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Eigenfunctions of Laplace operator on a squere with finite differences

  1. Apr 6, 2009 #1
    1. The problem statement, all variables and given/known data

    I want to numerically compute the eigenfunctions and eigenvalues of Laplace operator on a square with Dirichlet boundary conditions (i.e. [tex]u|_{\partial}=0[/tex]). Exact analytical solutions are well known sinusoidal modes: [tex]u_{m,n}(x,y)=\sin(k_mx)\sin(k_ny) [/tex], where [tex]k_m=m\frac{\pi}{a}[/tex] and .[tex]k_n=n\frac{\pi}{a}[/tex], the length of a square side is [tex]a[/tex].

    2. Relevant equations

    We are numerically solving the following equation:
    [tex]\nabla^2u(x,y)=\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}=k^2u(x,y)[/tex]
    on a square grid of size [tex]N\times N[/tex], where points are separated by [tex]h=a/N[/tex].

    3. The attempt at a solution

    Two dimensional Laplace operator is replaced with simple finite differences on a square grid.
    [tex]\nabla^2u(x,y)=\frac{u_{i,j}-2u_{i,j}+u_{i-1,j}}{h^2}+\frac{u_{i,j+1}-2u_{i,j}+u_{i,j-1}}{h^2}[/tex]
    The problem thus gets the form:
    [tex]u_{i+1,j}+u_{i-1,j}+u_{i,j+1}+u_{i,j-1}-4u_{i,j}=h^2k^2u_{i,j}[/tex]
    Which can be written as a typical matrix eigenvalue problem:
    [tex]{\bf A}{\bf u}=\lambda{\bf u}[/tex]
    The matrix [tex]\bf A[/tex] is square and has the following form for [tex]N=3[/tex]:
    [tex]
    \left(
    \begin{array}{ccccccccc}
    -4 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
    1 & -4 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\
    0 & 1 & -4 & 1 & 0 & 1 & 0 & 0 & 0 \\
    1 & 0 & 1 & -4 & 1 & 0 & 1 & 0 & 0 \\
    0 & 1 & 0 & 1 & -4 & 1 & 0 & 1 & 0 \\
    0 & 0 & 1 & 0 & 1 & -4 & 1 & 0 & 1 \\
    0 & 0 & 0 & 1 & 0 & 1 & -4 & 1 & 0 \\
    0 & 0 & 0 & 0 & 1 & 0 & 1 & -4 & 1 \\
    0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & -4
    \end{array}
    \right)
    [/tex]
    We can now use some numerical library to find eigenvalues and eigenfunctions of [tex]\bf A[/tex].

    What i don't know is where and how to take the boundary conditions into account.
     
  2. jcsd
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