2D Case:

Let e be the error function that satisfies

△_{h}e = \{ \begin{matrix}h, & on the interface \\ 0, & elsewhere. \end{matrix}

Claim: ∥e∥_{L^{2}} = O(h^{2}| log h|).

Let △x = △y = h = \frac{1}{N} and x_{i} = ih, y_{j} = jh. Define the O.N.B for the sine transformation

f_{kℓ}(x, y) = \frac{2}{N} · sin(kxπ) sin(ℓyπ),

then (f_{kℓ})_{ij} = f_{kℓ}(x_{i}, y_{j}) = \frac{2}{N} sin(kx_{i}π) sin(ℓy_{j}π). Let G^{h} satisfies △_{h}G^{h} = δ_{i_{0},j_{0}}, where (i_{0}, j_{0}) ∈ interface. Then

< △_{h}G^{h}, f_{kℓ} >=< δ_{i_{0},j_{0}}, f_{kℓ} >= (f_{kℓ})_{i_{0},j_{0}} .  (I)

On the other hand, we can write

(G^{h})_{ij} = ∑_{k,ℓ=1}^{N−1} Ĝ^{h}_{kℓ} (f_{kℓ})_{ij} .

And hence

< △_{h}G^{h}, f_{mn} > = ∑_{i,j} (∑_{k,ℓ} (λ_{k} + λ_{ℓ}) Ĝ^{h}_{kℓ} (f_{kℓ})_{ij} (f_{mn})_{ij})
= (λ_{m} + λ_{n}) Ĝ^{h}_{mn}  (II)

compare (I) and (II), we have

Ĝ^{h}_{kℓ} = \frac{(f_{kℓ})_{i_{0},j_{0}}}{(λ_{k} + λ_{ℓ})},

where λ_{k} = \frac{−4 sin^{2}(\frac{khπ}{2})}{h^{2}} is the eigenvalue for △_{h}. Thus

(G^{h})_{ij} = ∑_{k,ℓ=1}^{N−1} \frac{(f_{kℓ})_{i_{0},j_{0}}}{(λ_{k} + λ_{ℓ})} · (f_{kℓ})_{ij}
= \frac{−h^{2}}{N^{2}} ∑_{k,ℓ=1}^{N−1} \frac{sin(kx_{i}π) sin(ℓy_{j}π) · sin(kx_{i_{0}}π) sin(ℓy_{j_{0}}π)}{sin^{2} (\frac{khπ}{2} ) + sin^{2} (\frac{ℓhπ}{2} )}

and then

|(G^{h})_{ij}| ≤ \frac{h^{2}}{N^{2}} ∑_{k,ℓ=1}^{N−1} \frac{1}{sin (\frac{khπ}{2} ) + sin^{2} (\frac{ℓhπ}{2} )}

∼ \frac{1}{N^{2}} ∫_{h}^{1} ∫_{h}^{1} \frac{dxdy}{sin^{2} (\frac{πx}{2}) + sin^{2} (\frac{πy}{2} )}

( because of sin^{2} (\frac{πx}{2}) ≥ x^{2}) , ≤ \frac{1}{N^{2}} ∫0^{\frac{π}{2}} ∫_{\sqrt{2}h}^{√\overline{2}} \frac{rdrdθ}{r^{2}} = \frac{\frac{π}{2}}{N_{2}} | log h|
= O (h^{2}| log h|) .

Hence ∥e∥_{L^{2}} = O (hG^{h}) · O(N) = O (G^{h}) = O (h^{2}| log h|). 3D Case:

The mesh point is defined by (x_{i}, y_{j}, z_{k}) = (ih, jh, kh), where h = \frac{1}{N}. Let Γ be the interface in Ω ⊂ IR^{3}, then

#{ mesh points ∈ Γ} = O(N^{2})

Also, let e be the error function with

△_{h}e = \{ \begin{matrix}h, & on Γ \\ 0, & otherwise \end{matrix}

Claim: ∥e∥_{L^{2}(Γ)} = O(h)

First, we define G^{h} satisfies

△_{h}G^{h} = δ_{i_{0},j_{0},k_{0}}, for (i_{0}, j_{0}, k_{0}) ∈ Γ.

Second, we define the O.N.B for the sine transformation

f_{ℓmn} = (\frac{2}{N} )^{\frac{3}{2}} sin (ℓxπ) sin (myπ) sin (nzπ) ,

then

(f_{ℓmn})_{ijk} = (\frac{2}{N} )^{\frac{3}{2}} sin(ℓx_{i}π) sin(my_{j}π) sin(nz_{k}π).

Note that

< △_{h}G^{h}, f_{ℓmn} >= f_{ℓmn} (i_{0}, j_{0}, k_{0}) ,  (III)

On the other hand, we write

G^{h}(i, j, k) = ∑_{ℓmn} Ĝ^{h}(ℓ, m, n)f_{ℓmn}(i, j, k).

Therefore

△_{h}G^{h}(i, j, k) = ∑_{ℓmn} (λ_{ℓ} + ℓ_{m} + ℓ_{n}) Ĝ^{h} (ℓ, m, n) f_{ℓmn}(i, j, k)

and

< △_{h}G^{h}, f_{ℓmn} >= (λ_{ℓ} + λ_{m} + λ_{n}) Ĝ^{h} (ℓ, m, n) .  (IV)

According to (III) and (IV), we have

Ĝ^{h}(ℓ, m, n) = \frac{f_{ℓmn} (i_{0}, j_{0}, k_{0})}{λ_{ℓ} + λ_{m} + λ_{n}}.

Therefore

|G^{h}(i, j, k)| ≤ ∑_{ℓmn} |\frac{f_{ℓmn}(i_{0}, j_{0}, k_{0})}{λ_{ℓ} + λ_{m} + λ_{n}}f_{ℓmn}(i, j, k)|
≤ ∑_{ℓmn} \frac{(\frac{2}{N} )^{3} · 1 · h^{2}}{4 [sin^{2} (\frac{ℓhπ}{2} ) + sin^{2} (\frac{mhπ}{2} ) + sin^{2} (\frac{nhπ}{2} )]}
≃ \frac{2}{hN^{3}} ∫_{h}^{1} ∫_{h}^{1} ∫_{h}^{1} \frac{dxdydz}{x^{2} + y^{2} + z^{2}}
≤ 2h^{2} ∫0^{\frac{π}{2}} ∫0^{\frac{π}{2}} ∫_{\sqrt{3}h}^{√\overline{3}} \frac{r^{2} sin φ}{r^{2}}drdφdθ
≈ O(h^{2}).

Finally ∥e∥_{L^{2}} = O(hG^{h}) · O(N^{2}) = O(h).