Paper VI: Green operators and finite difference transfer

E12 Green operators and finite difference transfer

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Consider

\[ -u''=f \quad\hbox{on }(0,1), \qquad u(0)=u(1)=0. \]

The continuum Green kernel is

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

Thus

\[ u(x)=\int_0^1G(x,y)f(y){\,\mathrm d} y. \]

For \(n\) interior nodes, let

\[ h=\frac{1}{n+1}, \qquad x_i=ih, \]

and define

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

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Theorem E12.1: Exact nodal Green reconstruction

For \(1\leq i,j\leq n\),

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

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Proof of Theorem E12.1

The inverse of the unscaled tridiagonal matrix has entry

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

Multiplying by \(h^2\) and 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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Corollary E12.2: Discrete Green quadrature

If \(u^h\) solves

\[ L_hu^h=f^h, \]

then every nodal value has the exact representation

\[ u_i^h = h\sum_{j=1}^nG(x_i,x_j)f_j^h. \]

Thus the discrete Green operator is the nodal rectangle quadrature of the continuum Green kernel.

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Proof of Corollary E12.2

By Theorem E12.1,

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

Multiply the inverse matrix by the source vector.

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Proposition E12.3: Discrete maximum bound

The inverse \(L_h^{-1}\) has nonnegative entries and

\[ {\left\lVert L_h^{-1}\right\rVert}_{\infty\to\infty} = \max_{1\leq i\leq n} \frac{x_i(1-x_i)}{2} \leq \frac18. \]

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Proof of Proposition E12.3

Nonnegativity follows from the entry formula in Theorem E12.1. Let \(v_i=x_i(1-x_i)/2\). Since the second difference is exact on quadratics,

\[ L_hv={{\boldsymbol{1}}}. \]

Hence the \(i\)th row sum of \(L_h^{-1}\) is \(v_i\). For a nonnegative matrix, the infinity operator norm is the largest row sum.

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Theorem E12.4: Second order Poisson solution error

Suppose \(u\in C^4[0,1]\) solves the Dirichlet problem above. Let

\[ f^h_i=f(x_i) \]

and let \(u^h\) solve

\[ L_hu^h=f^h. \]

Then

\[ \max_i{\left\lvert u^h_i-u(x_i)\right\rvert} \leq \frac{h^2}{96} {\left\lVert u^{(4)}\right\rVert}_\infty. \]

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Proof of Theorem E12.4

Taylor expansion with remainder gives

\[ {\left\lVert L_h(u(x_i))_i-f^h\right\rVert}_\infty \leq \frac{h^2}{12}{\left\lVert u^{(4)}\right\rVert}_\infty. \]

Apply \(L_h^{-1}\) and use Proposition E12.3.

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The continuum Dirichlet Green section and the scaled discrete inverse. The nodal values agree exactly.

The discrete eigenpairs are

\[ \lambda_{k,h} = \frac{4}{h^2} \sin^2\left(\frac{k\pi h}{2}\right), \qquad v_{k,h}(i)=\sin(k\pi x_i). \]

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Theorem E12.5: Low mode eigenvalue control

For \(kh\leq1\),

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

Consequently, for every fixed \(k\),

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

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Proof of Theorem E12.5

Set \(x=k\pi h/2\). The condition \(kh\leq1\) gives \(0\leq x\leq\pi/2\). On this interval,

\[ \sin x\geq x-\frac{x^3}{6}, \]

\[ 0\leq x^2-\sin^2x\leq\frac{x^4}{3}. \]

Multiply by \(4/h^2\).

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Corollary E12.6: Heat and fractional mode transfer

Let \(\lambda_k=(k\pi)^2\). For \(t\geq0\),

\[ 0\leq {\mathrm e}^{-t\lambda_{k,h}}-{\mathrm e}^{-t\lambda_k} \leq \frac{t(k\pi)^4}{12}h^2{\mathrm e}^{-t\lambda_{k,h}}. \]

For \(0<\alpha\leq1\) and fixed \(k\),

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

and

\[ {\mathrm e}^{-t\lambda_{k,h}^\alpha} = {\mathrm e}^{-t\lambda_k^\alpha}+O(th^2) \]

uniformly for \(t\) in bounded intervals.

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Proof of Corollary E12.6

The ordinary heat estimate follows from the mean value theorem and Theorem E12.5. For fixed \(k\), the eigenvalues stay uniformly away from zero once \(h\) is small. The function \(x^\alpha\) is then Lipschitz on a common compact interval containing both eigenvalues. Apply the mean value theorem twice.

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Finite difference schemes and their stability theory are treated more broadly in [Strikwerda2004]. The present result uses the special one dimensional Green structure to obtain an explicit constant.

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