j,k);
          ligne++ ;



    
          // Values in 1: beginning with solution in eta...
            
          systeme.set(ligne, colonne) 
              += sol_hom_un_eta.val_out_bound_jk(zone1, j, k)  ;
          systeme.set(ligne, colonne+1) 
              += sol_hom_deux_eta.val_out_bound_jk(zone1, j, k)  ;
          systeme.set(ligne, colonne+2) 
              += sol_hom_trois_eta.val_out_bound_jk(zone1, j, k)  ;
          systeme.set(ligne, colonne+3) 
              += sol_hom_quatre_eta.val_out_bound_jk(zone1, j, k)  ;
          sec_membre.set(ligne) -= sol_part_eta.val_out_bound_jk(zone1, j, k) ;
          ligne++ ;
          // ... and their counterparts for V^r
          systeme.set(ligne, colonne) 
              += sol_hom_un_vr.val_out_bound_jk(zone1, j, k)  ;
          systeme.set(ligne, colonne+1) 
              += sol_hom_deux_vr.val_out_bound_jk(zone1, j, k)  ;
          systeme.set(ligne, colonne+2) 
              += sol_hom_trois_vr.val_out_bound_jk(zone1, j, k)  ;
          systeme.set(ligne, colonne+3) 
              += sol_hom_quatre_vr.val_out_bound_jk(zone1, j, k)  ;
          sec_membre.set(ligne) -= sol_part_vr.val_out_bound_jk(zone1, j, k) ;
          ligne++ ;

          //derivative at 1 
          pari = 1 ;
          for (int i=0; i<nr; i++) {
              systeme.set(ligne, colonne) 
              += pari*i*i*sol_hom_un_eta(zone1, k, j, i)/alpha ;
              systeme.set(ligne, colonne+1) 
              += pari*i*i*sol_hom_deux_eta(zone1, k, j, i)/alpha ;
              systeme.set(ligne, colonne+2) 
              += pari*i*i*sol_hom_trois_eta(zone1, k, j, i)/alpha ;
              systeme.set(ligne, colonne+3) 
              += pari*i*i*sol_hom_quatre_eta(zone1, k, j, i)/alpha ;
              sec_membre.set(ligne) 
              -= pari*i*i* sol_part_eta(zone1, k, j, i)/alpha ;
              pari = pari ;
          }
          ligne++ ;
          // ... and their counterparts for V^r
          pari = 1 ;
          for (int i=0; i<nr; i++) {
              systeme.set(ligne, colonne) 
              += pari*i*i*sol_hom_un_vr(zone1, k, j, i)/alpha ;
              systeme.set(ligne, colonne+1) 
              += pari*i*i*sol_hom_deux_vr(zone1, k, j, i)/alpha ;
              systeme.set(ligne, colonne+2) 
              += pari*i*i*sol_hom_trois_vr(zone1, k, j, i)/alpha ;
              systeme.set(ligne, colonne+3) 
              += pari*i*i*sol_hom_quatre_vr(zone1, k, j, i)/alpha ;
              sec_membre.set(ligne) 
              -= pari*i*i* sol_part_vr(zone1, k, j, i)/alpha ;
              pari = pari ;
          }
          colonne += 4 ;

          // Other shells
          if ( nz <= 2){
            cout <<"WARNING!! Mapping must contain at least 2 shells!!" << endl;}
          for (int zone=2 ; zone<nzm1 ; zone++) {

      nr = mg.get_nr(zone) ;
          alpha = mpaff->get_alpha()[zone] ;
          ligne -= 3 ;
          systeme.set(ligne, colonne) 
              = -sol_hom_un_eta.val_in_bound_jk(zone, j, k)  ;
          systeme.set(ligne, colonne+1) 
              = -sol_hom_deux_eta.val_in_bound_jk(zone, j, k)  ;
          systeme.set(ligne, colonne+2) 
              = -sol_hom_trois_eta.val_in_bound_jk(zone, j, k)  ;
          systeme.set(ligne, colonne+3) 
              = -sol_hom_quatre_eta.val_in_bound_jk(zone, j, k)  ;
          sec_membre.set(ligne) += sol_part_eta.val_in_bound_jk(zone, j, k) ;
          ligne++ ;
          // ... and their counterparts for V^r
          systeme.set(ligne, colonne) 
              = -sol_hom_un_vr.val_in_bound_jk(zone, j, k)  ;
          systeme.set(ligne, colonne+1) 
              = -sol_hom_deux_vr.val_in_bound_jk(zone, j, k)  ;
          systeme.set(ligne, colonne+2) 
              = -sol_hom_trois_vr.val_in_bound_jk(zone, j, k)  ;
          systeme.set(ligne, colonne+3) 
              = -sol_hom_quatre_vr.val_in_bound_jk(zone, j, k)  ;
          sec_membre.set(ligne) += sol_part_vr.val_in_bound_jk(zone, j, k) ;
          ligne++ ;

          //derivative of (x+echelle)^(l-1) at -1 
          pari = -1 ;
          for (int i=0; i<nr; i++) {
              systeme.set(ligne, colonne) 
              -= pari*i*i*sol_hom_un_eta(zone, k, j, i)/alpha ;
              systeme.set(ligne, colonne+1) 
              -= pari*i*i*sol_hom_deux_eta(zone, k, j, i)/alpha ;
              systeme.set(ligne, colonne+2) 
              -= pari*i*i*sol_hom_trois_eta(zone, k, j, i)/alpha ;
              systeme.set(ligne, colonne+3) 
              -= pari*i*i*sol_hom_quatre_eta(zone, k, j, i)/alpha ;
              sec_membre.set(ligne) 
              += pari*i*i* sol_part_eta(zone, k, j, i)/alpha ;
              pari = -pari ;
          }
          ligne++ ;
          // ... and their counterparts for V^r
          pari = -1 ;
          for (int i=0; i<nr; i++) {
              systeme.set(ligne, colonne) 
              -= pari*i*i*sol_hom_un_vr(zone, k, j, i)/alpha ;
              systeme.set(ligne, colonne+1) 
              -= pari*i*i*sol_hom_deux_vr(zone, k, j, i)/alpha ;
              systeme.set(ligne, colonne+2) 
              -= pari*i*i*sol_hom_trois_vr(zone, k, j, i)/alpha ;
              systeme.set(ligne, colonne+3) 
              -= pari*i*i*sol_hom_quatre_vr(zone, k, j, i)/alpha ;
              sec_membre.set(ligne) 
              += pari*i*i* sol_part_vr(zone, k, j, i)/alpha ;
              pari = -pari ;
          }
          ligne++ ;
            
          systeme.set(ligne, colonne) 
              += sol_hom_un_eta.val_out_bound_jk(zone, j, k)  ;
          systeme.set(ligne, colonne+1) 
              += sol_hom_deux_eta.val_out_bound_jk(zone, j, k)  ;
          systeme.set(ligne, colonne+2) 
              += sol_hom_trois_eta.val_out_bound_jk(zone, j, k)  ;
          systeme.set(ligne, colonne+3) 
              += sol_hom_quatre_eta.val_out_bound_jk(zone, j, k)  ;
          sec_membre.set(ligne) -= sol_part_eta.val_out_bound_jk(zone, j, k) ;
          ligne++ ;
          // ... and their counterparts for V^r
          systeme.set(ligne, colonne) 
              += sol_hom_un_vr.val_out_bound_jk(zone, j, k)  ;
          systeme.set(ligne, colonne+1) 
              += sol_hom_deux_vr.val_out_bound_jk(zone, j, k)  ;
          systeme.set(ligne, colonne+2) 
              += sol_hom_trois_vr.val_out_bound_jk(zone, j, k)  ;
          systeme.set(ligne, colonne+3) 
              += sol_hom_quatre_vr.val_out_bound_jk(zone, j, k)  ;
          sec_membre.set(ligne) -= sol_part_vr.val_out_bound_jk(zone, j, k) ;
          ligne++ ;

          //derivative at 1 
          pari = 1 ;
          for (int i=0; i<nr; i++) {
              systeme.set(ligne, colonne) 
              += pari*i*i*sol_hom_un_eta(zone, k, j, i)/alpha ;
              systeme.set(ligne, colonne+1) 
              += pari*i*i*sol_hom_deux_eta(zone, k, j, i)/alpha ;
              systeme.set(ligne, colonne+2) 
              += pari*i*i*sol_hom_trois_eta(zone, k, j, i)/alpha ;
              systeme.set(ligne, colonne+3) 
              += pari*i*i*sol_hom_quatre_eta(zone, k, j, i)/alpha ;
              sec_membre.set(ligne) 
              -= pari*i*i* sol_part_eta(zone, k, j, i)/alpha ;
              pari = pari ;
          }
          ligne++ ;
          // ... and their couterparts for V^r
          pari = 1 ;
          for (int i=0; i<nr; i++) {
              systeme.set(ligne, colonne) 
              += pari*i*i*sol_hom_un_vr(zone, k, j, i)/alpha ;
              systeme.set(ligne, colonne+1) 
              += pari*i*i*sol_hom_deux_vr(zone, k, j, i)/alpha ;
              systeme.set(ligne, colonne+2) 
              += pari*i*i*sol_hom_trois_vr(zone, k, j, i)/alpha ;
              systeme.set(ligne, colonne+3) 
              += pari*i*i*sol_hom_quatre_vr(zone, k, j, i)/alpha ;
              sec_membre.set(ligne) 
              -= pari*i*i* sol_part_vr(zone, k, j, i)/alpha ;
              pari = pari ;
          }
          colonne += 4 ;
          }         
                    
          //Compactified external domain
          nr = mg.get_nr(nzm1) ;

          alpha = mpaff->get_alpha()[nzm1] ;
          ligne -= 3 ;
          //value of (x-1)^(l+2) at -1 :
          systeme.set(ligne, colonne) 
          -= sol_hom_deux_eta.val_in_bound_jk(nzm1, j, k) ;
          systeme.set(ligne, colonne+1) 
          -= sol_hom_quatre_eta.val_in_bound_jk(nzm1, j, k) ;
          sec_membre.set(ligne) += sol_part_eta.val_in_bound_jk(nzm1, j, k) ;
          //... and of its counterpart for V^r
          systeme.set(ligne+1, colonne) 
          -= sol_hom_deux_vr.val_in_bound_jk(nzm1, j, k) ;
          systeme.set(ligne+1, colonne+1) 
          -= sol_hom_quatre_vr.val_in_bound_jk(nzm1, j, k) ;
          sec_membre.set(ligne+1) += sol_part_vr.val_in_bound_jk(nzm1, j, k) ;
            
          ligne += 2 ;
          //derivative of (x-1)^(l+2) at -1 :
          pari = 1 ;
          for (int i=0; i<nr; i++) {
              systeme.set(ligne, colonne) 
              -= 4*alpha*pari*i*i*sol_hom_deux_eta(nzm1, k, j, i) ;
              systeme.set(ligne, colonne+1) 
              -= 4*alpha*pari*i*i*sol_hom_quatre_eta(nzm1, k, j, i) ;
              sec_membre.set(ligne) 
              += 4*alpha*pari*i*i* sol_part_eta(nzm1, k, j, i) ;
              pari = -pari ;
          }
          ligne++ ;
          // ... and their counterparts for V^r
          pari = 1 ;
          for (int i=0; i<nr; i++) {
              systeme.set(ligne, colonne) 
              -= 4*alpha*pari*i*i*sol_hom_deux_vr(nzm1, k, j, i) ;
              systeme.set(ligne, colonne+1) 
              -= 4*alpha*pari*i*i*sol_hom_quatre_vr(nzm1, k, j, i) ;
              sec_membre.set(ligne) 
              += 4*alpha*pari*i*i* sol_part_vr(nzm1, k, j, i) ;
              pari = -pari ;
          }
            
          // Solution of the system giving the coefficients for the homogeneous 
          // solutions
          //-------------------------------------------------------------------
          systeme.set_lu() ;
          Tbl facteurs(systeme.inverse(sec_membre)) ;
          int conte = 0 ;

          // everything is put to the right place, the same combination of hom.
          // solutions (with some l or -(l+1) factors) must be used for V^r
          //-------------------------------------------------------------------
          nr = mg.get_nr(0) ; //nucleus
          for (int i=0 ; i<nr ; i++) {
        cf_eta.set(0, k, j, i) = 0.;
        cf_vr.set(0, k, j, i) = 0.;
          }

          for (int zone=1 ; zone<nzm1 ; zone++) { //shells
          nr = mg.get_nr(zone) ;
          for (int i=0 ; i<nr ; i++) {
              cf_eta.set(zone, k, j, i) = 
              sol_part_eta(zone, k, j, i)
              +facteurs(conte)*sol_hom_un_eta(zone, k, j, i) 
              +facteurs(conte+1)*sol_hom_deux_eta(zone, k, j, i) 
              +facteurs(conte+2)*sol_hom_trois_eta(zone, k, j, i) 
              +facteurs(conte+3)*sol_hom_quatre_eta(zone, k, j, i) ;
              cf_vr.set(zone, k, j, i) = sol_part_vr(zone, k, j, i)
              +facteurs(conte)*sol_hom_un_vr(zone, k, j, i) 
              +facteurs(conte+1)*sol_hom_deux_vr(zone, k, j, i) 
              +facteurs(conte+2)*sol_hom_trois_vr(zone, k, j, i) 
              +facteurs(conte+3)*sol_hom_quatre_vr(zone, k, j, i) ;
          }
          conte+=4 ;
          }
          nr = mg.get_nr(nz-1) ; //compactified external domain
          for (int i=0 ; i<nr ; i++) {
          cf_eta.set(nzm1, k, j, i) = sol_part_eta(nzm1, k, j, i)
              +facteurs(conte)*sol_hom_deux_eta(nzm1, k, j, i) 
              +facteurs(conte+1)*sol_hom_quatre_eta(nzm1, k, j, i) ;
          cf_vr.set(nzm1, k, j, i) = sol_part_vr(nzm1, k, j, i)
              +facteurs(conte)*sol_hom_deux_vr(nzm1, k, j, i) 
              +facteurs(conte+1)*sol_hom_quatre_vr(nzm1, k, j, i) ;

          }
      } // End of nullite_plm  
      } //End of loop on theta
  vr.set_spectral_va().ylm_i() ;
  vr += vrl0 ;
  het.set_spectral_va().ylm_i() ;


  
  Scalar pipo(*mpaff); 
  pipo.annule_hard(); 
  pipo.std_spectral_base(); 
  pipo.set_dzpuis(3);
  Param_elliptic param_mu(mu());


  

    resu.set_vr_eta_mu(vr, het, mu().sol_elliptic_boundary(param_mu, (*bound_mu), dir_mu , neum_mu)) ;

  return ;
  
}

---