Paper VI: Green operators and the Poisson seed

11 Green operators and the Poisson seed

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On the interval \((0,1)\) with Dirichlet boundary conditions, the Green kernel for \(-u''=f\) is

\[ \boxed{G(x,y)=\min(x,y)-xy.} \]

Thus the point source \(\delta_y\) generates the solution kernel, and a general source is reconstructed by

\[ u(x)=\int_0^1G(x,y)f(y)\,dy. \]

For \(n\) interior points with spacing \(h=1/(n+1)\), define

\[ L_h=\frac1{h^2} \begin{pmatrix} 2&-1&&\\ -1&2&\ddots&\\ &\ddots&\ddots&-1\\ &&-1&2 \end{pmatrix}. \]

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Theorem 11.1: Exact discrete Green reconstruction at grid nodes

Let \(x_i=ih\). Then

\[ \boxed{ \frac1h(L_h^{-1})_{ij} =G(x_i,x_j). } \]

Thus the scaled discrete inverse reproduces the continuum Dirichlet Green kernel exactly at the grid nodes.

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Proof of Theorem 11.1

The inverse of the unscaled tridiagonal matrix has entries

\[ \frac{\min(i,j)(n+1-\max(i,j))}{n+1}. \]

Multiplying by \(h^2\) and then dividing by \(h\) gives

\[ h\min(i,j)\left(1-\frac{\max(i,j)}{n+1}\right) =\min(x_i,x_j)-x_ix_j. \]

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A section of the continuum Green kernel and the scaled discrete inverse. They agree at the grid nodes.

11.1 Low-mode convergence

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The \(k\)th discrete eigenvalue is

\[ \lambda_{k,h}=\frac4{h^2}\sin^2\left(\frac{k\pi h}{2}\right), \]

whereas the continuum eigenvalue is \((k\pi)^2\).

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Theorem 11.2: Second-order low-mode convergence

For fixed \(k\),

\[ \lambda_{k,h}=(k\pi)^2+O(h^2). \]

More precisely, for sufficiently small \(h\),

\[ 0\le (k\pi)^2-\lambda_{k,h} \le \frac{(k\pi)^4}{12}h^2. \]

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Proof of Theorem 11.2

For \(x\ge0\), \(0\le x^2-\sin^2x\le x^4/3\). Set \(x=k\pi h/2\) and multiply by \(4/h^2\).

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The first finite-difference eigenvalue converges with observed order approximately two.

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