ACCEPTED MANUSCRIPT This completes the proof of the lemma. :0≤t≤3,1≤1≤2 follows that。≤M. PROOF.By Eq.(5).w hae +(w,a+p1o十pz)， b=忘p++pu+pu+), bu-pa+p+号n+2p小+D+pa tfocmsthnt Iboo-2b+=(Pu-2PLa+P2o) )( ≤M e一b+bas专，te Ed子aithep金tae Theorem 3 For(,)we hve IS(a.uy-F(w.e)l5(u(1-w)+o(1-) ma8u-85分-， anb6lgthnednsaadFu,o)s ACCEPTED MANUSCRIPT ACCEPTED MANUSCRIPT This completes the proof of the lemma. ✷ Lemma 2 For a regular CCSS patch S as defined in Eq. (3), the second order norm is M = max{{pi−1,j − 2pi,j + pi+1,j : 1 ≤ i ≤ 2, 0 ≤ j ≤ 3} ∪{pi,j−1 − 2pi,j + pi,j+1 : 0 ≤ i ≤ 3, 1 ≤ j ≤ 2}} . It follows that Mb ≤ 1 6M. PROOF. By Eq. (5), we have: b0,0 = 1 36(p0,0 + 4p1,0 + p2,0) + 1 9 (p0,1 + 4p1,1 + p2,1) + 1 36(p0,0 + 4p1,0 + p2,0) , b1,0 = 1 18(2p1,0 + p2,0) + 2 9 (2p1,1 + p2,1) + 1 18(2p1,2 + p2,2) , b2,0 = 1 18(p1,0 + 2p2,0) + 2 9 (p1,1 + 2p2,1) + 1 18(p1,2 + 2p2,2) . It follows that: b0,0 − 2b1,0 + b2,0 = 1 36(p0,0 − 2p1,0 + p2,0) +4(p0,0 − 2p1,0 + p2,0)+(p0,0 − 2p1,0 + p2,0) ≤ 1 6 M . Similarly, we have b1,0 − 2b1,1 + b1,2 ≤ 1 6M. By symmetry, the result fol￾lows. ✷ Combining Lemmas 1 and 2, we obtain a bound on the pointwise distance between S(u, v) and F(u, v): Theorem 3 For (u, v) ∈ Ω, we have S(u, v) − F(u, v) ≤ 1 2 (u(1 − u) + v(1 − v))M . In the above theorem, B(u, v) = 1 2 (u(1 − u) + v(1 − v)) is called the distance bound function of S(u, v) with respect to F(u, v). Since max (u,v)∈Ω B(u, v) = B( 1 2 , 1 2 ) = 1 4 , we have a bound on the maximal distance between S(u, v) and F(u, v) as stated in the following theorem: 10