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m (→‎Condição de Dirichlet em todos os lados)
m (→‎Condição de Neumann em todos os lados)
Linha 160: Linha 160:
  
 
==Condição de Neumann em todos os lados==
 
==Condição de Neumann em todos os lados==
−
O espectro é real com espectro σ⊂(-1,1).
+
Usando as CC
 +
  pnew( :,1 ) = pnew(:,2)
 +
  pnew( :,ny) = pnew(:,ny-1)
 +
  pnew( 1,: ) = pnew(2,:)
 +
  pnew(nx,: ) = pnew(nx-1,:)
  
−
Incluindo CC na matriz        Eliminando CC na matriz
+
O espectro é real com σ⊂(-1,1).
  
−
  lambda(1) = 1.0000E+00        lambda(1)=  0.955274744954830
+
  Incluindo CC na matriz                Eliminando CC na matriz
−
lambda(n) = 0.0000E+00        lambda(n)=  4.218228659674831E-003
 
  
−
  ( 0.100E+01, 0.0      )     (-0.955E+00, 0.0      )
+
  lambda(1)=  1.00000000000000          lambda(1)=  0.955274744954830
−
  ( 0.976E+00, 0.0      )     ( 0.955E+00, 0.0      )
+
  lambda(n)=  0.000000000000000E+000    lambda(n)=  4.218228659674831E-003 
−
  ( 0.970E+00, 0.0      )     (-0.896E+00, 0.0      )
+
      Eigenvalues                            Eigenvalues                   
−
  (-0.945E+00, 0.0      )     ( 0.896E+00, 0.0      )
+
  ( 0.100E+01, 0.000E+00)               (-0.955E+00, 0.000E+00)            
−
  ( 0.945E+00, 0.0      )     ( 0.884E+00, 0.0      )
+
  ( 0.976E+00, 0.000E+00)               ( 0.955E+00, 0.000E+00)            
−
  ( 0.905E+00, 0.0      )     (-0.884E+00, 0.0      )
+
  ( 0.970E+00, 0.000E+00)               (-0.896E+00, 0.000E+00)            
−
  ( 0.883E+00, 0.0      )     ( 0.825E+00, 0.0      )
+
  (-0.945E+00, 0.000E+00)               ( 0.896E+00, 0.000E+00)            
−
  (-0.874E+00, 0.0      )     (-0.825E+00, 0.0      )
+
  ( 0.945E+00, 0.000E+00)               ( 0.884E+00, 0.000E+00)            
−
  ( 0.874E+00, 0.0      )     (-0.803E+00, 0.0      )
+
  ( 0.905E+00, 0.000E+00)               (-0.884E+00, 0.000E+00)            
−
  ( 0.859E+00, 0.0      )     ( 0.803E+00, 0.0      )
+
  ( 0.883E+00, 0.000E+00)               ( 0.825E+00, 0.000E+00)            
−
  (-0.859E+00, 0.0      )     ( 0.774E+00, 0.0      )
+
  (-0.874E+00, 0.000E+00)               (-0.825E+00, 0.000E+00)            
−
  ( 0.794E+00, 0.0      )     (-0.774E+00, 0.0      )
+
  ( 0.874E+00, 0.000E+00)               (-0.803E+00, 0.000E+00)            
−
  (-0.788E+00, 0.0      )     ( 0.732E+00, 0.0      )
+
  ( 0.859E+00, 0.000E+00)               ( 0.803E+00, 0.000E+00)            
−
  ( 0.788E+00, 0.0      )     (-0.732E+00, 0.0      )
+
  (-0.859E+00, 0.000E+00)               ( 0.774E+00, 0.000E+00)            
−
  ( 0.764E+00, 0.0      )     ( 0.715E+00, 0.0      )
+
  ( 0.794E+00, 0.000E+00)               (-0.774E+00, 0.000E+00)            
−
  (-0.764E+00, 0.0      )     (-0.715E+00, 0.0      )
+
  (-0.788E+00, 0.000E+00)               ( 0.732E+00, 0.000E+00)            
−
  ( 0.750E+00, 0.0      )     ( 0.683E+00, 0.0      )
+
  ( 0.788E+00, 0.000E+00)               (-0.732E+00, 0.000E+00)            
−
  (-0.726E+00, 0.0      )     (-0.683E+00, 0.0      )
+
  ( 0.764E+00, 0.000E+00)               ( 0.715E+00, 0.000E+00)            
−
  ( 0.726E+00, 0.0      )     (-0.634E+00, 0.0      )
+
  (-0.764E+00, 0.000E+00)               (-0.715E+00, 0.000E+00)            
−
  ( 0.677E+00, 0.0      )     ( 0.634E+00, 0.0      )
+
  ( 0.750E+00, 0.000E+00)               ( 0.683E+00, 0.000E+00)            
−
  (-0.677E+00, 0.0      )     ( 0.621E+00, 0.0      )
+
  (-0.726E+00, 0.000E+00)               (-0.683E+00, 0.000E+00)            
−
  (-0.655E+00, 0.0      )     (-0.621E+00, 0.0      )
+
  ( 0.726E+00, 0.000E+00)               (-0.634E+00, 0.000E+00)            
−
  ( 0.655E+00, 0.0      )     ( 0.612E+00, 0.0      )
+
  ( 0.677E+00, 0.000E+00)               ( 0.634E+00, 0.000E+00)            
−
  ( 0.655E+00, 0.0      )     (-0.612E+00, 0.0      )
+
  (-0.677E+00, 0.000E+00)               ( 0.621E+00, 0.000E+00)            
−
  ( 0.624E+00, 0.0      )     ( 0.575E+00, 0.0      )
+
  (-0.655E+00, 0.000E+00)               (-0.621E+00, 0.000E+00)            
−
  (-0.624E+00, 0.0      )     (-0.575E+00, 0.0      )
+
  ( 0.655E+00, 0.000E+00)               ( 0.612E+00, 0.000E+00)            
−
  ( 0.587E+00, 0.0      )     ( 0.547E+00, 0.0      )
+
  ( 0.655E+00, 0.000E+00)               (-0.612E+00, 0.000E+00)            
−
  (-0.562E+00, 0.0      )     (-0.547E+00, 0.0      )
+
  ( 0.624E+00, 0.000E+00)               ( 0.575E+00, 0.000E+00)            
−
  ( 0.562E+00, 0.0      )     ( 0.502E+00, 0.0      )
+
  (-0.624E+00, 0.000E+00)               (-0.575E+00, 0.000E+00)            
−
  (-0.544E+00, 0.0      )     (-0.502E+00, 0.0      )
+
  ( 0.587E+00, 0.000E+00)               ( 0.547E+00, 0.000E+00)            
−
  ( 0.544E+00, 0.0      )     ( 0.482E+00, 0.0      )
+
  (-0.562E+00, 0.000E+00)               (-0.547E+00, 0.000E+00)            
−
  ( 0.538E+00, 0.0      )     (-0.482E+00, 0.0      )
+
  ( 0.562E+00, 0.000E+00)               ( 0.502E+00, 0.000E+00)            
−
  (-0.538E+00, 0.0      )     ( 0.480E+00, 0.0      )
+
  (-0.544E+00, 0.000E+00)               (-0.502E+00, 0.000E+00)            
−
  ( 0.500E+00, 0.0      )     (-0.480E+00, 0.0      )
+
  ( 0.544E+00, 0.000E+00)               ( 0.482E+00, 0.000E+00)            
−
  ( 0.491E+00, 0.0      )     ( 0.476E+00, 0.0      )
+
  ( 0.538E+00, 0.000E+00)               (-0.482E+00, 0.000E+00)            
−
  (-0.491E+00, 0.0      )     (-0.476E+00, 0.0      )
+
  (-0.538E+00, 0.000E+00)               ( 0.480E+00, 0.000E+00)            
−
  (-0.470E+00, 0.0      )     ( 0.421E+00, 0.0      )
+
  ( 0.500E+00, 0.000E+00)               (-0.480E+00, 0.000E+00)            
−
  ( 0.470E+00, 0.0      )     (-0.421E+00, 0.0      )
+
  ( 0.491E+00, 0.000E+00)               ( 0.476E+00, 0.000E+00)            
−
  ( 0.413E+00, 0.0      )     ( 0.404E+00, 0.0      )
+
  (-0.491E+00, 0.000E+00)               (-0.476E+00, 0.000E+00)            
−
  (-0.405E+00, 0.0      )     (-0.404E+00, 0.0      )
+
  (-0.470E+00, 0.000E+00)               ( 0.421E+00, 0.000E+00)            
−
  ( 0.405E+00, 0.0      )     ( 0.365E+00, 0.0      )
+
  ( 0.470E+00, 0.000E+00)               (-0.421E+00, 0.000E+00)            
−
  (-0.389E+00, 0.0      )     (-0.365E+00, 0.0      )
+
  ( 0.413E+00, 0.000E+00)               ( 0.404E+00, 0.000E+00)            
−
  ( 0.389E+00, 0.0      )     ( 0.362E+00, 0.0      )
+
  (-0.405E+00, 0.000E+00)               (-0.404E+00, 0.000E+00)            
−
  (-0.383E+00, 0.0      )     (-0.362E+00, 0.0      )
+
  ( 0.405E+00, 0.000E+00)               ( 0.365E+00, 0.000E+00)            
−
  ( 0.383E+00, 0.0      )     (-0.333E+00, 0.0      )
+
  (-0.389E+00, 0.000E+00)               (-0.365E+00, 0.000E+00)            
−
  (-0.381E+00, 0.0      )     ( 0.333E+00, 0.0      )
+
  ( 0.389E+00, 0.000E+00)               ( 0.362E+00, 0.000E+00)            
−
  ( 0.381E+00, 0.0      )     ( 0.327E+00, 0.0      )
+
  (-0.383E+00, 0.000E+00)               (-0.362E+00, 0.000E+00)            
−
  ( 0.345E+00, 0.0      )     (-0.327E+00, 0.0      )
+
  ( 0.383E+00, 0.000E+00)               (-0.333E+00, 0.000E+00)            
−
  (-0.318E+00, 0.0      )     (-0.325E+00, 0.0      )
+
  (-0.381E+00, 0.000E+00)               ( 0.333E+00, 0.000E+00)            
−
  ( 0.318E+00, 0.0      )     ( 0.325E+00, 0.0      )
+
  ( 0.381E+00, 0.000E+00)               ( 0.327E+00, 0.000E+00)            
−
  ( 0.315E+00, 0.0      )     (-0.268E+00, 0.0      )
+
  ( 0.345E+00, 0.000E+00)               (-0.327E+00, 0.000E+00)            
−
  (-0.315E+00, 0.0      )     ( 0.268E+00, 0.0      )
+
  (-0.318E+00, 0.000E+00)               (-0.325E+00, 0.000E+00)            
−
  (-0.250E+00, 0.0      )     (-0.266E+00, 0.0      )
+
  ( 0.318E+00, 0.000E+00)               ( 0.325E+00, 0.000E+00)            
−
  ( 0.250E+00, 0.0      )     ( 0.266E+00, 0.0      )
+
  ( 0.315E+00, 0.000E+00)               (-0.268E+00, 0.000E+00)            
−
  ( 0.250E+00, 0.0      )     ( 0.226E+00, 0.0      )
+
  (-0.315E+00, 0.000E+00)               ( 0.268E+00, 0.000E+00)            
−
  (-0.241E+00, 0.0      )     (-0.226E+00, 0.0      )
+
  (-0.250E+00, 0.000E+00)               (-0.266E+00, 0.000E+00)            
−
  ( 0.241E+00, 0.0      )     ( 0.223E+00, 0.0      )
+
  ( 0.250E+00, 0.000E+00)               ( 0.266E+00, 0.000E+00)            
−
  (-0.229E+00, 0.0      )     (-0.223E+00, 0.0      )
+
  ( 0.250E+00, 0.000E+00)               ( 0.226E+00, 0.000E+00)            
−
  ( 0.229E+00, 0.0      )     (-0.208E+00, 0.0      )
+
  (-0.241E+00, 0.000E+00)               (-0.226E+00, 0.000E+00)            
−
  (-0.226E+00, 0.0      )     ( 0.208E+00, 0.0      )
+
  ( 0.241E+00, 0.000E+00)               ( 0.223E+00, 0.000E+00)            
−
  ( 0.226E+00, 0.0      )     ( 0.197E+00, 0.0      )
+
  (-0.229E+00, 0.000E+00)               (-0.223E+00, 0.000E+00)            
−
  ( 0.207E+00, 0.0      )     (-0.197E+00, 0.0      )
+
  ( 0.229E+00, 0.000E+00)               (-0.208E+00, 0.000E+00)            
−
  (-0.207E+00, 0.0      )     ( 0.186E+00, 0.0      )
+
  (-0.226E+00, 0.000E+00)               ( 0.208E+00, 0.000E+00)            
−
  ( 0.206E+00, 0.0      )     (-0.186E+00, 0.0      )
+
  ( 0.226E+00, 0.000E+00)               ( 0.197E+00, 0.000E+00)            
−
  ( 0.176E+00, 0.0      )     ( 0.173E+00, 0.0      )
+
  ( 0.207E+00, 0.000E+00)               (-0.197E+00, 0.000E+00)            
−
  (-0.176E+00, 0.0      )     (-0.173E+00, 0.0      )
+
  (-0.207E+00, 0.000E+00)               ( 0.186E+00, 0.000E+00)            
−
  (-0.155E+00, 0.0      )     ( 0.148E+00, 0.0      )
+
  ( 0.206E+00, 0.000E+00)               (-0.186E+00, 0.000E+00)            
−
  ( 0.155E+00, 0.0      )     (-0.148E+00, 0.0      )
+
  ( 0.176E+00, 0.000E+00)               ( 0.173E+00, 0.000E+00)            
−
  ( 0.117E+00, 0.0      )     ( 0.127E+00, 0.0      )
+
  (-0.176E+00, 0.000E+00)               (-0.173E+00, 0.000E+00)            
−
  ( 0.955E-01, 0.0      )     (-0.127E+00, 0.0      )
+
  (-0.155E+00, 0.000E+00)               ( 0.148E+00, 0.000E+00)            
−
  ( 0.955E-01, 0.0      )     (-0.862E-01, 0.0      )
+
  ( 0.155E+00, 0.000E+00)               (-0.148E+00, 0.000E+00)            
−
  (-0.955E-01, 0.0      )     ( 0.862E-01, 0.0      )
+
  ( 0.117E+00, 0.000E+00)               ( 0.127E+00, 0.000E+00)            
−
  (-0.925E-01, 0.0      )     (-0.834E-01, 0.0      )
+
  ( 0.955E-01, 0.000E+00)               (-0.127E+00, 0.000E+00)            
−
  ( 0.925E-01, 0.0      )     ( 0.834E-01, 0.0      )
+
  ( 0.955E-01, 0.000E+00)               (-0.862E-01, 0.000E+00)            
−
  (-0.891E-01, 0.0      )     (-0.771E-01, 0.0      )
+
  (-0.955E-01, 0.000E+00)               ( 0.862E-01, 0.000E+00)            
−
  ( 0.891E-01, 0.0      )     ( 0.771E-01, 0.0      )
+
  (-0.925E-01, 0.000E+00)               (-0.834E-01, 0.000E+00)            
−
  (-0.868E-01, 0.0      )     (-0.752E-01, 0.0      )
+
  ( 0.925E-01, 0.000E+00)               ( 0.834E-01, 0.000E+00)            
−
  ( 0.868E-01, 0.0      )     ( 0.752E-01, 0.0      )
+
  (-0.891E-01, 0.000E+00)               (-0.771E-01, 0.000E+00)            
−
  ( 0.677E-01, 0.0      )     (-0.712E-01, 0.0      )
+
  ( 0.891E-01, 0.000E+00)               ( 0.771E-01, 0.000E+00)            
−
  (-0.677E-01, 0.0      )     ( 0.712E-01, 0.0      )
+
  (-0.868E-01, 0.000E+00)               (-0.752E-01, 0.000E+00)            
−
  (-0.653E-01, 0.0      )     ( 0.549E-01, 0.0      )
+
  ( 0.868E-01, 0.000E+00)               ( 0.752E-01, 0.000E+00)            
−
  ( 0.653E-01, 0.0      )     (-0.549E-01, 0.0      )
+
  ( 0.677E-01, 0.000E+00)               (-0.712E-01, 0.000E+00)            
−
  ( 0.439E-01, 0.0      )     (-0.532E-01, 0.0      )
+
  (-0.677E-01, 0.000E+00)               ( 0.712E-01, 0.000E+00)            
−
  (-0.439E-01, 0.0      )     ( 0.532E-01, 0.0      )
+
  (-0.653E-01, 0.000E+00)               ( 0.549E-01, 0.000E+00)            
−
  ( 0.302E-01, 0.0      )     ( 0.335E-01, 0.0      )
+
  ( 0.653E-01, 0.000E+00)               (-0.549E-01, 0.000E+00)            
−
  ( 0.245E-01, 0.0      )     (-0.335E-01, 0.0      )
+
  ( 0.439E-01, 0.000E+00)               (-0.532E-01, 0.000E+00)            
−
  ( 0.215E-01, 0.0      )     ( 0.161E-01, 0.0      )
+
  (-0.439E-01, 0.000E+00)               ( 0.532E-01, 0.000E+00)            
−
  (-0.215E-01, 0.0      )     (-0.161E-01, 0.0      )
+
  ( 0.302E-01, 0.000E+00)               ( 0.335E-01, 0.000E+00)            
−
  (-0.568E-02, 0.0      )     ( 0.422E-02, 0.0      )
+
  ( 0.245E-01, 0.000E+00)               (-0.335E-01, 0.000E+00)            
−
  ( 0.568E-02, 0.0      )     (-0.422E-02, 0.0      )
+
  ( 0.215E-01, 0.000E+00)               ( 0.161E-01, 0.000E+00)            
 +
  (-0.215E-01, 0.000E+00)               (-0.161E-01, 0.000E+00)            
 +
  (-0.568E-02, 0.000E+00)               ( 0.422E-02, 0.000E+00)            
 +
  ( 0.568E-02, 0.000E+00)               (-0.422E-02, 0.000E+00)            
 
  ( 0.251E-14, 0.857E-15)
 
  ( 0.251E-14, 0.857E-15)
 
  ( 0.251E-14,-0.857E-15)
 
  ( 0.251E-14,-0.857E-15)
−
  (-0.255E-14, 0.0      )
+
  (-0.255E-14, 0.000E+00)
 
  ( 0.233E-15, 0.235E-14)
 
  ( 0.233E-15, 0.235E-14)
 
  ( 0.233E-15,-0.235E-14)
 
  ( 0.233E-15,-0.235E-14)
 
  (-0.124E-14, 0.565E-15)
 
  (-0.124E-14, 0.565E-15)
 
  (-0.124E-14,-0.565E-15)
 
  (-0.124E-14,-0.565E-15)
−
  ( 0.113E-14, 0.0      )
+
  ( 0.113E-14, 0.000E+00)
−
  (-0.772E-15, 0.0      )
+
  (-0.772E-15, 0.000E+00)
 
  ( 0.632E-15, 0.207E-15)
 
  ( 0.632E-15, 0.207E-15)
 
  ( 0.632E-15,-0.207E-15)
 
  ( 0.632E-15,-0.207E-15)
Linha 272: Linha 278:
 
  ( 0.165E-15, 0.488E-15)
 
  ( 0.165E-15, 0.488E-15)
 
  ( 0.165E-15,-0.488E-15)
 
  ( 0.165E-15,-0.488E-15)
−
  ( 0.458E-15, 0.0      )
+
  ( 0.458E-15, 0.000E+00)
−
  (-0.433E-15, 0.0      )
+
  (-0.433E-15, 0.000E+00)
 
  (-0.330E-15, 0.101E-15)
 
  (-0.330E-15, 0.101E-15)
 
  (-0.330E-15,-0.101E-15)
 
  (-0.330E-15,-0.101E-15)
Linha 286: Linha 292:
 
  (-0.217E-16, 0.252E-16)
 
  (-0.217E-16, 0.252E-16)
 
  (-0.217E-16,-0.252E-16)
 
  (-0.217E-16,-0.252E-16)
−
  ( 0.327E-16, 0.0      )
+
  ( 0.327E-16, 0.000E+00)
−
  (-0.230E-16, 0.0      )
+
  (-0.230E-16, 0.000E+00)
 
  (-0.824E-30, 0.885E-30)
 
  (-0.824E-30, 0.885E-30)
 
  (-0.824E-30,-0.885E-30)
 
  (-0.824E-30,-0.885E-30)
−
  ( 0.680E-30, 0.0      )
+
  ( 0.680E-30, 0.000E+00)
−
  (-0.684E-31, 0.0      )
+
  (-0.684E-31, 0.000E+00)
−
  ( 0.317E-31, 0.0      )
+
  ( 0.317E-31, 0.000E+00)
−
  ( 0.0      , 0.0      )
+
  ( 0.000E+00, 0.000E+00)
−
  ( 0.0      , 0.0      )
+
  ( 0.000E+00, 0.000E+00)
−
  ( 0.0      , 0.0      )
+
  ( 0.000E+00, 0.000E+00)
−
  ( 0.0      , 0.0      )
+
  ( 0.000E+00, 0.000E+00)
−
  ( 0.0      , 0.0      )
+
  ( 0.000E+00, 0.000E+00)
−
  ( 0.0      , 0.0      )
+
  ( 0.000E+00, 0.000E+00)
  
 
===Condição de Neumann em todos os lados e subtraindo P(1,2) ===
 
===Condição de Neumann em todos os lados e subtraindo P(1,2) ===

Edição das 14h53min de 30 de julho de 2009

A análise dos autovalores de uma matriz de iteração pode ser usada para estudar a estabilidade de um método iterativo.

Vamos relatar um estudo para um problema específico.

O problema

Queremos aproximar a solução da equação de Navier Stokes em um duto. Para isso devemos resolver a cada passo de tempo uma equação de Poisson como <math>\frac{\partial^2 p}{\partial x^2}+\frac{\partial^2 p}{\partial y^2}= f(u)\,\!</math>

que é discretizada como

<math>A p_{i+1,j}+A p_{i-1,j}+C p_{i,j} +B p_{i,j+1}+B p_{i,j-1} = f_{i,j} \,\!</math>

  • A matriz possui o estêncil [...B,0...0,A,C,A,0...0,B,0...] onde

<math>A=\frac{-1}{dx^2}, B=\frac{-1}{dy^2}, C=-2A-2B \,\!</math>

  • O espectro do problema não depende de dt, Re ou U0. Depende apenas de A, B e C

Condição de Dirichlet em todos os lados

Para obter a matriz de iteração foi necessário usar as CC. 

 pnew( :, 1)= p( :, 1)
 pnew( :,ny)= p( :,ny)
 pnew( 1, :)= p( 1, :)
 pnew(nx, :)= p(nx, :)

O espectro é real com 0<|λ|<1. A condição de contorno contribui com autovalores λ=1. 

 lambda(1)=   1.00000000000000           lambda(1)=  0.955274744954830
 lambda(n)=  4.218228659668808E-003      lambda(n)=  4.218228659674831E-003
      Eigenvalues                             Eigenvalues
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
( 0.100E+01, 0.000E+00)                 
(-0.955E+00, 0.000E+00)                 (-0.955E+00, 0.000E+00)
( 0.955E+00, 0.000E+00)                 ( 0.955E+00, 0.000E+00)
( 0.896E+00, 0.000E+00)                 (-0.896E+00, 0.000E+00)
(-0.896E+00, 0.000E+00)                 ( 0.896E+00, 0.000E+00)
(-0.884E+00, 0.000E+00)                 ( 0.884E+00, 0.000E+00)
( 0.884E+00, 0.000E+00)                 (-0.884E+00, 0.000E+00)
( 0.825E+00, 0.000E+00)                 ( 0.825E+00, 0.000E+00)
(-0.825E+00, 0.000E+00)                 (-0.825E+00, 0.000E+00)
( 0.803E+00, 0.000E+00)                 (-0.803E+00, 0.000E+00)
(-0.803E+00, 0.000E+00)                 ( 0.803E+00, 0.000E+00)
(-0.774E+00, 0.000E+00)                 ( 0.774E+00, 0.000E+00)
( 0.774E+00, 0.000E+00)                 (-0.774E+00, 0.000E+00)
( 0.732E+00, 0.000E+00)                 ( 0.732E+00, 0.000E+00)
(-0.732E+00, 0.000E+00)                 (-0.732E+00, 0.000E+00)
(-0.715E+00, 0.000E+00)                 ( 0.715E+00, 0.000E+00)
( 0.715E+00, 0.000E+00)                 (-0.715E+00, 0.000E+00)
( 0.683E+00, 0.000E+00)                 ( 0.683E+00, 0.000E+00)
(-0.683E+00, 0.000E+00)                 (-0.683E+00, 0.000E+00)
(-0.634E+00, 0.000E+00)                 (-0.634E+00, 0.000E+00)
( 0.634E+00, 0.000E+00)                 ( 0.634E+00, 0.000E+00)
( 0.621E+00, 0.000E+00)                 ( 0.621E+00, 0.000E+00)
(-0.621E+00, 0.000E+00)                 (-0.621E+00, 0.000E+00)
(-0.612E+00, 0.000E+00)                 ( 0.612E+00, 0.000E+00)
( 0.612E+00, 0.000E+00)                 (-0.612E+00, 0.000E+00)
( 0.575E+00, 0.000E+00)                 ( 0.575E+00, 0.000E+00)
(-0.575E+00, 0.000E+00)                 (-0.575E+00, 0.000E+00)
(-0.547E+00, 0.000E+00)                 ( 0.547E+00, 0.000E+00)
( 0.547E+00, 0.000E+00)                 (-0.547E+00, 0.000E+00)
( 0.502E+00, 0.000E+00)                 ( 0.502E+00, 0.000E+00)
(-0.502E+00, 0.000E+00)                 (-0.502E+00, 0.000E+00)
(-0.482E+00, 0.000E+00)                 ( 0.482E+00, 0.000E+00)
( 0.482E+00, 0.000E+00)                 (-0.482E+00, 0.000E+00)
(-0.480E+00, 0.000E+00)                 ( 0.480E+00, 0.000E+00)
( 0.480E+00, 0.000E+00)                 (-0.480E+00, 0.000E+00)
(-0.476E+00, 0.000E+00)                 ( 0.476E+00, 0.000E+00)
( 0.476E+00, 0.000E+00)                 (-0.476E+00, 0.000E+00)
( 0.421E+00, 0.000E+00)                 ( 0.421E+00, 0.000E+00)
(-0.421E+00, 0.000E+00)                 (-0.421E+00, 0.000E+00)
(-0.404E+00, 0.000E+00)                 ( 0.404E+00, 0.000E+00)
( 0.404E+00, 0.000E+00)                 (-0.404E+00, 0.000E+00)
(-0.365E+00, 0.000E+00)                 ( 0.365E+00, 0.000E+00)
( 0.365E+00, 0.000E+00)                 (-0.365E+00, 0.000E+00)
( 0.362E+00, 0.000E+00)                 ( 0.362E+00, 0.000E+00)
(-0.362E+00, 0.000E+00)                 (-0.362E+00, 0.000E+00)
( 0.333E+00, 0.000E+00)                 (-0.333E+00, 0.000E+00)
(-0.333E+00, 0.000E+00)                 ( 0.333E+00, 0.000E+00)
(-0.327E+00, 0.000E+00)                 ( 0.327E+00, 0.000E+00)
( 0.327E+00, 0.000E+00)                 (-0.327E+00, 0.000E+00)
( 0.325E+00, 0.000E+00)                 (-0.325E+00, 0.000E+00)
(-0.325E+00, 0.000E+00)                 ( 0.325E+00, 0.000E+00)
( 0.268E+00, 0.000E+00)                 (-0.268E+00, 0.000E+00)
(-0.268E+00, 0.000E+00)                 ( 0.268E+00, 0.000E+00)
( 0.266E+00, 0.000E+00)                 (-0.266E+00, 0.000E+00)
(-0.266E+00, 0.000E+00)                 ( 0.266E+00, 0.000E+00)
( 0.226E+00, 0.000E+00)                 ( 0.226E+00, 0.000E+00)
(-0.226E+00, 0.000E+00)                 (-0.226E+00, 0.000E+00)
(-0.223E+00, 0.000E+00)                 ( 0.223E+00, 0.000E+00)
( 0.223E+00, 0.000E+00)                 (-0.223E+00, 0.000E+00)
(-0.208E+00, 0.000E+00)                 (-0.208E+00, 0.000E+00)
( 0.208E+00, 0.000E+00)                 ( 0.208E+00, 0.000E+00)
(-0.197E+00, 0.000E+00)                 ( 0.197E+00, 0.000E+00)
( 0.197E+00, 0.000E+00)                 (-0.197E+00, 0.000E+00)
(-0.186E+00, 0.000E+00)                 ( 0.186E+00, 0.000E+00)
( 0.186E+00, 0.000E+00)                 (-0.186E+00, 0.000E+00)
( 0.173E+00, 0.000E+00)                 ( 0.173E+00, 0.000E+00)
(-0.173E+00, 0.000E+00)                 (-0.173E+00, 0.000E+00)
( 0.148E+00, 0.000E+00)                 ( 0.148E+00, 0.000E+00)
(-0.148E+00, 0.000E+00)                 (-0.148E+00, 0.000E+00)
(-0.127E+00, 0.000E+00)                 ( 0.127E+00, 0.000E+00)
( 0.127E+00, 0.000E+00)                 (-0.127E+00, 0.000E+00)
(-0.862E-01, 0.000E+00)                 (-0.862E-01, 0.000E+00)
( 0.862E-01, 0.000E+00)                 ( 0.862E-01, 0.000E+00)
(-0.834E-01, 0.000E+00)                 (-0.834E-01, 0.000E+00)
( 0.834E-01, 0.000E+00)                 ( 0.834E-01, 0.000E+00)
( 0.771E-01, 0.000E+00)                 (-0.771E-01, 0.000E+00)
(-0.771E-01, 0.000E+00)                 ( 0.771E-01, 0.000E+00)
( 0.752E-01, 0.000E+00)                 (-0.752E-01, 0.000E+00)
(-0.752E-01, 0.000E+00)                 ( 0.752E-01, 0.000E+00)
(-0.712E-01, 0.000E+00)                 (-0.712E-01, 0.000E+00)
( 0.712E-01, 0.000E+00)                 ( 0.712E-01, 0.000E+00)
(-0.549E-01, 0.000E+00)                 ( 0.549E-01, 0.000E+00)
( 0.549E-01, 0.000E+00)                 (-0.549E-01, 0.000E+00)
( 0.532E-01, 0.000E+00)                 (-0.532E-01, 0.000E+00)
(-0.532E-01, 0.000E+00)                 ( 0.532E-01, 0.000E+00)
(-0.335E-01, 0.000E+00)                 ( 0.335E-01, 0.000E+00)
( 0.335E-01, 0.000E+00)                 (-0.335E-01, 0.000E+00)
(-0.161E-01, 0.000E+00)                 ( 0.161E-01, 0.000E+00)
( 0.161E-01, 0.000E+00)                 (-0.161E-01, 0.000E+00)
( 0.422E-02, 0.000E+00)                 ( 0.422E-02, 0.000E+00)
(-0.422E-02, 0.000E+00)                 (-0.422E-02, 0.000E+00)

Condição de Neumann em todos os lados

Usando as CC 

 pnew( :,1 ) = pnew(:,2)
 pnew( :,ny) = pnew(:,ny-1)
 pnew( 1,: ) = pnew(2,:)
 pnew(nx,: ) = pnew(nx-1,:)

O espectro é real com σ⊂(-1,1). 

Incluindo CC na matriz                 Eliminando CC na matriz

 lambda(1)=   1.00000000000000          lambda(1)=  0.955274744954830
 lambda(n)=  0.000000000000000E+000     lambda(n)=  4.218228659674831E-003  
      Eigenvalues                            Eigenvalues                    
( 0.100E+01, 0.000E+00)                (-0.955E+00, 0.000E+00)              
( 0.976E+00, 0.000E+00)                ( 0.955E+00, 0.000E+00)              
( 0.970E+00, 0.000E+00)                (-0.896E+00, 0.000E+00)              
(-0.945E+00, 0.000E+00)                ( 0.896E+00, 0.000E+00)              
( 0.945E+00, 0.000E+00)                ( 0.884E+00, 0.000E+00)              
( 0.905E+00, 0.000E+00)                (-0.884E+00, 0.000E+00)              
( 0.883E+00, 0.000E+00)                ( 0.825E+00, 0.000E+00)              
(-0.874E+00, 0.000E+00)                (-0.825E+00, 0.000E+00)              
( 0.874E+00, 0.000E+00)                (-0.803E+00, 0.000E+00)              
( 0.859E+00, 0.000E+00)                ( 0.803E+00, 0.000E+00)              
(-0.859E+00, 0.000E+00)                ( 0.774E+00, 0.000E+00)              
( 0.794E+00, 0.000E+00)                (-0.774E+00, 0.000E+00)              
(-0.788E+00, 0.000E+00)                ( 0.732E+00, 0.000E+00)              
( 0.788E+00, 0.000E+00)                (-0.732E+00, 0.000E+00)              
( 0.764E+00, 0.000E+00)                ( 0.715E+00, 0.000E+00)              
(-0.764E+00, 0.000E+00)                (-0.715E+00, 0.000E+00)              
( 0.750E+00, 0.000E+00)                ( 0.683E+00, 0.000E+00)              
(-0.726E+00, 0.000E+00)                (-0.683E+00, 0.000E+00)              
( 0.726E+00, 0.000E+00)                (-0.634E+00, 0.000E+00)              
( 0.677E+00, 0.000E+00)                ( 0.634E+00, 0.000E+00)              
(-0.677E+00, 0.000E+00)                ( 0.621E+00, 0.000E+00)              
(-0.655E+00, 0.000E+00)                (-0.621E+00, 0.000E+00)              
( 0.655E+00, 0.000E+00)                ( 0.612E+00, 0.000E+00)              
( 0.655E+00, 0.000E+00)                (-0.612E+00, 0.000E+00)              
( 0.624E+00, 0.000E+00)                ( 0.575E+00, 0.000E+00)              
(-0.624E+00, 0.000E+00)                (-0.575E+00, 0.000E+00)              
( 0.587E+00, 0.000E+00)                ( 0.547E+00, 0.000E+00)              
(-0.562E+00, 0.000E+00)                (-0.547E+00, 0.000E+00)              
( 0.562E+00, 0.000E+00)                ( 0.502E+00, 0.000E+00)              
(-0.544E+00, 0.000E+00)                (-0.502E+00, 0.000E+00)              
( 0.544E+00, 0.000E+00)                ( 0.482E+00, 0.000E+00)              
( 0.538E+00, 0.000E+00)                (-0.482E+00, 0.000E+00)              
(-0.538E+00, 0.000E+00)                ( 0.480E+00, 0.000E+00)              
( 0.500E+00, 0.000E+00)                (-0.480E+00, 0.000E+00)              
( 0.491E+00, 0.000E+00)                ( 0.476E+00, 0.000E+00)              
(-0.491E+00, 0.000E+00)                (-0.476E+00, 0.000E+00)              
(-0.470E+00, 0.000E+00)                ( 0.421E+00, 0.000E+00)              
( 0.470E+00, 0.000E+00)                (-0.421E+00, 0.000E+00)              
( 0.413E+00, 0.000E+00)                ( 0.404E+00, 0.000E+00)              
(-0.405E+00, 0.000E+00)                (-0.404E+00, 0.000E+00)              
( 0.405E+00, 0.000E+00)                ( 0.365E+00, 0.000E+00)              
(-0.389E+00, 0.000E+00)                (-0.365E+00, 0.000E+00)              
( 0.389E+00, 0.000E+00)                ( 0.362E+00, 0.000E+00)              
(-0.383E+00, 0.000E+00)                (-0.362E+00, 0.000E+00)              
( 0.383E+00, 0.000E+00)                (-0.333E+00, 0.000E+00)              
(-0.381E+00, 0.000E+00)                ( 0.333E+00, 0.000E+00)              
( 0.381E+00, 0.000E+00)                ( 0.327E+00, 0.000E+00)              
( 0.345E+00, 0.000E+00)                (-0.327E+00, 0.000E+00)              
(-0.318E+00, 0.000E+00)                (-0.325E+00, 0.000E+00)              
( 0.318E+00, 0.000E+00)                ( 0.325E+00, 0.000E+00)              
( 0.315E+00, 0.000E+00)                (-0.268E+00, 0.000E+00)              
(-0.315E+00, 0.000E+00)                ( 0.268E+00, 0.000E+00)              
(-0.250E+00, 0.000E+00)                (-0.266E+00, 0.000E+00)              
( 0.250E+00, 0.000E+00)                ( 0.266E+00, 0.000E+00)              
( 0.250E+00, 0.000E+00)                ( 0.226E+00, 0.000E+00)              
(-0.241E+00, 0.000E+00)                (-0.226E+00, 0.000E+00)              
( 0.241E+00, 0.000E+00)                ( 0.223E+00, 0.000E+00)              
(-0.229E+00, 0.000E+00)                (-0.223E+00, 0.000E+00)              
( 0.229E+00, 0.000E+00)                (-0.208E+00, 0.000E+00)              
(-0.226E+00, 0.000E+00)                ( 0.208E+00, 0.000E+00)              
( 0.226E+00, 0.000E+00)                ( 0.197E+00, 0.000E+00)              
( 0.207E+00, 0.000E+00)                (-0.197E+00, 0.000E+00)              
(-0.207E+00, 0.000E+00)                ( 0.186E+00, 0.000E+00)              
( 0.206E+00, 0.000E+00)                (-0.186E+00, 0.000E+00)              
( 0.176E+00, 0.000E+00)                ( 0.173E+00, 0.000E+00)              
(-0.176E+00, 0.000E+00)                (-0.173E+00, 0.000E+00)              
(-0.155E+00, 0.000E+00)                ( 0.148E+00, 0.000E+00)              
( 0.155E+00, 0.000E+00)                (-0.148E+00, 0.000E+00)              
( 0.117E+00, 0.000E+00)                ( 0.127E+00, 0.000E+00)              
( 0.955E-01, 0.000E+00)                (-0.127E+00, 0.000E+00)              
( 0.955E-01, 0.000E+00)                (-0.862E-01, 0.000E+00)              
(-0.955E-01, 0.000E+00)                ( 0.862E-01, 0.000E+00)              
(-0.925E-01, 0.000E+00)                (-0.834E-01, 0.000E+00)              
( 0.925E-01, 0.000E+00)                ( 0.834E-01, 0.000E+00)              
(-0.891E-01, 0.000E+00)                (-0.771E-01, 0.000E+00)              
( 0.891E-01, 0.000E+00)                ( 0.771E-01, 0.000E+00)              
(-0.868E-01, 0.000E+00)                (-0.752E-01, 0.000E+00)              
( 0.868E-01, 0.000E+00)                ( 0.752E-01, 0.000E+00)              
( 0.677E-01, 0.000E+00)                (-0.712E-01, 0.000E+00)              
(-0.677E-01, 0.000E+00)                ( 0.712E-01, 0.000E+00)              
(-0.653E-01, 0.000E+00)                ( 0.549E-01, 0.000E+00)              
( 0.653E-01, 0.000E+00)                (-0.549E-01, 0.000E+00)              
( 0.439E-01, 0.000E+00)                (-0.532E-01, 0.000E+00)              
(-0.439E-01, 0.000E+00)                ( 0.532E-01, 0.000E+00)              
( 0.302E-01, 0.000E+00)                ( 0.335E-01, 0.000E+00)              
( 0.245E-01, 0.000E+00)                (-0.335E-01, 0.000E+00)              
( 0.215E-01, 0.000E+00)                ( 0.161E-01, 0.000E+00)              
(-0.215E-01, 0.000E+00)                (-0.161E-01, 0.000E+00)              
(-0.568E-02, 0.000E+00)                ( 0.422E-02, 0.000E+00)              
( 0.568E-02, 0.000E+00)                (-0.422E-02, 0.000E+00)              
( 0.251E-14, 0.857E-15)
( 0.251E-14,-0.857E-15)
(-0.255E-14, 0.000E+00)
( 0.233E-15, 0.235E-14)
( 0.233E-15,-0.235E-14)
(-0.124E-14, 0.565E-15)
(-0.124E-14,-0.565E-15)
( 0.113E-14, 0.000E+00)
(-0.772E-15, 0.000E+00)
( 0.632E-15, 0.207E-15)
( 0.632E-15,-0.207E-15)
(-0.490E-15, 0.439E-15)
(-0.490E-15,-0.439E-15)
( 0.165E-15, 0.488E-15)
( 0.165E-15,-0.488E-15)
( 0.458E-15, 0.000E+00)
(-0.433E-15, 0.000E+00)
(-0.330E-15, 0.101E-15)
(-0.330E-15,-0.101E-15)
( 0.212E-15, 0.141E-15)
( 0.212E-15,-0.141E-15)
(-0.123E-15, 0.217E-15)
(-0.123E-15,-0.217E-15)
( 0.750E-17, 0.128E-15)
( 0.750E-17,-0.128E-15)
( 0.277E-16, 0.412E-16)
( 0.277E-16,-0.412E-16)
(-0.217E-16, 0.252E-16)
(-0.217E-16,-0.252E-16)
( 0.327E-16, 0.000E+00)
(-0.230E-16, 0.000E+00)
(-0.824E-30, 0.885E-30)
(-0.824E-30,-0.885E-30)
( 0.680E-30, 0.000E+00)
(-0.684E-31, 0.000E+00)
( 0.317E-31, 0.000E+00)
( 0.000E+00, 0.000E+00)
( 0.000E+00, 0.000E+00)
( 0.000E+00, 0.000E+00)
( 0.000E+00, 0.000E+00)
( 0.000E+00, 0.000E+00)
( 0.000E+00, 0.000E+00)

Condição de Neumann em todos os lados e subtraindo P(1,2)

Usando Pnew=Pnew-P(1,2) temos o espectro 

 Incluindo CC na Matriz                 Excluindo CC na Matriz

lambda(1)=  0.975528258147575          lambda(1)=  0.955274744954828
lambda(n)=  0.000000000000000E+000     lambda(n)=  0.000000000000000E+000

( 0.976E+00, 0.0      )     (-0.955E+00, 0.0      )
( 0.970E+00, 0.0      )     ( 0.911E+00, 0.0      )
( 0.945E+00, 0.0      )     ( 0.896E+00, 0.0      )
(-0.945E+00, 0.0      )     (-0.896E+00, 0.0      )
( 0.905E+00, 0.0      )     ( 0.884E+00, 0.0      )
( 0.883E+00, 0.0      )     (-0.884E+00, 0.0      )
(-0.874E+00, 0.0      )     (-0.825E+00, 0.0      )
( 0.874E+00, 0.0      )     ( 0.825E+00, 0.0      )
( 0.859E+00, 0.0      )     (-0.803E+00, 0.0      )
(-0.859E+00, 0.0      )     ( 0.789E+00, 0.0      )
( 0.794E+00, 0.0      )     (-0.774E+00, 0.0      )
( 0.788E+00, 0.0      )     (-0.732E+00, 0.0      )
(-0.788E+00, 0.0      )     ( 0.732E+00, 0.0      )
( 0.764E+00, 0.0      )     ( 0.715E+00, 0.0      )
(-0.764E+00, 0.0      )     (-0.712E+00, 0.0      )
( 0.750E+00, 0.0      )     ( 0.693E+00, 0.0      )
(-0.726E+00, 0.0      )     ( 0.683E+00, 0.0      )
( 0.726E+00, 0.0      )     (-0.682E+00, 0.0      )
(-0.677E+00, 0.0      )     (-0.634E+00, 0.0      )
( 0.677E+00, 0.0      )     ( 0.634E+00, 0.0      )
( 0.655E+00, 0.0      )     (-0.621E+00, 0.0      )
(-0.655E+00, 0.0      )     (-0.612E+00, 0.0      )
( 0.655E+00, 0.0      )     ( 0.612E+00, 0.0      )
(-0.624E+00, 0.0      )     ( 0.583E+00, 0.0      )
( 0.624E+00, 0.0      )     (-0.575E+00, 0.0      )
( 0.587E+00, 0.0      )     ( 0.575E+00, 0.0      )
( 0.562E+00, 0.0      )     (-0.547E+00, 0.0      )
(-0.562E+00, 0.0      )     ( 0.507E+00, 0.0      )
( 0.544E+00, 0.0      )     ( 0.502E+00, 0.0      )
(-0.544E+00, 0.0      )     (-0.494E+00, 0.0      )
(-0.538E+00, 0.0      )     ( 0.482E+00, 0.0      )
( 0.538E+00, 0.0      )     (-0.482E+00, 0.0      )
( 0.500E+00, 0.0      )     (-0.480E+00, 0.0      )
( 0.491E+00, 0.0      )     (-0.476E+00, 0.0      )
(-0.491E+00, 0.0      )     ( 0.476E+00, 0.0      )
(-0.470E+00, 0.0      )     ( 0.421E+00, 0.0      )
( 0.470E+00, 0.0      )     (-0.416E+00, 0.0      )
( 0.413E+00, 0.0      )     ( 0.412E+00, 0.0      )
(-0.405E+00, 0.0      )     ( 0.404E+00, 0.0      )
( 0.405E+00, 0.0      )     (-0.402E+00, 0.0      )
(-0.389E+00, 0.0      )     (-0.365E+00, 0.0      )
( 0.389E+00, 0.0      )     (-0.362E+00, 0.0      )
( 0.383E+00, 0.0      )     ( 0.362E+00, 0.0      )
(-0.383E+00, 0.0      )     ( 0.343E+00, 0.0      )
(-0.381E+00, 0.0      )     ( 0.333E+00, 0.0      )
( 0.381E+00, 0.0      )     (-0.333E+00, 0.0      )
( 0.345E+00, 0.0      )     (-0.327E+00, 0.0      )
( 0.318E+00, 0.0      )     (-0.325E+00, 0.0      )
(-0.318E+00, 0.0      )     ( 0.325E+00, 0.0      )
( 0.315E+00, 0.0      )     ( 0.283E+00, 0.0      )
(-0.315E+00, 0.0      )     (-0.268E+00, 0.0      )
( 0.250E+00, 0.0      )     ( 0.266E+00, 0.0      )
(-0.250E+00, 0.0      )     (-0.266E+00, 0.0      )
( 0.250E+00, 0.0      )     (-0.226E+00, 0.0      )
( 0.241E+00, 0.0      )     ( 0.226E+00, 0.0      )
(-0.241E+00, 0.0      )     ( 0.223E+00, 0.0      )
(-0.229E+00, 0.0      )     (-0.218E+00, 0.0      )
( 0.229E+00, 0.0      )     ( 0.208E+00, 0.0      )
(-0.226E+00, 0.0      )     ( 0.202E+00, 0.0      )
( 0.226E+00, 0.0      )     (-0.197E+00, 0.0      )
( 0.207E+00, 0.0      )     ( 0.197E+00, 0.0      )
(-0.207E+00, 0.0      )     (-0.190E+00, 0.0      )
( 0.206E+00, 0.0      )     (-0.186E+00, 0.0      )
( 0.176E+00, 0.0      )     (-0.173E+00, 0.0      )
(-0.176E+00, 0.0      )     ( 0.173E+00, 0.0      )
( 0.155E+00, 0.0      )     ( 0.148E+00, 0.0      )
(-0.155E+00, 0.0      )     (-0.146E+00, 0.0      )
( 0.117E+00, 0.0      )     ( 0.127E+00, 0.0      )
(-0.955E-01, 0.0      )     ( 0.122E+00, 0.0      )
( 0.955E-01, 0.411E-15)     (-0.120E+00, 0.0      )
( 0.955E-01,-0.411E-15)     (-0.862E-01, 0.0      )
( 0.925E-01, 0.0      )     (-0.834E-01, 0.0      )
(-0.925E-01, 0.0      )     ( 0.834E-01, 0.0      )
(-0.891E-01, 0.0      )     ( 0.789E-01, 0.0      )
( 0.891E-01, 0.0      )     (-0.771E-01, 0.0      )
( 0.868E-01, 0.0      )     ( 0.771E-01, 0.0      )
(-0.868E-01, 0.0      )     (-0.752E-01, 0.0      )
(-0.677E-01, 0.0      )     ( 0.752E-01, 0.0      )
( 0.677E-01, 0.0      )     (-0.712E-01, 0.0      )
( 0.653E-01, 0.0      )     ( 0.573E-01, 0.0      )
(-0.653E-01, 0.0      )     (-0.549E-01, 0.0      )
(-0.439E-01, 0.0      )     (-0.532E-01, 0.0      )
( 0.439E-01, 0.0      )     ( 0.532E-01, 0.0      )
( 0.302E-01, 0.0      )     ( 0.395E-01, 0.0      )
( 0.245E-01, 0.0      )     (-0.335E-01, 0.0      )
(-0.215E-01, 0.0      )     (-0.161E-01, 0.0      )
( 0.215E-01, 0.0      )     ( 0.161E-01, 0.0      )
( 0.568E-02, 0.0      )     ( 0.523E-02, 0.0      )
(-0.568E-02, 0.0      )     (-0.422E-02, 0.0      )
(-0.444E-14, 0.0      )     ( 0.0      , 0.0      )
( 0.214E-14, 0.128E-14)
( 0.214E-14,-0.128E-14)
( 0.217E-14, 0.0      )
(-0.168E-14, 0.0      )
(-0.140E-14, 0.798E-15)
(-0.140E-14,-0.798E-15)
(-0.890E-15, 0.126E-14)
(-0.890E-15,-0.126E-14)
( 0.565E-15, 0.105E-14)
( 0.565E-15,-0.105E-14)
( 0.876E-15, 0.0      )
(-0.773E-15, 0.208E-15)
(-0.773E-15,-0.208E-15)
( 0.587E-15, 0.0      )
(-0.150E-16, 0.567E-15)
(-0.150E-16,-0.567E-15)
( 0.352E-15, 0.337E-15)
( 0.352E-15,-0.337E-15)
( 0.127E-15, 0.324E-15)
( 0.127E-15,-0.324E-15)
(-0.137E-16, 0.199E-15)
(-0.137E-16,-0.199E-15)
( 0.172E-15, 0.372E-16)
( 0.172E-15,-0.372E-16)
( 0.719E-16, 0.157E-15)
( 0.719E-16,-0.157E-15)
(-0.153E-15, 0.661E-16)
(-0.153E-15,-0.661E-16)
(-0.304E-16, 0.683E-16)
(-0.304E-16,-0.683E-16)
(-0.192E-16, 0.0      )
(-0.955E-28, 0.0      )
(-0.281E-30, 0.325E-30)
(-0.281E-30,-0.325E-30)
( 0.0      , 0.0      )
( 0.0      , 0.0      )
( 0.0      , 0.0      )
( 0.0      , 0.0      )
( 0.0      , 0.0      )
( 0.0      , 0.0      )
( 0.0      , 0.0      )
( 0.0      , 0.0      )

Condição de Neumann em todos os lados fixando um ponto, P(1,2)=1

O espectro tem autovalores próximos do eixo real (talvez devido a erros de ponto flutuante) com σ⊂ 1.726 ∪ (-1,1) incluindo 0. 

Incluindo CC na matriz                    Excluindo CC na matriz
lambda(1)=   1.72605097442453             lambda(1)=  0.955274744954830
lambda(n)=  0.000000000000000E+000        lambda(n)=  4.218228659674831E-003
( 1.726E+00, 0.0      )     (-0.955E+00, 0.0      )
( 0.975E+00, 0.0      )     ( 0.955E+00, 0.0      )
( 0.969E+00, 0.0      )     (-0.896E+00, 0.0      )
(-0.945E+00, 0.0      )     ( 0.896E+00, 0.0      )
( 0.942E+00, 0.589E-02)     ( 0.884E+00, 0.0      )
( 0.942E+00,-0.589E-02)     (-0.884E+00, 0.0      )
( 0.893E+00, 0.0      )     ( 0.825E+00, 0.0      )
(-0.875E+00, 0.0      )     (-0.825E+00, 0.0      )
( 0.874E+00, 0.0      )     (-0.803E+00, 0.0      )
(-0.859E+00, 0.0      )     ( 0.803E+00, 0.0      )
( 0.857E+00, 0.0      )     ( 0.774E+00, 0.0      )
( 0.795E+00, 0.244E-02)     (-0.774E+00, 0.0      )
( 0.795E+00,-0.244E-02)     ( 0.732E+00, 0.0      )
(-0.788E+00, 0.0      )     (-0.732E+00, 0.0      )
(-0.764E+00, 0.0      )     ( 0.715E+00, 0.0      )
( 0.762E+00, 0.0      )     (-0.715E+00, 0.0      )
( 0.748E+00, 0.0      )     ( 0.683E+00, 0.0      )
(-0.726E+00, 0.0      )     (-0.683E+00, 0.0      )
( 0.715E+00, 0.774E-02)     (-0.634E+00, 0.0      )
( 0.715E+00,-0.774E-02)     ( 0.634E+00, 0.0      )
( 0.679E+00, 0.0      )     ( 0.621E+00, 0.0      )
(-0.677E+00, 0.0      )     (-0.621E+00, 0.0      )
(-0.656E+00, 0.0      )     ( 0.612E+00, 0.0      )
( 0.655E+00, 0.0      )     (-0.612E+00, 0.0      )
(-0.625E+00, 0.0      )     ( 0.575E+00, 0.0      )
( 0.620E+00, 0.362E-02)     (-0.575E+00, 0.0      )
( 0.620E+00,-0.362E-02)     ( 0.547E+00, 0.0      )
(-0.562E+00, 0.0      )     (-0.547E+00, 0.0      )
( 0.561E+00, 0.0      )     ( 0.502E+00, 0.0      )
(-0.545E+00, 0.0      )     (-0.502E+00, 0.0      )
( 0.544E+00, 0.277E-02)     ( 0.482E+00, 0.0      )
( 0.544E+00,-0.277E-02)     (-0.482E+00, 0.0      )
(-0.538E+00, 0.0      )     ( 0.480E+00, 0.0      )
( 0.500E+00, 0.219E-02)     (-0.480E+00, 0.0      )
( 0.500E+00,-0.219E-02)     ( 0.476E+00, 0.0      )
(-0.492E+00, 0.0      )     (-0.476E+00, 0.0      )
(-0.470E+00, 0.0      )     ( 0.421E+00, 0.0      )
( 0.469E+00, 0.0      )     (-0.421E+00, 0.0      )
(-0.418E+00, 0.0      )     ( 0.404E+00, 0.0      )
( 0.412E+00, 0.0      )     (-0.404E+00, 0.0      )
( 0.401E+00, 0.0      )     ( 0.365E+00, 0.0      )
(-0.389E+00, 0.0      )     (-0.365E+00, 0.0      )
( 0.387E+00, 0.137E-01)     ( 0.362E+00, 0.0      )
( 0.387E+00,-0.137E-01)     (-0.362E+00, 0.0      )
( 0.386E+00, 0.0      )     (-0.333E+00, 0.0      )
(-0.384E+00, 0.0      )     ( 0.333E+00, 0.0      )
( 0.382E+00, 0.0      )     ( 0.327E+00, 0.0      )
(-0.381E+00, 0.0      )     (-0.327E+00, 0.0      )
(-0.332E+00, 0.0      )     (-0.325E+00, 0.0      )
( 0.317E+00, 0.0      )     ( 0.325E+00, 0.0      )
(-0.316E+00, 0.0      )     (-0.268E+00, 0.0      )
( 0.314E+00, 0.0      )     ( 0.268E+00, 0.0      )
( 0.290E+00, 0.0      )     (-0.266E+00, 0.0      )
(-0.256E+00, 0.0      )     ( 0.266E+00, 0.0      )
( 0.250E+00, 0.0      )     ( 0.226E+00, 0.0      )
(-0.245E+00, 0.0      )     (-0.226E+00, 0.0      )
( 0.244E+00, 0.0      )     ( 0.223E+00, 0.0      )
(-0.233E+00, 0.0      )     (-0.223E+00, 0.0      )
( 0.232E+00, 0.0      )     (-0.208E+00, 0.0      )
(-0.226E+00, 0.0      )     ( 0.208E+00, 0.0      )
( 0.225E+00, 0.0      )     ( 0.197E+00, 0.0      )
(-0.221E+00, 0.0      )     (-0.197E+00, 0.0      )
( 0.207E+00, 0.0      )     ( 0.186E+00, 0.0      )
( 0.206E+00, 0.0      )     (-0.186E+00, 0.0      )
(-0.204E+00, 0.0      )     ( 0.173E+00, 0.0      )
(-0.176E+00, 0.0      )     (-0.173E+00, 0.0      )
( 0.176E+00, 0.0      )     ( 0.148E+00, 0.0      )
( 0.163E+00, 0.0      )     (-0.148E+00, 0.0      )
(-0.154E+00, 0.0      )     ( 0.127E+00, 0.0      )
( 0.117E+00, 0.0      )     (-0.127E+00, 0.0      )
( 0.107E+00, 0.0      )     (-0.862E-01, 0.0      )
( 0.955E-01, 0.0      )     ( 0.862E-01, 0.0      )
(-0.925E-01, 0.0      )     (-0.834E-01, 0.0      )
( 0.925E-01, 0.0      )     ( 0.834E-01, 0.0      )
(-0.894E-01, 0.0      )     (-0.771E-01, 0.0      )
( 0.891E-01, 0.0      )     ( 0.771E-01, 0.0      )
(-0.878E-01, 0.0      )     (-0.752E-01, 0.0      )
( 0.867E-01, 0.0      )     ( 0.752E-01, 0.0      )
(-0.855E-01, 0.0      )     (-0.712E-01, 0.0      )
( 0.675E-01, 0.0      )     ( 0.712E-01, 0.0      )
(-0.656E-01, 0.0      )     ( 0.549E-01, 0.0      )
( 0.652E-01, 0.0      )     (-0.549E-01, 0.0      )
(-0.457E-01, 0.0      )     (-0.532E-01, 0.0      )
( 0.436E-01, 0.0      )     ( 0.532E-01, 0.0      )
(-0.427E-01, 0.0      )     ( 0.335E-01, 0.0      )
( 0.372E-01, 0.0      )     (-0.335E-01, 0.0      )
( 0.245E-01, 0.0      )     ( 0.161E-01, 0.0      )
( 0.215E-01, 0.0      )     (-0.161E-01, 0.0      )
(-0.107E-01, 0.0      )     ( 0.422E-02, 0.0      )
( 0.568E-02, 0.0      )     (-0.422E-02, 0.0      )
(-0.567E-02, 0.0      )
( 0.701E-14, 0.566E-15)
( 0.701E-14,-0.566E-15)
(-0.579E-14, 0.0      )
(-0.219E-14, 0.0      )
(-0.161E-14, 0.0      )
( 0.497E-15, 0.120E-14)
( 0.497E-15,-0.120E-14)
(-0.683E-16, 0.975E-15)
(-0.683E-16,-0.975E-15)
(-0.478E-15, 0.405E-15)
(-0.478E-15,-0.405E-15)
( 0.624E-15, 0.0      )
(-0.431E-15, 0.0      )
( 0.147E-15, 0.360E-15)
( 0.147E-15,-0.360E-15)
(-0.183E-15, 0.295E-15)
(-0.183E-15,-0.295E-15)
( 0.317E-15, 0.0      )
( 0.124E-15, 0.232E-15)
( 0.124E-15,-0.232E-15)
( 0.197E-15, 0.0      )
( 0.165E-16, 0.190E-15)
( 0.165E-16,-0.190E-15)
(-0.188E-15, 0.0      )
( 0.104E-15, 0.0      )
( 0.322E-16, 0.971E-16)
( 0.322E-16,-0.971E-16)
(-0.187E-16, 0.857E-16)
(-0.187E-16,-0.857E-16)
(-0.229E-16, 0.0      )
(-0.904E-17, 0.887E-17)
(-0.904E-17,-0.887E-17)
( 0.651E-28, 0.0      )
( 0.264E-29, 0.0      )
( 0.522E-30, 0.0      )
(-0.923E-31, 0.0      )
( 0.121E-31, 0.0      )
( 0.0      , 0.0      )
( 0.0      , 0.0      )
( 0.0      , 0.0      )
( 0.0      , 0.0      )

Condição de Neumann em 3 lados fixando P=1 na entrada (ou na saída)

O espectro tem a maioria dos autovalores próximos do eixo real com alguns autovalores complexos (um pequeno circulo de valores complexos dentro do círculo unitário) com σ ⊂ 11.26 ∪ (-1,1) incluindo 0. 

Incluindo CC na matriz                  Excluindo CC na matriz
lambda(1)=   11.2617565379434           lambda(1)=  0.955274744954830
lambda(n)=  0.000000000000000E+000      lambda(n)=  4.218228659674831E-003
(11.262E+00, 0.0      )                 (-0.955E+00, 0.0      )
( 0.975E+00, 0.0      )                 ( 0.955E+00, 0.0      )
( 0.964E+00, 0.0      )                 (-0.896E+00, 0.0      )
(-0.948E+00, 0.0      )                 ( 0.896E+00, 0.0      )
( 0.932E+00, 0.0      )                 ( 0.884E+00, 0.0      )
( 0.920E+00, 0.0      )                 (-0.884E+00, 0.0      )
(-0.883E+00, 0.0      )                 ( 0.825E+00, 0.0      )
( 0.877E+00, 0.0      )                 (-0.825E+00, 0.0      )
(-0.861E+00, 0.0      )                 (-0.803E+00, 0.0      )
( 0.836E+00, 0.0      )                 ( 0.803E+00, 0.0      )
( 0.834E+00, 0.0      )                 ( 0.774E+00, 0.0      )
(-0.796E+00, 0.0      )                 (-0.774E+00, 0.0      )
( 0.790E+00, 0.0      )                 ( 0.732E+00, 0.0      )
(-0.782E+00, 0.0      )                 (-0.732E+00, 0.0      )
( 0.750E+00, 0.0      )                 ( 0.715E+00, 0.0      )
( 0.750E+00, 0.0      )                 (-0.715E+00, 0.0      )
( 0.744E+00, 0.0      )                 ( 0.683E+00, 0.0      )
(-0.728E+00, 0.0      )                 (-0.683E+00, 0.0      )
( 0.720E+00, 0.0      )                 (-0.634E+00, 0.0      )
( 0.700E+00, 0.0      )                 ( 0.634E+00, 0.0      )
(-0.695E+00, 0.0      )                 ( 0.621E+00, 0.0      )
(-0.663E+00, 0.0      )                 (-0.621E+00, 0.0      )
(-0.653E+00, 0.0      )                 ( 0.612E+00, 0.0      )
( 0.633E+00, 0.0      )                 (-0.612E+00, 0.0      )
( 0.617E+00, 0.0      )                 ( 0.575E+00, 0.0      )
( 0.581E+00, 0.0      )                 (-0.575E+00, 0.0      )
( 0.581E+00, 0.0      )                 ( 0.547E+00, 0.0      )
(-0.566E+00, 0.0      )                 (-0.547E+00, 0.0      )
(-0.565E+00, 0.0      )                 ( 0.502E+00, 0.0      )
(-0.562E+00, 0.0      )                 (-0.502E+00, 0.0      )
( 0.537E+00, 0.0      )                 ( 0.482E+00, 0.0      )
(-0.507E+00, 0.0      )                 (-0.482E+00, 0.0      )
( 0.500E+00, 0.0      )                 ( 0.480E+00, 0.0      )
(-0.500E+00, 0.0      )                 (-0.480E+00, 0.0      )
( 0.498E+00, 0.138E-01)                 ( 0.476E+00, 0.0      )
( 0.498E+00,-0.138E-01)                 (-0.476E+00, 0.0      )
( 0.494E+00, 0.0      )                 ( 0.421E+00, 0.0      )
( 0.453E+00, 0.0      )                 (-0.421E+00, 0.0      )
(-0.433E+00, 0.0      )                 ( 0.404E+00, 0.0      )
( 0.432E+00, 0.0      )                 (-0.404E+00, 0.0      )
(-0.420E+00, 0.0      )                 ( 0.365E+00, 0.0      )
( 0.408E+00, 0.0      )                 (-0.365E+00, 0.0      )
(-0.399E+00, 0.0      )                 ( 0.362E+00, 0.0      )
(-0.391E+00, 0.0      )                 (-0.362E+00, 0.0      )
( 0.364E+00, 0.0      )                 (-0.333E+00, 0.0      )
( 0.361E+00, 0.0      )                 ( 0.333E+00, 0.0      )
(-0.359E+00, 0.0      )                 ( 0.327E+00, 0.0      )
( 0.346E+00, 0.0      )                 (-0.327E+00, 0.0      )
( 0.337E+00, 0.0      )                 (-0.325E+00, 0.0      )
(-0.326E+00, 0.0      )                 ( 0.325E+00, 0.0      )
(-0.287E+00, 0.0      )                 (-0.268E+00, 0.0      )
( 0.287E+00, 0.0      )                 ( 0.268E+00, 0.0      )
( 0.280E+00, 0.0      )                 (-0.266E+00, 0.0      )
(-0.272E+00, 0.0      )                 ( 0.266E+00, 0.0      )
(-0.269E+00, 0.0      )                 ( 0.226E+00, 0.0      )
( 0.247E+00, 0.0      )                 (-0.226E+00, 0.0      )
( 0.244E+00, 0.0      )                 ( 0.223E+00, 0.0      )
(-0.228E+00, 0.0      )                 (-0.223E+00, 0.0      )
(-0.225E+00, 0.0      )                 (-0.208E+00, 0.0      )
(-0.220E+00, 0.0      )                 ( 0.208E+00, 0.0      )
( 0.213E+00, 0.0      )                 ( 0.197E+00, 0.0      )
( 0.207E+00, 0.0      )                 (-0.197E+00, 0.0      )
( 0.200E+00, 0.0      )                 ( 0.186E+00, 0.0      )
( 0.200E+00, 0.0      )                 (-0.186E+00, 0.0      )
( 0.198E+00, 0.0      )                 ( 0.173E+00, 0.0      )
( 0.163E+00, 0.0      )                 (-0.173E+00, 0.0      )
(-0.163E+00, 0.0      )                 ( 0.148E+00, 0.0      )
( 0.158E+00, 0.0      )                 (-0.148E+00, 0.0      )
(-0.139E+00, 0.0      )                 ( 0.127E+00, 0.0      )
(-0.133E+00, 0.0      )                 (-0.127E+00, 0.0      )
(-0.124E+00, 0.0      )                 (-0.862E-01, 0.0      )
( 0.117E+00, 0.0      )                 ( 0.862E-01, 0.0      )
( 0.111E+00, 0.0      )                 (-0.834E-01, 0.0      )
(-0.103E+00, 0.0      )                 ( 0.834E-01, 0.0      )
(-0.958E-01, 0.0      )                 (-0.771E-01, 0.0      )
(-0.948E-01, 0.0      )                 ( 0.771E-01, 0.0      )
( 0.713E-01, 0.0      )                 (-0.752E-01, 0.0      )
( 0.675E-01, 0.0      )                 ( 0.752E-01, 0.0      )
( 0.673E-01, 0.0      )                 (-0.712E-01, 0.0      )
( 0.668E-01, 0.0      )                 ( 0.712E-01, 0.0      )
(-0.617E-01, 0.0      )                 ( 0.549E-01, 0.0      )
( 0.567E-01, 0.0      )                 (-0.549E-01, 0.0      )
( 0.495E-01, 0.0      )                 (-0.532E-01, 0.0      )
(-0.301E-01, 0.0      )                 ( 0.532E-01, 0.0      )
( 0.246E-01, 0.0      )                 ( 0.335E-01, 0.0      )
( 0.245E-01, 0.0      )                 (-0.335E-01, 0.0      )
( 0.244E-01, 0.0      )                 ( 0.161E-01, 0.0      )
(-0.194E-01, 0.0      )                 (-0.161E-01, 0.0      )
(-0.165E-01, 0.0      )                 ( 0.422E-02, 0.0      )
(-0.794E-02, 0.0      )                 (-0.422E-02, 0.0      )
(-0.568E-08, 0.0      )
( 0.568E-08, 0.0      )
( 0.308E-14, 0.0      )
(-0.178E-14, 0.136E-14)
(-0.178E-14,-0.136E-14)
(-0.126E-14, 0.0      )
( 0.882E-15, 0.828E-15)
( 0.882E-15,-0.828E-15)
( 0.518E-15, 0.109E-14)
( 0.518E-15,-0.109E-14)
(-0.466E-16, 0.552E-15)
(-0.466E-16,-0.552E-15)
( 0.508E-15, 0.0      )
(-0.336E-15, 0.361E-15)
(-0.336E-15,-0.361E-15)
( 0.154E-15, 0.417E-15)
( 0.154E-15,-0.417E-15)
(-0.333E-15, 0.0      )
(-0.275E-15, 0.112E-15)
(-0.275E-15,-0.112E-15)
( 0.156E-15, 0.208E-15)
( 0.156E-15,-0.208E-15)
( 0.253E-16, 0.658E-16)
( 0.253E-16,-0.658E-16)
( 0.666E-16, 0.0      )
(-0.536E-16, 0.0      )
(-0.205E-16, 0.0      )
(-0.119E-16, 0.0      )
( 0.259E-18, 0.106E-16)
( 0.259E-18,-0.106E-16)
(-0.216E-29, 0.0      )
( 0.171E-29, 0.0      )
( 0.357E-30, 0.914E-31)
( 0.357E-30,-0.914E-31)
(-0.246E-30, 0.0      )
(-0.204E-31, 0.0      )
(-0.392E-32, 0.313E-32)
(-0.392E-32,-0.313E-32)
( 0.0      , 0.0      )
( 0.0      , 0.0      )
( 0.0      , 0.0      )
( 0.0      , 0.0      )

Teste 5

Disponível em "http://hunter.mat.ufrgs.br/mediawiki/index.php?title=Análise_espectral_e_estabilidade&oldid=1671"