One-level point-patch Schwarz loses uniformity on 1D Poisson

Direct expansion gives $x^{\mathsf{T}}{A}_{N}x=h^{-1}{\sum }_{j=1}^N(x_j-x_{j-1}{)}^2$ with $x_0=x_N=0$. It is positive for every nonzero vector because vanishing of all successive differences and $x_0=0$ forces every $x_j=0$. For the point-patch injection $R_i=e_i$, the local matrix is $A_i=e_i^{\mathsf{T}}{A}_N{e}_i=2\mathrm{/}h$. Thus [F1] gives $B_N={\sum }_i{e}_i(h\mathrm{/}2)e_i^{\mathsf{T}}=(h\mathrm{/}2)I$. Each corresponding nodal hat is supported on two adjacent mesh intervals, so neighboring local supports overlap and at most three closed supports meet at a point.

For $k=1,\dots ,N-1$, set $v_j^{(k)}=\sin (jk\pi \mathrm{/}N)$ for $j=0,\dots ,N$. The endpoints vanish. With $\theta =k\pi \mathrm{/}N$, the identity $v_{j-1}^{(k)}+v_{j+1}^{(k)}=2\cos (\theta )v_j^{(k)}$ gives $A_N{v}^{(k)}=\frac{2}{h}(1-\cos \theta )v^{(k)}\mathrm{.}$ These eigenvalues are distinct because cosine is strictly decreasing on $(0,\pi )$. Since $A_N$ is symmetric, eigenvectors for distinct eigenvalues are orthogonal: for $k\mathrm{\ne }l$, $({\lambda }_k-{\lambda }_l)(v^{(k)}{)}^{\mathsf{T}}{v}^{(l)}=(A_N{v}^{(k)}{)}^{\mathsf{T}}{v}^{(l)}-(v^{(k)}{)}^{\mathsf{T}}{A}_N{v}^{(l)}=0$. The vectors are nonzero, so the $N-1$ orthogonal vectors form a basis of ${\mathbb{R}}^{N-1}$. Multiplication by $B_N=(h\mathrm{/}2)I$ gives eigenvalues $1-\cos (k\pi \mathrm{/}N)$ for $B_N{A}_N$.

The smallest and largest values occur at $k=1$ and $k=N-1$, respectively. Therefore ${\kappa }_A(B_N{A}_N)=\frac{1-\cos ((N-1)\pi \mathrm{/}N)}{1-\cos (\pi \mathrm{/}N)}=\frac{1+\cos (\pi \mathrm{/}N)}{1-\cos (\pi \mathrm{/}N)}\mathrm{.}$ As $N\to \mathrm{\infty }$ the numerator tends to $2$ and the positive denominator tends to $0$, so the condition numbers diverge.