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A análise dos autovalores de uma matriz de iteração pode ser usada para estudar a estabilidade de um método iterativo.
 
A análise dos autovalores de uma matriz de iteração pode ser usada para estudar a estabilidade de um método iterativo.
  
 +
 +
<math>\frac{\partial^2 p}{\partial x^2}+\frac{\partial^2 p}{\partial y^2}= f(u)\,\!</math>
  
 
==O problema==
 
==O problema==

Edição das 14h40min de 10 de maio de 2010

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


<math>\frac{\partial^2 p}{\partial x^2}+\frac{\partial^2 p}{\partial y^2}= f(u)\,\!</math>

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.
  • Para montar a matriz de iteração corretamente usamos como condição de Dirichlet
 p(contorno) = p0
e dentro do loop de iteração temos
 pnew(contorno)= p(contorno)
ou seja, p continua sempre fixo.

Método Iterativo

Para solucionar o sistema

<math>M p = f</math>

usaremos um método iterativo da forma

<math> p^{(k+1)} = G p^{(k)}+g </math>

Para que esse método seja convergente é necessário que o raio espectral de G seja estritamente menor do que 1,

<math> \rho(G) < 1</math>

Método de Jacobi

Tomando

<math>M=D+L+U</math>

o sistema pode ser reescrito como

<math>D p = -(L+U)p +f </math>
<math> p^{(k+1)} = - D^{-1}(L+U) p^{(k)}+D^{-1}f </math>
<math> p^{(k+1)} = G p^{(k)}+D^{-1}f </math>

onde

<math> G = - D^{-1}(L+U)</math>

Método de Gauss-Seidel

Tomando

<math>M=D+L+U</math>

o sistema pode ser reescrito como

<math>(D+L)p = -Up +f </math>
<math> p^{(k+1)} = - (D+L)^{-1}U p^{(k)}+(D+L)^{-1}f </math>
<math> p^{(k+1)} = G p^{(k)}+(D+L)^{-1}f </math>

onde

<math> G_{GS} = - (D+L)^{-1}U</math>

Método SOR

Usando o método de Gauss-Seidel como base, temos

<math> p^{(k+1)} =(1-w)p^{(k)}+w( G p^{(k)}+(D+L)^{-1}f) </math>

onde

<math> G_{SOR} = (1-w)I+wG = I + w(G-I)</math>

Comparação

Comparação do número de iterações para a convergência dos métodos de Jacobi, Gauss-Seidel e SOR (com ω=1.7) para a equação de Poisson com [m, n]=[11, 11]

Configuração A
  • Condição de contorno de Neumann
  • pnew=pnew-p(1,1)
    • |λ[1]|<1
  • int(f,dv) não é zero (não é divergente free)
Configuração B
  • Condição de contorno de Neumann
    • |λ[1]|=1
  • int(f,dv)=zero (é divergente free)

Os gráficos abaixo são para a configuração A, porém com a configuração B temos o mesmo comportamento. SorxGaussSeidelxJacobi1.jpg SorxGaussSeidelxJacobi.jpg

Condição de Dirichlet em todos os lados

Antes de entrar no loop fixamos

 p( :, 1)= p_sul
 p( :,ny)= p_norte
 p( 1, :)= p_oeste
 p(nx, :)= p_leste

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
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
(-0.955    , 0        )                 (-0.955    , 0        )
( 0.955    , 0        )                 ( 0.955    , 0        )
( 0.896    , 0        )                 (-0.896    , 0        )
(-0.896    , 0        )                 ( 0.896    , 0        )
(-0.884    , 0        )                 ( 0.884    , 0        )
( 0.884    , 0        )                 (-0.884    , 0        )
( 0.825    , 0        )                 ( 0.825    , 0        )
(-0.825    , 0        )                 (-0.825    , 0        )
( 0.803    , 0        )                 (-0.803    , 0        )
(-0.803    , 0        )                 ( 0.803    , 0        )
(-0.774    , 0        )                 ( 0.774    , 0        )
( 0.774    , 0        )                 (-0.774    , 0        )
( 0.732    , 0        )                 ( 0.732    , 0        )
(-0.732    , 0        )                 (-0.732    , 0        )
(-0.715    , 0        )                 ( 0.715    , 0        )
( 0.715    , 0        )                 (-0.715    , 0        )
( 0.683    , 0        )                 ( 0.683    , 0        )
(-0.683    , 0        )                 (-0.683    , 0        )
(-0.634    , 0        )                 (-0.634    , 0        )
( 0.634    , 0        )                 ( 0.634    , 0        )
( 0.621    , 0        )                 ( 0.621    , 0        )
(-0.621    , 0        )                 (-0.621    , 0        )
(-0.612    , 0        )                 ( 0.612    , 0        )
( 0.612    , 0        )                 (-0.612    , 0        )
( 0.575    , 0        )                 ( 0.575    , 0        )
(-0.575    , 0        )                 (-0.575    , 0        )
(-0.547    , 0        )                 ( 0.547    , 0        )
( 0.547    , 0        )                 (-0.547    , 0        )
( 0.502    , 0        )                 ( 0.502    , 0        )
(-0.502    , 0        )                 (-0.502    , 0        )
(-0.482    , 0        )                 ( 0.482    , 0        )
( 0.482    , 0        )                 (-0.482    , 0        )
(-0.480    , 0        )                 ( 0.480    , 0        )
( 0.480    , 0        )                 (-0.480    , 0        )
(-0.476    , 0        )                 ( 0.476    , 0        )
( 0.476    , 0        )                 (-0.476    , 0        )
( 0.421    , 0        )                 ( 0.421    , 0        )
(-0.421    , 0        )                 (-0.421    , 0        )
(-0.404    , 0        )                 ( 0.404    , 0        )
( 0.404    , 0        )                 (-0.404    , 0        )
(-0.365    , 0        )                 ( 0.365    , 0        )
( 0.365    , 0        )                 (-0.365    , 0        )
( 0.362    , 0        )                 ( 0.362    , 0        )
(-0.362    , 0        )                 (-0.362    , 0        )
( 0.333    , 0        )                 (-0.333    , 0        )
(-0.333    , 0        )                 ( 0.333    , 0        )
(-0.327    , 0        )                 ( 0.327    , 0        )
( 0.327    , 0        )                 (-0.327    , 0        )
( 0.325    , 0        )                 (-0.325    , 0        )
(-0.325    , 0        )                 ( 0.325    , 0        )
( 0.268    , 0        )                 (-0.268    , 0        )
(-0.268    , 0        )                 ( 0.268    , 0        )
( 0.266    , 0        )                 (-0.266    , 0        )
(-0.266    , 0        )                 ( 0.266    , 0        )
( 0.226    , 0        )                 ( 0.226    , 0        )
(-0.226    , 0        )                 (-0.226    , 0        )
(-0.223    , 0        )                 ( 0.223    , 0        )
( 0.223    , 0        )                 (-0.223    , 0        )
(-0.208    , 0        )                 (-0.208    , 0        )
( 0.208    , 0        )                 ( 0.208    , 0        )
(-0.197    , 0        )                 ( 0.197    , 0        )
( 0.197    , 0        )                 (-0.197    , 0        )
(-0.186    , 0        )                 ( 0.186    , 0        )
( 0.186    , 0        )                 (-0.186    , 0        )
( 0.173    , 0        )                 ( 0.173    , 0        )
(-0.173    , 0        )                 (-0.173    , 0        )
( 0.148    , 0        )                 ( 0.148    , 0        )
(-0.148    , 0        )                 (-0.148    , 0        )
(-0.127    , 0        )                 ( 0.127    , 0        )
( 0.127    , 0        )                 (-0.127    , 0        )
(-0.862E-01, 0        )                 (-0.862E-01, 0        )
( 0.862E-01, 0        )                 ( 0.862E-01, 0        )
(-0.834E-01, 0        )                 (-0.834E-01, 0        )
( 0.834E-01, 0        )                 ( 0.834E-01, 0        )
( 0.771E-01, 0        )                 (-0.771E-01, 0        )
(-0.771E-01, 0        )                 ( 0.771E-01, 0        )
( 0.752E-01, 0        )                 (-0.752E-01, 0        )
(-0.752E-01, 0        )                 ( 0.752E-01, 0        )
(-0.712E-01, 0        )                 (-0.712E-01, 0        )
( 0.712E-01, 0        )                 ( 0.712E-01, 0        )
(-0.549E-01, 0        )                 ( 0.549E-01, 0        )
( 0.549E-01, 0        )                 (-0.549E-01, 0        )
( 0.532E-01, 0        )                 (-0.532E-01, 0        )
(-0.532E-01, 0        )                 ( 0.532E-01, 0        )
(-0.335E-01, 0        )                 ( 0.335E-01, 0        )
( 0.335E-01, 0        )                 (-0.335E-01, 0        )
(-0.161E-01, 0        )                 ( 0.161E-01, 0        )
( 0.161E-01, 0        )                 (-0.161E-01, 0        )
( 0.422E-02, 0        )                 ( 0.422E-02, 0        )
(-0.422E-02, 0        )                 (-0.422E-02, 0        )

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)=  0.000000000000000    0     lambda(n)=  4.218228659674831E-003
      Eigenvalues                            Eigenvalues
( 1        , 0        )                (-0.955    , 0        )
( 0.976    , 0        )                ( 0.955    , 0        )
( 0.970    , 0        )                (-0.896    , 0        )
(-0.945    , 0        )                ( 0.896    , 0        )
( 0.945    , 0        )                ( 0.884    , 0        )
( 0.905    , 0        )                (-0.884    , 0        )
( 0.883    , 0        )                ( 0.825    , 0        )
(-0.874    , 0        )                (-0.825    , 0        )
( 0.874    , 0        )                (-0.803    , 0        )
( 0.859    , 0        )                ( 0.803    , 0        )
(-0.859    , 0        )                ( 0.774    , 0        )
( 0.794    , 0        )                (-0.774    , 0        )
(-0.788    , 0        )                ( 0.732    , 0        )
( 0.788    , 0        )                (-0.732    , 0        )
( 0.764    , 0        )                ( 0.715    , 0        )
(-0.764    , 0        )                (-0.715    , 0        )
( 0.750    , 0        )                ( 0.683    , 0        )
(-0.726    , 0        )                (-0.683    , 0        )
( 0.726    , 0        )                (-0.634    , 0        )
( 0.677    , 0        )                ( 0.634    , 0        )
(-0.677    , 0        )                ( 0.621    , 0        )
(-0.655    , 0        )                (-0.621    , 0        )
( 0.655    , 0        )                ( 0.612    , 0        )
( 0.655    , 0        )                (-0.612    , 0        )
( 0.624    , 0        )                ( 0.575    , 0        )
(-0.624    , 0        )                (-0.575    , 0        )
( 0.587    , 0        )                ( 0.547    , 0        )
(-0.562    , 0        )                (-0.547    , 0        )
( 0.562    , 0        )                ( 0.502    , 0        )
(-0.544    , 0        )                (-0.502    , 0        )
( 0.544    , 0        )                ( 0.482    , 0        )
( 0.538    , 0        )                (-0.482    , 0        )
(-0.538    , 0        )                ( 0.480    , 0        )
( 0.500    , 0        )                (-0.480    , 0        )
( 0.491    , 0        )                ( 0.476    , 0        )
(-0.491    , 0        )                (-0.476    , 0        )
(-0.470    , 0        )                ( 0.421    , 0        )
( 0.470    , 0        )                (-0.421    , 0        )
( 0.413    , 0        )                ( 0.404    , 0        )
(-0.405    , 0        )                (-0.404    , 0        )
( 0.405    , 0        )                ( 0.365    , 0        )
(-0.389    , 0        )                (-0.365    , 0        )
( 0.389    , 0        )                ( 0.362    , 0        )
(-0.383    , 0        )                (-0.362    , 0        )
( 0.383    , 0        )                (-0.333    , 0        )
(-0.381    , 0        )                ( 0.333    , 0        )
( 0.381    , 0        )                ( 0.327    , 0        )
( 0.345    , 0        )                (-0.327    , 0        )
(-0.318    , 0        )                (-0.325    , 0        )
( 0.318    , 0        )                ( 0.325    , 0        )
( 0.315    , 0        )                (-0.268    , 0        )
(-0.315    , 0        )                ( 0.268    , 0        )
(-0.250    , 0        )                (-0.266    , 0        )
( 0.250    , 0        )                ( 0.266    , 0        )
( 0.250    , 0        )                ( 0.226    , 0        )
(-0.241    , 0        )                (-0.226    , 0        )
( 0.241    , 0        )                ( 0.223    , 0        )
(-0.229    , 0        )                (-0.223    , 0        )
( 0.229    , 0        )                (-0.208    , 0        )
(-0.226    , 0        )                ( 0.208    , 0        )
( 0.226    , 0        )                ( 0.197    , 0        )
( 0.207    , 0        )                (-0.197    , 0        )
(-0.207    , 0        )                ( 0.186    , 0        )
( 0.206    , 0        )                (-0.186    , 0        )
( 0.176    , 0        )                ( 0.173    , 0        )
(-0.176    , 0        )                (-0.173    , 0        )
(-0.155    , 0        )                ( 0.148    , 0        )
( 0.155    , 0        )                (-0.148    , 0        )
( 0.117    , 0        )                ( 0.127    , 0        )
( 0.955E-01, 0        )                (-0.127    , 0        )
( 0.955E-01, 0        )                (-0.862E-01, 0        )
(-0.955E-01, 0        )                ( 0.862E-01, 0        )
(-0.925E-01, 0        )                (-0.834E-01, 0        )
( 0.925E-01, 0        )                ( 0.834E-01, 0        )
(-0.891E-01, 0        )                (-0.771E-01, 0        )
( 0.891E-01, 0        )                ( 0.771E-01, 0        )
(-0.868E-01, 0        )                (-0.752E-01, 0        )
( 0.868E-01, 0        )                ( 0.752E-01, 0        )
( 0.677E-01, 0        )                (-0.712E-01, 0        )
(-0.677E-01, 0        )                ( 0.712E-01, 0        )
(-0.653E-01, 0        )                ( 0.549E-01, 0        )
( 0.653E-01, 0        )                (-0.549E-01, 0        )
( 0.439E-01, 0        )                (-0.532E-01, 0        )
(-0.439E-01, 0        )                ( 0.532E-01, 0        )
( 0.302E-01, 0        )                ( 0.335E-01, 0        )
( 0.245E-01, 0        )                (-0.335E-01, 0        )
( 0.215E-01, 0        )                ( 0.161E-01, 0        )
(-0.215E-01, 0        )                (-0.161E-01, 0        )
(-0.568E-02, 0        )                ( 0.422E-02, 0        )
( 0.568E-02, 0        )                (-0.422E-02, 0        )
( 0.251E-14, 0.857E-15)
( 0.251E-14,-0.857E-15)
(-0.255E-14, 0        )
( 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        )
(-0.772E-15, 0        )
( 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        )
(-0.433E-15, 0        )
(-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        )
(-0.230E-16, 0        )
(-0.824E-30, 0.885E-30)
(-0.824E-30,-0.885E-30)
( 0.680E-30, 0        )
(-0.684E-31, 0        )
( 0.317E-31, 0        )
( 0        , 0        )
( 0        , 0        )
( 0        , 0        )
( 0        , 0        )
( 0        , 0        )
( 0        , 0        )

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

Usando

 Pnew=Pnew -P(1,2)

temos o espectro abaixo.

 Incluindo CC na Matriz                 Excluindo CC na Matriz
lambda(1)=  0.975528258147575          lambda(1)=  0.955274744954828
lambda(n)=  0.000000000000000    0     lambda(n)=  0.000000000000000    0
( 0.976    , 0.0      )     (-0.955    , 0.0      )
( 0.970    , 0.0      )     ( 0.911    , 0.0      )
( 0.945    , 0.0      )     ( 0.896    , 0.0      )
(-0.945    , 0.0      )     (-0.896    , 0.0      )
( 0.905    , 0.0      )     ( 0.884    , 0.0      )
( 0.883    , 0.0      )     (-0.884    , 0.0      )
(-0.874    , 0.0      )     (-0.825    , 0.0      )
( 0.874    , 0.0      )     ( 0.825    , 0.0      )
( 0.859    , 0.0      )     (-0.803    , 0.0      )
(-0.859    , 0.0      )     ( 0.789    , 0.0      )
( 0.794    , 0.0      )     (-0.774    , 0.0      )
( 0.788    , 0.0      )     (-0.732    , 0.0      )
(-0.788    , 0.0      )     ( 0.732    , 0.0      )
( 0.764    , 0.0      )     ( 0.715    , 0.0      )
(-0.764    , 0.0      )     (-0.712    , 0.0      )
( 0.750    , 0.0      )     ( 0.693    , 0.0      )
(-0.726    , 0.0      )     ( 0.683    , 0.0      )
( 0.726    , 0.0      )     (-0.682    , 0.0      )
(-0.677    , 0.0      )     (-0.634    , 0.0      )
( 0.677    , 0.0      )     ( 0.634    , 0.0      )
( 0.655    , 0.0      )     (-0.621    , 0.0      )
(-0.655    , 0.0      )     (-0.612    , 0.0      )
( 0.655    , 0.0      )     ( 0.612    , 0.0      )
(-0.624    , 0.0      )     ( 0.583    , 0.0      )
( 0.624    , 0.0      )     (-0.575    , 0.0      )
( 0.587    , 0.0      )     ( 0.575    , 0.0      )
( 0.562    , 0.0      )     (-0.547    , 0.0      )
(-0.562    , 0.0      )     ( 0.507    , 0.0      )
( 0.544    , 0.0      )     ( 0.502    , 0.0      )
(-0.544    , 0.0      )     (-0.494    , 0.0      )
(-0.538    , 0.0      )     ( 0.482    , 0.0      )
( 0.538    , 0.0      )     (-0.482    , 0.0      )
( 0.500    , 0.0      )     (-0.480    , 0.0      )
( 0.491    , 0.0      )     (-0.476    , 0.0      )
(-0.491    , 0.0      )     ( 0.476    , 0.0      )
(-0.470    , 0.0      )     ( 0.421    , 0.0      )
( 0.470    , 0.0      )     (-0.416    , 0.0      )
( 0.413    , 0.0      )     ( 0.412    , 0.0      )
(-0.405    , 0.0      )     ( 0.404    , 0.0      )
( 0.405    , 0.0      )     (-0.402    , 0.0      )
(-0.389    , 0.0      )     (-0.365    , 0.0      )
( 0.389    , 0.0      )     (-0.362    , 0.0      )
( 0.383    , 0.0      )     ( 0.362    , 0.0      )
(-0.383    , 0.0      )     ( 0.343    , 0.0      )
(-0.381    , 0.0      )     ( 0.333    , 0.0      )
( 0.381    , 0.0      )     (-0.333    , 0.0      )
( 0.345    , 0.0      )     (-0.327    , 0.0      )
( 0.318    , 0.0      )     (-0.325    , 0.0      )
(-0.318    , 0.0      )     ( 0.325    , 0.0      )
( 0.315    , 0.0      )     ( 0.283    , 0.0      )
(-0.315    , 0.0      )     (-0.268    , 0.0      )
( 0.250    , 0.0      )     ( 0.266    , 0.0      )
(-0.250    , 0.0      )     (-0.266    , 0.0      )
( 0.250    , 0.0      )     (-0.226    , 0.0      )
( 0.241    , 0.0      )     ( 0.226    , 0.0      )
(-0.241    , 0.0      )     ( 0.223    , 0.0      )
(-0.229    , 0.0      )     (-0.218    , 0.0      )
( 0.229    , 0.0      )     ( 0.208    , 0.0      )
(-0.226    , 0.0      )     ( 0.202    , 0.0      )
( 0.226    , 0.0      )     (-0.197    , 0.0      )
( 0.207    , 0.0      )     ( 0.197    , 0.0      )
(-0.207    , 0.0      )     (-0.190    , 0.0      )
( 0.206    , 0.0      )     (-0.186    , 0.0      )
( 0.176    , 0.0      )     (-0.173    , 0.0      )
(-0.176    , 0.0      )     ( 0.173    , 0.0      )
( 0.155    , 0.0      )     ( 0.148    , 0.0      )
(-0.155    , 0.0      )     (-0.146    , 0.0      )
( 0.117    , 0.0      )     ( 0.127    , 0.0      )
(-0.955E-01, 0.0      )     ( 0.122    , 0.0      )
( 0.955E-01, 0.411E-15)     (-0.120    , 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

Usando CC de Neumann e fixando em um ponto

 Pnew(1,2)=P(1,2)

Temos o espectro abaixo.

 lambda(1)=   1.00000000000000         lambda(1)=  0.955274744954830
 lambda(n)=  0.000000000000000    0    lambda(n)=  4.218228659674831E-003
      Eigenvalues                           Eigenvalues
( 1        , 0        )               (-0.955    , 0        )
( 0.999    , 0        )               ( 0.955    , 0        )
( 0.974    , 0        )               (-0.896    , 0        )
( 0.967    , 0        )               ( 0.896    , 0        )
(-0.945    , 0        )               ( 0.884    , 0        )
( 0.939    , 0        )               (-0.884    , 0        )
( 0.902    , 0        )               ( 0.825    , 0        )
( 0.881    , 0        )               (-0.825    , 0        )
(-0.874    , 0        )               (-0.803    , 0        )
( 0.869    , 0        )               ( 0.803    , 0        )
(-0.859    , 0        )               ( 0.774    , 0        )
( 0.849    , 0        )               (-0.774    , 0        )
( 0.793    , 0        )               ( 0.732    , 0        )
(-0.788    , 0        )               (-0.732    , 0        )
( 0.782    , 0        )               ( 0.715    , 0        )
(-0.764    , 0        )               (-0.715    , 0        )
( 0.758    , 0        )               ( 0.683    , 0        )
( 0.746    , 0        )               (-0.683    , 0        )
(-0.726    , 0        )               (-0.634    , 0        )
( 0.718    , 0        )               ( 0.634    , 0        )
(-0.677    , 0        )               ( 0.621    , 0        )
( 0.672    , 0        )               (-0.621    , 0        )
(-0.655    , 0        )               ( 0.612    , 0        )
( 0.655    , 0        )               (-0.612    , 0        )
( 0.645    , 0        )               ( 0.575    , 0        )
(-0.625    , 0        )               (-0.575    , 0        )
( 0.617    , 0        )               ( 0.547    , 0        )
( 0.584    , 0        )               (-0.547    , 0        )
(-0.563    , 0        )               ( 0.502    , 0        )
( 0.558    , 0        )               (-0.502    , 0        )
(-0.545    , 0        )               ( 0.482    , 0        )
( 0.541    , 0        )               (-0.482    , 0        )
(-0.538    , 0        )               ( 0.480    , 0        )
( 0.527    , 0        )               (-0.480    , 0        )
( 0.498    , 0        )               ( 0.476    , 0        )
(-0.492    , 0        )               (-0.476    , 0        )
( 0.485    , 0        )               ( 0.421    , 0        )
(-0.470    , 0        )               (-0.421    , 0        )
( 0.462    , 0        )               ( 0.404    , 0        )
( 0.412    , 0        )               (-0.404    , 0        )
(-0.406    , 0        )               ( 0.365    , 0        )
( 0.401    , 0        )               (-0.365    , 0        )
(-0.389    , 0        )               ( 0.362    , 0        )
( 0.387    , 0        )               (-0.362    , 0        )
(-0.384    , 0        )               (-0.333    , 0        )
( 0.382    , 0        )               ( 0.333    , 0        )
(-0.382    , 0        )               ( 0.327    , 0        )
( 0.366    , 0        )               (-0.327    , 0        )
( 0.343    , 0        )               (-0.325    , 0        )
(-0.319    , 0        )               ( 0.325    , 0        )
( 0.317    , 0        )               (-0.268    , 0        )
(-0.316    , 0        )               ( 0.268    , 0        )
( 0.306    , 0        )               (-0.266    , 0        )
(-0.253    , 0        )               ( 0.266    , 0        )
( 0.250    , 0        )               ( 0.226    , 0        )
( 0.247    , 0        )               (-0.226    , 0        )
(-0.243    , 0        )               ( 0.223    , 0        )
( 0.237    , 0        )               (-0.223    , 0        )
(-0.230    , 0        )               (-0.208    , 0        )
( 0.227    , 0        )               ( 0.208    , 0        )
(-0.226    , 0        )               ( 0.197    , 0        )
( 0.219    , 0        )               (-0.197    , 0        )
(-0.209    , 0        )               ( 0.186    , 0        )
( 0.206    , 0        )               (-0.186    , 0        )
( 0.197    , 0        )               ( 0.173    , 0        )
(-0.176    , 0        )               (-0.173    , 0        )
( 0.172    , 0        )               ( 0.148    , 0        )
(-0.156    , 0        )               (-0.148    , 0        )
( 0.150    , 0        )               ( 0.127    , 0        )
( 0.116    , 0        )               (-0.127    , 0        )
(-0.102    , 0        )               (-0.862E-01, 0        )
( 0.955E-01, 0        )               ( 0.862E-01, 0        )
( 0.941E-01, 0        )               (-0.834E-01, 0        )
(-0.930E-01, 0        )               ( 0.834E-01, 0        )
( 0.916E-01, 0        )               (-0.771E-01, 0        )
(-0.911E-01, 0        )               ( 0.771E-01, 0        )
( 0.882E-01, 0        )               (-0.752E-01, 0        )
(-0.881E-01, 0        )               ( 0.752E-01, 0        )
( 0.795E-01, 0        )               (-0.712E-01, 0        )
(-0.708E-01, 0        )               ( 0.712E-01, 0        )
( 0.660E-01, 0        )               ( 0.549E-01, 0        )
(-0.655E-01, 0        )               (-0.549E-01, 0        )
( 0.587E-01, 0        )               (-0.532E-01, 0        )
(-0.463E-01, 0        )               ( 0.532E-01, 0        )
( 0.389E-01, 0        )               ( 0.335E-01, 0        )
( 0.298E-01, 0        )               (-0.335E-01, 0        )
( 0.243E-01, 0        )               ( 0.161E-01, 0        )
(-0.229E-01, 0        )               (-0.161E-01, 0        )
( 0.186E-01, 0        )               ( 0.422E-02, 0        )
(-0.615E-02, 0        )               (-0.422E-02, 0        )
( 0.500E-02, 0        )
(-0.799E-14, 0        )
( 0.595E-14, 0        )
(-0.228E-14, 0        )
(-0.173E-14, 0.900E-15)
(-0.173E-14,-0.900E-15)
( 0.176E-14, 0        )
( 0.906E-15, 0.488E-15)
( 0.906E-15,-0.488E-15)
(-0.669E-15, 0.726E-15)
(-0.669E-15,-0.726E-15)
( 0.756E-15, 0        )
(-0.694E-15, 0        )
(-0.592E-16, 0.509E-15)
(-0.592E-16,-0.509E-15)
( 0.451E-15, 0.209E-15)
( 0.451E-15,-0.209E-15)
( 0.995E-16, 0.355E-15)
( 0.995E-16,-0.355E-15)
(-0.226E-15, 0.206E-15)
(-0.226E-15,-0.206E-15)
(-0.240E-15, 0        )
( 0.157E-15, 0        )
( 0.110E-16, 0.108E-15)
( 0.110E-16,-0.108E-15)
(-0.407E-16, 0.954E-16)
(-0.407E-16,-0.954E-16)
( 0.798E-16, 0.494E-16)
( 0.798E-16,-0.494E-16)
(-0.372E-16, 0        )
( 0.172E-17, 0.943E-17)
( 0.172E-17,-0.943E-17)
( 0.454E-26, 0        )
(-0.800E-30, 0.213E-30)
(-0.800E-30,-0.213E-30)
( 0.683E-30, 0        )
(-0.556E-31, 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)

Fixando a entrada oeste com

 Pnew(1,:)=P(1,:)

e o resto com CC de Neumann temos o espectro

 lambda(1)=   1.00000000000000         lambda(1)=  0.955274744954830
 lambda(n)=  0.000000000000000    0    lambda(n)=  4.218228659674831E-003
      Eigenvalues                           Eigenvalues
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 1        , 0        )
( 0.994    , 0        )               (-0.955    , 0        )
( 0.964    , 0        )               ( 0.955    , 0        )
( 0.950    , 0        )               (-0.896    , 0        )
(-0.948    , 0        )               ( 0.896    , 0        )
( 0.920    , 0        )               ( 0.884    , 0        )
(-0.883    , 0        )               (-0.884    , 0        )
( 0.877    , 0        )               ( 0.825    , 0        )
( 0.867    , 0        )               (-0.825    , 0        )
(-0.861    , 0        )               (-0.803    , 0        )
( 0.836    , 0        )               ( 0.803    , 0        )
( 0.834    , 0        )               ( 0.774    , 0        )
(-0.796    , 0        )               (-0.774    , 0        )
(-0.782    , 0        )               ( 0.732    , 0        )
( 0.750    , 0        )               (-0.732    , 0        )
( 0.750    , 0        )               ( 0.715    , 0        )
( 0.744    , 0        )               (-0.715    , 0        )
(-0.728    , 0        )               ( 0.683    , 0        )
( 0.720    , 0        )               (-0.683    , 0        )
( 0.700    , 0        )               (-0.634    , 0        )
(-0.695    , 0        )               ( 0.634    , 0        )
(-0.663    , 0        )               ( 0.621    , 0        )
(-0.653    , 0        )               (-0.621    , 0        )
( 0.633    , 0        )               ( 0.612    , 0        )
( 0.617    , 0        )               (-0.612    , 0        )
( 0.611    , 0        )               ( 0.575    , 0        )
( 0.581    , 0        )               (-0.575    , 0        )
( 0.581    , 0        )               ( 0.547    , 0        )
(-0.566    , 0        )               (-0.547    , 0        )
(-0.565    , 0        )               ( 0.502    , 0        )
(-0.562    , 0        )               (-0.502    , 0        )
( 0.537    , 0        )               ( 0.482    , 0        )
(-0.507    , 0        )               (-0.482    , 0        )
( 0.500    , 0        )               ( 0.480    , 0        )
(-0.500    , 0        )               (-0.480    , 0        )
( 0.494    , 0        )               ( 0.476    , 0        )
( 0.463    , 0        )               (-0.476    , 0        )
( 0.453    , 0        )               ( 0.421    , 0        )
(-0.433    , 0        )               (-0.421    , 0        )
( 0.432    , 0        )               ( 0.404    , 0        )
(-0.420    , 0        )               (-0.404    , 0        )
( 0.408    , 0        )               ( 0.365    , 0        )
(-0.399    , 0        )               (-0.365    , 0        )
(-0.391    , 0        )               ( 0.362    , 0        )
( 0.364    , 0        )               (-0.362    , 0        )
( 0.361    , 0        )               (-0.333    , 0        )
(-0.359    , 0        )               ( 0.333    , 0        )
( 0.346    , 0        )               ( 0.327    , 0        )
( 0.337    , 0        )               (-0.327    , 0        )
(-0.326    , 0        )               (-0.325    , 0        )
( 0.317    , 0        )               ( 0.325    , 0        )
(-0.287    , 0        )               (-0.268    , 0        )
( 0.287    , 0        )               ( 0.268    , 0        )
( 0.280    , 0        )               (-0.266    , 0        )
(-0.272    , 0        )               ( 0.266    , 0        )
(-0.269    , 0        )               ( 0.226    , 0        )
( 0.244    , 0        )               (-0.226    , 0        )
(-0.228    , 0        )               ( 0.223    , 0        )
(-0.225    , 0        )               (-0.223    , 0        )
(-0.220    , 0        )               (-0.208    , 0        )
( 0.213    , 0        )               ( 0.208    , 0        )
( 0.200    , 0        )               ( 0.197    , 0        )
( 0.200    , 0        )               (-0.197    , 0        )
( 0.198    , 0        )               ( 0.186    , 0        )
( 0.188    , 0        )               (-0.186    , 0        )
( 0.163    , 0        )               ( 0.173    , 0        )
(-0.163    , 0        )               (-0.173    , 0        )
( 0.158    , 0        )               ( 0.148    , 0        )
(-0.139    , 0        )               (-0.148    , 0        )
(-0.133    , 0        )               ( 0.127    , 0        )
(-0.124    , 0        )               (-0.127    , 0        )
( 0.117    , 0        )               (-0.862E-01, 0        )
( 0.111    , 0        )               ( 0.862E-01, 0        )
(-0.103    , 0        )               (-0.834E-01, 0        )
(-0.958E-01, 0        )               ( 0.834E-01, 0        )
(-0.948E-01, 0        )               (-0.771E-01, 0        )
( 0.869E-01, 0        )               ( 0.771E-01, 0        )
( 0.713E-01, 0        )               (-0.752E-01, 0        )
( 0.675E-01, 0        )               ( 0.752E-01, 0        )
( 0.673E-01, 0        )               (-0.712E-01, 0        )
(-0.617E-01, 0        )               ( 0.712E-01, 0        )
( 0.567E-01, 0        )               ( 0.549E-01, 0        )
( 0.495E-01, 0        )               (-0.549E-01, 0        )
(-0.301E-01, 0        )               (-0.532E-01, 0        )
( 0.246E-01, 0        )               ( 0.532E-01, 0        )
( 0.244E-01, 0        )               ( 0.335E-01, 0        )
( 0.222E-01, 0        )               (-0.335E-01, 0        )
(-0.194E-01, 0        )               ( 0.161E-01, 0        )
(-0.165E-01, 0        )               (-0.161E-01, 0        )
(-0.794E-02, 0        )               ( 0.422E-02, 0        )
( 0.841E-09, 0        )               (-0.422E-02, 0        )
(-0.841E-09, 0        )
(-0.308E-13, 0        )
( 0.297E-14, 0.401E-15)
( 0.297E-14,-0.401E-15)
(-0.245E-14, 0        )
( 0.124E-14, 0        )
(-0.702E-15, 0.693E-15)
(-0.702E-15,-0.693E-15)
( 0.671E-15, 0.438E-15)
( 0.671E-15,-0.438E-15)
(-0.404E-15, 0.517E-15)
(-0.404E-15,-0.517E-15)
( 0.230E-15, 0.290E-15)
( 0.230E-15,-0.290E-15)
(-0.317E-15, 0        )
(-0.302E-15, 0        )
(-0.589E-16, 0.240E-15)
(-0.589E-16,-0.240E-15)
(-0.138E-15, 0.790E-16)
(-0.138E-15,-0.790E-16)
( 0.146E-15, 0        )
(-0.599E-16, 0.168E-16)
(-0.599E-16,-0.168E-16)
( 0.309E-16, 0        )
( 0.123E-16, 0        )
(-0.442E-17, 0.108E-16)
(-0.442E-17,-0.108E-16)
(-0.940E-30, 0        )
( 0        , 0        )
( 0        , 0        )
( 0        , 0        )

Resumo

  • Condição de Dirichlet em todos os lados
λ( 1.. 90)={+- 0.955,..., +- 0.422E-02}    λ(91..132)=1.0     
λ( 1.. 90)={+- 0.955,..., +- 0.422E-02}


  • Condição de Neumann em todos os lados
λ( 1.. 90)= { 1, 0.976, 0.970, +-0.945,..., +-0.568E-02 }  λ(91..132)= (eps,...,eps**2,...,0)
λ( 1.. 90)={+- 0.955,..., +- 0.422E-02}
  • Condição de Neumann em todos os lados e subtraindo P(1,2)
λ={0.976, 0.97,+-0.945,0.905,0.883,...,+-0.568E-2,-0.44E-14} U {eps,...0}
λ={-0.955,0.911,+-0.896,+-0.884,...,0.523E-02,-0.422E-02, 0.0)
  • Condição de Neumann em todos os lados fixando um ponto, P(1,2)=1
λ( 1.. 90)={+- 0.955,..., +- 0.422E-02}
λ(1)=1   λ(2..91)=( 0.999, 0.974, 0.967,-0.945, 0.939,0.902...-0.615E-02, 0.500E-02 ) λ=(eps,...0)

  • Condição de Neumann em 3 lados fixando P=1 na entrada (ou na saída)
 lambda(1)=   1.00000000000000         lambda(1)=  0.955274744954830
 lambda(n)=  0.000000000000000    0    lambda(n)=  4.218228659674831E-003
      Eigenvalues                           Eigenvalues
λ(1..11)=1
( 0.994    , 0        )               (-0.955    , 0        )
( 0.964    , 0        )               ( 0.955    , 0        )
( 0.950    , 0        )               (-0.896    , 0        )
(-0.948    , 0        )               ( 0.896    , 0        )
( 0.920    , 0        )               ( 0.884    , 0        )
(-0.883    , 0        )               (-0.884    , 0        )
. . .
(-0.194E-01, 0        )               ( 0.161E-01, 0        )
(-0.165E-01, 0        )               (-0.161E-01, 0        )
(-0.794E-02, 0        )               ( 0.422E-02, 0        )
( 0.841E-09, 0        )               (-0.422E-02, 0        )
(-0.841E-09, 0        )
(-0.308E-13, 0        )
( 0.297E-14, 0.401E-15)
. . .
( 0        , 0        )

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