前面介紹的P \mathbf{P} P P d \mathbf{Pd} P d
因爲同參映射P \mathbf{P} P L L L (64) L − 1 ∣ x 1 − x 2 ∣ ≤ ∣ x ~ 1 − x ~ 2 ∣ ≤ L ∣ x 1 − x 2 ∣ , x ~ i = P ( x i ) , i = 1 , 2
\begin{array}{ll}{\text { (64) }} & {L^{-1}\left|\mathrm{x}_{1}-\mathrm{x}_{2}\right| \leq\left|\widetilde{\mathrm{x}}_{1}-\widetilde{\mathrm{x}}_{2}\right| \leq L\left|\mathrm{x}_{1}-\mathrm{x}_{2}\right|, \quad \tilde{\mathrm{x}}_{i}=\mathrm{P}\left(\mathrm{x}_{i}\right), i=1,2}\end{array}
(64) L − 1 ∣ x 1 − x 2 ∣ ≤ ∣ x 1 − x 2 ∣ ≤ L ∣ x 1 − x 2 ∣ , x ~ i = P ( x i ) , i = 1 , 2
定義T T T T T T γ \gamma γ Γ \Gamma Γ ( 65 ) ω T = ∪ { T ′ : T ′ ∩ T ≠ ∅ } , ω ~ T = P ( ω T )
(65) \quad \omega_{T}=\cup\left\{T^{\prime}: T^{\prime} \cap T \neq \emptyset\right\}, \quad \tilde{\omega}_{T}=\mathbf{P}\left(\omega_{T}\right)
( 6 5 ) ω T = ∪ { T ′ : T ′ ∩ T = ∅ } , ω ~ T = P ( ω T )
定義形正則化常數，這裏h T : = ∣ T ∣ 1 n h_{T}:=|T|^{\frac{1}{n}} h T : = ∣ T ∣ n 1 ( 66 ) σ : = sup T max T ∈ T diam ( T ) h T
(66) \quad \sigma:=\sup _{\mathcal{T}} \max _{T \in \mathcal{T}} \frac{\operatorname{diam}(T)}{h_{T}}
( 6 6 ) σ : = T sup T ∈ T max h T d i a m ( T )
我們用T ^ \widehat{T} T T = X T ( T ^ ) T=\mathbf{X}_{T}(\widehat{T}) T = X T ( T ) D X T D \mathbf{X}_{T} D X T T T T diam ( T ) \operatorname{diam}(T) d i a m ( T ) D X T D \mathbf{X}_{T} D X T h T h_T h T (67) h T ∣ w ∣ ≲ ∣ D X T w ∣ ≲ h T ∣ w ∣ , ∀ w ∈ R n
\begin{array}{ll}{\text { (67) }} & {h_{T}|\mathrm{w}| \lesssim\left|D \mathrm{X}_{T} \mathrm{w}\right| \lesssim h_{T}|\mathrm{w}|, \quad \forall \mathrm{w} \in \mathbb{R}^{n}}\end{array}
(67) h T ∣ w ∣ ≲ ∣ D X T w ∣ ≲ h T ∣ w ∣ , ∀ w ∈ R n
因爲投影的雙利普希茨性，我們也知道$和 和 和 ( 68 ) h T ∣ w ∣ ≲ ∣ D χ T ( y ) w ∣ ≲ h T ∣ w ∣ ∀ w ∈ R n , y ∈ T ^
(68) \quad h_{T}|\mathrm{w}| \lesssim\left|D \chi_{T}(\mathrm{y}) \mathrm{w}\right| \lesssim h_{T}|\mathrm{w}| \quad \forall \mathrm{w} \in \mathbb{R}^{n}, \mathrm{y} \in \widehat{T}
( 6 8 ) h T ∣ w ∣ ≲ ∣ D χ T ( y ) w ∣ ≲ h T ∣ w ∣ ∀ w ∈ R n , y ∈ T
我們依然用tilde符號來表示在曲面γ \gamma γ Γ \Gamma Γ X T \mathbf{X}_{T} X T γ \gamma γ χ T = P ∘ X T \chi_{T}=\mathbf{P} \circ \mathbf{X}_{T} χ T = P ∘ X T (69) L − 1 h T ∣ y 1 − y 2 ∣ ≤ ∣ x ~ 1 − x ~ 2 ∣ ≤ L h T ∣ y 1 − y 2 ∣
\begin{array}{ll}{\text { (69) }} & {L^{-1} h_{T}\left|\mathbf{y}_{1}-\mathbf{y}_{2}\right| \leq\left|\widetilde{\mathbf{x}}_{1}-\widetilde{\mathbf{x}}_{2}\right| \leq L h_{T}\left|\mathbf{y}_{1}-\mathbf{y}_{2}\right|}\end{array}
(69) L − 1 h T ∣ y 1 − y 2 ∣ ≤ ∣ x 1 − x 2 ∣ ≤ L h T ∣ y 1 − y 2 ∣
各個區域上的函數值定義如下： (70) v ^ T ( y ) : = v ~ T ( χ T ( y ) ) ∀ x ^ ∈ T ^ and v T ( x ) : = v ~ T ( P ( x ) ) ∀ x ∈ T
\begin{array}{ll}{\text { (70) } \quad \widehat{v}_{T}(\mathbf{y}):=\tilde{v}_{T}\left(\chi_{T}(\mathbf{y})\right) \quad \forall \widehat{\mathbf{x}} \in \widehat{T}} & {\text { and } \quad v_{T}(\mathbf{x}):=\tilde{v}_{T}(\mathbf{P}(\mathbf{x})) \quad \forall \mathbf{x} \in T}\end{array}
(70) v T ( y ) : = v ~ T ( χ T ( y ) ) ∀ x ∈ T and v T ( x ) : = v ~ T ( P ( x ) ) ∀ x ∈ T
同以前的方式，可以定義第一標準型和麪積元， (71) g T : = ( D X T ) t D X T , q T : = det g T , ∀ T ∈ T
\begin{array}{ll}{\text { (71) }} & {\mathrm{g}_{T}:=\left(D \mathrm{X}_{T}\right)^{t} D \mathrm{X}_{T}, \quad q_{T}:=\sqrt{\operatorname{det} \mathrm{g}_{T}}, \quad \forall T \in \mathcal{T}}\end{array}
(71) g T : = ( D X T ) t D X T , q T : = d e t g T , ∀ T ∈ T
且滿足， (72) eigen ( g T ) ≈ h T 2 , q T ≈ h T n , ∀ T ∈ T
\begin{array}{ll}{\text { (72) }} & {\text { eigen }\left(\mathrm{g}_{T}\right) \approx h_{T}^{2}, \quad q_{T} \approx h_{T}^{n}, \quad \forall T \in \mathcal{T}}\end{array}
(72) eigen ( g T ) ≈ h T 2 , q T ≈ h T n , ∀ T ∈ T
因爲D X T D \mathbf{X}_{T} D X T D χ T D \mathbf{\chi}_{T} D χ T ( 73 ) S χ ≈ 1
(73) \quad S_{\chi} \approx 1
( 7 3 ) S χ ≈ 1
因爲(72)我們知道q q q q Γ q_\Gamma q Γ C 1 ≤ q q Γ ≤ C 2
C_{1} \leq \frac{q}{q_{\Gamma}} \leq C_{2}
C 1 ≤ q Γ q ≤ C 2
回憶曲面梯度和拉普拉斯算子的定義，有， (75) ∇ v ^ = ( D X ) t ∇ T v , ∇ T v = ( D X ) g Γ − 1 ∇ v ^
\begin{array}{ll}{\text { (75) }} & {\nabla \widehat{v}=(D \mathrm{X})^{t} \nabla_{\mathrm{T}} v, \quad \nabla_{\mathrm{T}} v=(D \mathrm{X}) \mathrm{g}_{\mathrm{\Gamma}}^{-1} \nabla \widehat{v}}\end{array}
(75) ∇ v = ( D X ) t ∇ T v , ∇ T v = ( D X ) g Γ − 1 ∇ v (76) Δ Γ v = 1 q Γ div ( q Γ g Γ − 1 ∇ v ^ )
\begin{array}{ll}{\text { (76) }} & {\Delta_{\Gamma} v=\frac{1}{q_{\Gamma}} \operatorname{div}\left(q_{\Gamma} \mathbf{g}_{\Gamma}^{-1} \nabla \widehat{v}\right)}\end{array}
(76) Δ Γ v = q Γ 1 d i v ( q Γ g Γ − 1 ∇ v )
使用分佈積分公式，很容易看到：∫ Γ ∇ Γ v ⋅ ∇ Γ w = ∑ T ∈ T − ∫ T w Δ Γ v + ∫ ∂ T w ∇ Γ v ⋅ μ T = ∑ T ∈ T − ∫ T w Δ Γ v + ∑ S ∈ S ∫ S w [ ∇ Γ v ]
\begin{aligned} \int_{\Gamma} \nabla_{\Gamma} v \cdot \nabla_{\Gamma} w &=\sum_{T \in \mathcal{T}}-\int_{T} w \Delta_{\Gamma} v+\int_{\partial T} w \nabla_{\Gamma} v \cdot \boldsymbol{\mu}_{T} \\ &=\sum_{T \in \mathcal{T}}-\int_{T} w \Delta_{\Gamma} v+\sum_{S \in \mathcal{S}} \int_{S} w\left[\nabla_{\Gamma} v\right] \end{aligned}
∫ Γ ∇ Γ v ⋅ ∇ Γ w = T ∈ T ∑ − ∫ T w Δ Γ v + ∫ ∂ T w ∇ Γ v ⋅ μ T = T ∈ T ∑ − ∫ T w Δ Γ v + S ∈ S ∑ ∫ S w [ ∇ Γ v ] ( 77 ) [ ∇ Γ v ] : = ∇ Γ v + ⋅ μ + + ∇ Γ v − ⋅ μ −
(77) \quad\left[\nabla_{\Gamma} v\right]:=\nabla_{\Gamma} v_{+} \cdot \mu_{+}+\nabla_{\Gamma} v_{-} \cdot \mu_{-}
( 7 7 ) [ ∇ Γ v ] : = ∇ Γ v + ⋅ μ + + ∇ Γ v − ⋅ μ −
所謂的參數化有限元方法就是，尋找U : = U T ∈ V # ( T ) U:=U_{\mathcal{T}} \in \mathbb{V}_{\#}(\mathcal{T}) U : = U T ∈ V # ( T ) (78) ∫ Γ ∇ Γ U ⋅ ∇ Γ V = ∫ Γ F V ∀ V ∈ V # ( T )
\begin{array}{ll}{\text { (78) }} & {\int_{\Gamma} \nabla_{\Gamma} U \cdot \nabla_{\Gamma} V=\int_{\Gamma} F V \quad \forall V \in \mathbb{V}_{\#}(\mathcal{T})}\end{array}
(78) ∫ Γ ∇ Γ U ⋅ ∇ Γ V = ∫ Γ F V ∀ V ∈ V # ( T ) F ∈ L 2 , # ( Γ ) F \in L_{2, \#}(\Gamma) F ∈ L 2 , # ( Γ ) V ( T ) : = { V ∈ C 0 ( Γ ) ∣ V ∣ T = V ^ ∘ X − 1 for some V ^ ∈ P , T ∈ T }
\mathbb{V}(\mathcal{T}):=\left\{V \in C^{0}(\Gamma)|V|_{T}=\widehat{V} \circ \mathrm{X}^{-1} \text { for some } \widehat{V} \in \mathcal{P}, T \in \mathcal{T}\right\}
V ( T ) : = { V ∈ C 0 ( Γ ) ∣ V ∣ T = V ∘ X − 1 for some V ∈ P , T ∈ T } V # ( T ) : = V ( T ) ∩ L 2 , # ( Γ )
\mathbb{V}_{\#}(\mathcal{T}):=\mathbb{V}(\mathcal{T}) \cap L_{2, \#}(\Gamma)
V # ( T ) : = V ( T ) ∩ L 2 , # ( Γ ) P \mathcal{P} P
因爲F ∈ L 2 , # ( Γ ) F \in L_{2, \#}(\Gamma) F ∈ L 2 , # ( Γ ) (79) ∫ Γ ∇ Γ U ⋅ ∇ Γ V = ∫ Γ F V ∀ V ∈ V ( T )
\text { (79) } \quad \int_{\Gamma} \nabla_{\Gamma} U \cdot \nabla_{\Gamma} V=\int_{\Gamma} F V \quad \forall V \in \mathbb{V}(\mathcal{T})
(79) ∫ Γ ∇ Γ U ⋅ ∇ Γ V = ∫ Γ F V ∀ V ∈ V ( T )
介紹一個引理：γ \gamma γ Γ \Gamma Γ ∫ Γ ∇ Γ ( u − U ) ⋅ ∇ Γ V = ∫ γ ∇ γ u ~ ⋅ ∇ γ V ~ + ∫ Γ ∇ Γ u ⋅ E Γ ∇ Γ V − ∫ Γ F V
\int_{\Gamma} \nabla_{\Gamma}(u-U) \cdot \nabla_{\Gamma} V=\int_{\gamma} \nabla_{\gamma} \widetilde{u} \cdot \nabla_{\gamma} \widetilde{V}+\int_{\Gamma} \nabla_{\Gamma} u \cdot \mathbf{E}_{\Gamma} \nabla_{\Gamma} V-\int_{\Gamma} F V
∫ Γ ∇ Γ ( u − U ) ⋅ ∇ Γ V = ∫ γ ∇ γ u ⋅ ∇ γ V + ∫ Γ ∇ Γ u ⋅ E Γ ∇ Γ V − ∫ Γ F V
Γ \Gamma Γ 回憶一下龐加萊不等式： (82) ∥ v ∥ L 2 ( Γ ) ≲ ∥ ∇ v ∥ L 2 ( Γ ) ∀ v ∈ H # 1 ( Γ )
\begin{array}{ll}{\text { (82) }} & {\|v\|_{L_{2}(\Gamma)} \lesssim\|\nabla v\|_{L_{2}(\Gamma)} \quad \forall v \in H_{\#}^{1}(\Gamma)}\end{array}
(82) ∥ v ∥ L 2 ( Γ ) ≲ ∥ ∇ v ∥ L 2 ( Γ ) ∀ v ∈ H # 1 ( Γ )
定義一個幾何元指示子，反應的是曲面及其逼近的導數之間的距離： (83) λ T : = ∥ D ( P − I T P ) ∥ L ∞ ( T ) = ∥ D P − I ∥ L ∞ ( T ) ∀ T ∈ T
\text { (83) } \quad \lambda_{T}:=\left\|D\left(\mathbf{P}-\mathcal{I}_{\mathcal{T}} \mathbf{P}\right)\right\|_{L_{\infty}(T)}=\|D \mathbf{P}-\mathbf{I}\|_{L_{\infty}(T)} \quad \forall T \in \mathcal{T}
(83) λ T : = ∥ D ( P − I T P ) ∥ L ∞ ( T ) = ∥ D P − I ∥ L ∞ ( T ) ∀ T ∈ T
相應的幾何估計量就定義爲： (84) λ T ( Γ ) : = max T ∈ T λ T
\begin{array}{ll}{\text { (84) }} & {\lambda_{\mathcal{T}}(\Gamma):=\max _{T \in \mathcal{T}} \lambda_{T}}\end{array}
(84) λ T ( Γ ) : = max T ∈ T λ T
用一下複合函數求導，我們有：( 85 ) max y ∈ T ˉ ∣ D ( χ T − X T ) ( y ) ∣ min { ∣ D − χ T ( y ) ∣ , ∣ D − X T ( y ) ∣ } ≤ S χ λ T ∀ T ∈ T
(85) \quad \max _{\mathbf{y} \in \bar{T}} \frac{\left|D\left(\chi_{T}-\mathbf{X}_{T}\right)(\mathbf{y})\right|}{\min \left\{\left|D^{-} \chi_{T}(\mathbf{y})\right|,\left|D^{-} \mathbf{X}_{T}(\mathbf{y})\right|\right\}} \leq S_{\chi} \lambda_{T} \quad \forall T \in \mathcal{T}
( 8 5 ) y ∈ T ˉ max min { ∣ D − χ T ( y ) ∣ , ∣ D − X T ( y ) ∣ } ∣ D ( χ T − X T ) ( y ) ∣ ≤ S χ λ T ∀ T ∈ T
另外定義兩個量，他們是h T h_T h T (86) β T : = ∥ P − I T P ∥ L ∞ ( T ) , β T ( Γ ) : = max T ∈ T β T
\begin{array}{ll}{\text { (86) }} & {\beta_{T}:=\left\|\mathbf{P}-\mathcal{I}_{\mathcal{T}} \mathbf{P}\right\|_{L_{\infty}(T)}, \quad \beta_{\mathcal{T}}(\Gamma):=\max _{T \in \mathcal{T}} \beta_{T}}\end{array}
(86) β T : = ∥ P − I T P ∥ L ∞ ( T ) , β T ( Γ ) : = max T ∈ T β T ( 87 ) μ T : = β T + λ T 2 , μ T ( Γ ) : = max T ∈ T μ T
(87) \quad \mu_{T}:=\beta_{T}+\lambda_{T}^{2}, \quad \mu_{\mathcal{T}}(\Gamma):=\max _{T \in \mathcal{T}} \mu_{T}
( 8 7 ) μ T : = β T + λ T 2 , μ T ( Γ ) : = T ∈ T max μ T
給一個推論， Corollary 32 (geometric consistency errors for C 1 , α surfaces). If X and χ satisfy ( 67 ) and ( 68 ) , then for all T ∈ T we have (88) ∥ 1 − q − 1 q Γ ∥ L ∞ ( T ^ ) , ∥ I − grg − 1 ∥ L ∞ ( T ^ ) , ∥ ν Γ − ν ∥ L ∞ ( T ) ≲ λ T where the hidden constants depend on S χ ≈ 1 defined in ( 38 ) . Moreover, (89) ∥ E ∥ L ∞ ( T ^ ) + ∥ E Γ ∥ L ∞ ( T ^ ) ≲ λ T ∀ T ∈ T
\begin{array}{l}{\text { Corollary } 32 \text { (geometric consistency errors for } C^{1, \alpha} \text { surfaces). If } \mathrm{X} \text { and } \chi \text { satisfy }} \\ {(67) \text { and }(68), \text { then for all } T \in \mathcal{T} \text { we have }} \\ {\begin{array}{ll}{\text { (88) }\left\|1-q^{-1} q_{\Gamma}\right\|_{L_{\infty}(\widehat{T})},\left\|\mathbf{I}-\operatorname{grg}^{-1}\right\|_{L_{\infty}(\widehat{T})},\left\|\nu_{\Gamma}-\nu\right\|_{L_{\infty}(T)} \lesssim \lambda_{T}} \\ {\text { where the hidden constants depend on } S_{\chi} \approx 1 \text { defined in }(38) . \text { Moreover, }} \\ {\text { (89) }} & {\|\mathbf{E}\|_{L_{\infty}(\widehat{T})}+\left\|\mathbf{E}_{\Gamma}\right\|_{L_{\infty}(\widehat{T})} \lesssim \lambda_{T} \quad \forall T \in \mathcal{T}}\end{array}}\end{array}
Corollary 3 2 (geometric consistency errors for C 1 , α surfaces). If X and χ satisfy ( 6 7 ) and ( 6 8 ) , then for all T ∈ T we have (88) ∥ ∥ 1 − q − 1 q Γ ∥ ∥ L ∞ ( T ) , ∥ ∥ I − g r g − 1 ∥ ∥ L ∞ ( T ) , ∥ ν Γ − ν ∥ L ∞ ( T ) ≲ λ T where the hidden constants depend on S χ ≈ 1 defined in ( 3 8 ) . Moreover, (89) ∥ E ∥ L ∞ ( T ) + ∥ E Γ ∥ L ∞ ( T ) ≲ λ T ∀ T ∈ T λ ∞ \lambda_\infty λ ∞ E E E
首先要讓曲面和逼近曲面足夠近，使得， (90) β T ( Γ ) < 1 2 K ∞ ⇒ Γ ⊂ N
\begin{array}{ll}{\text { (90) }} & {{ \beta }_\mathcal{T}(\Gamma)<\frac{1}{2 K_{\infty}} \Rightarrow \Gamma \subset \mathcal{N}}\end{array}
(90) β T ( Γ ) < 2 K ∞ 1 ⇒ Γ ⊂ N ( 91 ) P d ∘ P − 1 ( T ~ ) ⊂ ω ~ T ∀ T ∈ T
(91) \quad \mathrm{P}_{d} \circ \mathrm{P}^{-1}(\widetilde{T}) \subset \widetilde{\omega}_{T} \quad \forall T \in \mathcal{T}
( 9 1 ) P d ∘ P − 1 ( T ) ⊂ ω T ∀ T ∈ T ( 92 ) ∣ x ~ − P d ∘ P − 1 ( x ~ ) ∣ = ∣ P ( x ) − P d ( x ) ∣ ≤ 2 β T ∀ x ∈ T
(92) \quad\left|\widetilde{\mathbf{x}}-\mathbf{P}_{d} \circ \mathbf{P}^{-1}(\widetilde{\mathbf{x}})\right|=\left|\mathbf{P}(\mathbf{x})-\mathbf{P}_{d}(\mathbf{x})\right| \leq 2 \beta_{T} \quad \forall \mathbf{x} \in T
( 9 2 ) ∣ ∣ x − P d ∘ P − 1 ( x ) ∣ ∣ = ∣ P ( x ) − P d ( x ) ∣ ≤ 2 β T ∀ x ∈ T
那麼，有這麼一個推論， Corollary 33 (geometric consistency errors for C 2 surfaces). If ( 90 ) and ( 74 ) hold, then so do the following estimates for all T ∈ T (93) ∥ d ∥ L ∞ ( T ) ≲ β T , ∥ ν − ν Γ ∥ L ∞ ( T ) ≲ λ T , ∥ 1 − q − 1 q Γ ∥ L ∞ ( T ) ≲ μ T where all the geometric quantities are defined using the parametrizations χ = P d ∘ X and X . Moreover, (94) ∥ E ∥ L ∞ ( T ) , ∥ E Γ ∥ L ∞ ( T ) ≲ μ T ∀ T ∈ T
\begin{array}{l}{\text { Corollary } 33 \text { (geometric consistency errors for } C^{2} \text { surfaces). If }(90) \text { and }(74)} \\ {\text { hold, then so do the following estimates for all } T \in \mathcal{T}} \\ {\text { (93) }\|d\|_{L_{\infty}(T)} \lesssim \beta_{T}, \quad\left\|\nu-\nu_{\Gamma}\right\|_{L_{\infty}(T)} \lesssim \lambda_{T}, \quad\left\|1-q^{-1} q_{\Gamma}\right\|_{L_{\infty}(T)} \lesssim \mu_{T}} \\ {\text { where all the geometric quantities are defined using the parametrizations } \chi=\mathbf{P}_{d} \circ \mathbf{X}} \\ {\text { and } \mathbf{X} . \text { Moreover, }} \\ {\begin{array}{llll}{\text { (94) }} & {\|\mathbf{E}\|_{L_{\infty}(T)},\left\|\mathbf{E}_{\Gamma}\right\|_{L_{\infty}(T)}} & {\lesssim \mu_{T}} & {\forall T \in \mathcal{T}}\end{array}}\end{array}
Corollary 3 3 (geometric consistency errors for C 2 surfaces). If ( 9 0 ) and ( 7 4 ) hold, then so do the following estimates for all T ∈ T (93) ∥ d ∥ L ∞ ( T ) ≲ β T , ∥ ν − ν Γ ∥ L ∞ ( T ) ≲ λ T , ∥ ∥ 1 − q − 1 q Γ ∥ ∥ L ∞ ( T ) ≲ μ T where all the geometric quantities are defined using the parametrizations χ = P d ∘ X and X . Moreover, (94) ∥ E ∥ L ∞ ( T ) , ∥ E Γ ∥ L ∞ ( T ) ≲ μ T ∀ T ∈ T ω ~ \tilde \omega ω ~ ∣ w ~ ( x ~ ) − w ~ ( P d ∘ P − 1 ( x ~ ) ) ∣ ≤ ∥ ∇ γ w ~ ∥ L ∞ ( ω ~ T ) ∣ x ~ − P d ∘ P − 1 ( x ~ ) ∣ ≤ 2 ∥ ∇ γ w ~ ∥ L ∞ ( ω ~ T ) β T
\left|\widetilde{w}(\widetilde{\mathbf{x}})-\widetilde{w}\left(\mathbf{P}_{d} \circ \mathbf{P}^{-1}(\widetilde{\mathbf{x}})\right)\right| \leq\left\|\nabla_{\gamma} \widetilde{w}\right\|_{L_{\infty}\left(\widetilde{\omega}_{T}\right)}\left|\widetilde{\mathbf{x}}-\mathbf{P}_{d} \circ \mathbf{P}^{-1}(\widetilde{\mathbf{x}})\right| \leq 2\left\|\nabla_{\gamma} \widetilde{w}\right\|_{L_{\infty}\left(\widetilde{\omega}_{T}\right)} \beta_{T}
∣ ∣ w ( x ) − w ( P d ∘ P − 1 ( x ) ) ∣ ∣ ≤ ∥ ∇ γ w ∥ L ∞ ( ω T ) ∣ ∣ x − P d ∘ P − 1 ( x ) ∣ ∣ ≤ 2 ∥ ∇ γ w ∥ L ∞ ( ω T ) β T
下面介紹一個命題， Proposition 34 (mismatch between P and P d ). Assume that (67) as well as the assumptions (74), (90) and (91) hold. Then there exists λ ∗ > 0 such for w ~ ∈ H 1 ( γ ) and T ∈ T we have ∥ w ~ − w ~ ∘ P d ∘ P − 1 ∥ L 2 ( T ~ ) ≲ β T ∥ w ~ ∥ H 1 ( ω ~ T ) provided λ T ≤ λ ∗ and ω ~ T is a patch in γ around T ~ .
\begin{array}{l}{\text { Proposition 34 (mismatch between } \mathbf{P} \text { and } \mathbf{P}_{d} \text { ). Assume that (67) as well as the }} \\ {\text { assumptions (74), (90) and (91) hold. Then there exists } \lambda_{*}>0 \text { such for } \widetilde{w} \in H^{1}(\gamma)} \\ {\text { and } T \in \mathcal{T} \text { we have }} \\ {\qquad\left\|\widetilde{w}-\widetilde{w} \circ \mathbf{P}_{d} \circ \mathbf{P}^{-1}\right\|_{L_{2}(\widetilde{T})} \lesssim \beta_{T}\|\widetilde{w}\|_{H^{1}\left(\widetilde{\omega}_{T}\right)}} \\ {\text { provided } \lambda_{T} \leq \lambda_{*} \text { and } \widetilde{\omega}_{T} \text { is a patch in } \gamma \text { around } \widetilde{T} .}\end{array}
Proposition 34 (mismatch between P and P d ). Assume that (67) as well as the assumptions (74), (90) and (91) hold. Then there exists λ ∗ > 0 such for w ∈ H 1 ( γ ) and T ∈ T we have ∥ ∥ w − w ∘ P d ∘ P − 1 ∥ ∥ L 2 ( T ) ≲ β T ∥ w ∥ H 1 ( ω T ) provided λ T ≤ λ ∗ and ω T is a patch in γ around T .
reduce到R n \mathbb{R}^{n} R n
光滑化
估計各項
找ϵ \epsilon ϵ
抓關鍵步驟，簡言之，就是先把要證的東西做個參數域的轉換，∥ w ~ − w ~ ∘ ψ ∥ L 2 ( T ~ ) ≲ h T n / 2 ∥ w ^ − w ^ ∘ ψ ^ ∥ L 2 ( T ^ )
\|\widetilde{w}-\widetilde{w} \circ \psi\|_{L_{2}(\widetilde{T})} \lesssim h_{T}^{n / 2}\|\widehat{w}-\widehat{w} \circ \widehat{\psi}\|_{L_{2}(\widehat{T})}
∥ w − w ∘ ψ ∥ L 2 ( T ) ≲ h T n / 2 ∥ w − w ∘ ψ ∥ L 2 ( T )
再把轉換出來的東西拆成幾部分：∥ w ^ − w ^ ∘ ψ ^ ∥ L 2 ( T ^ ) ≲ ∥ w ^ − w ^ ε ∥ L 2 ( T ^ ) + ∥ w ^ ε − w ^ ε ∘ ψ ^ ∥ L 2 ( T ^ ) + ∥ w ^ ε ∘ ψ ^ − w ^ ∘ ψ ^ ∥ L 2 ( T ^ )
\|\widehat{w}-\widehat{w} \circ \widehat{\psi}\|_{L_{2}(\widehat{T})} \lesssim\left\|\widehat{w}-\widehat{w}_{\varepsilon}\right\|_{L_{2}(\widehat{T})}+\left\|\widehat{w}_{\varepsilon}-\widehat{w}_{\varepsilon} \circ \widehat{\psi}\right\|_{L_{2}(\widehat{T})}+\left\|\widehat{w}_{\varepsilon} \circ \widehat{\psi}-\widehat{w} \circ \widehat{\psi}\right\|_{L_{2}(\widehat{T})}
∥ w − w ∘ ψ ∥ L 2 ( T ) ≲ ∥ w − w ε ∥ L 2 ( T ) + ∥ ∥ ∥ w ε − w ε ∘ ψ ∥ ∥ ∥ L 2 ( T ) + ∥ ∥ ∥ w ε ∘ ψ − w ∘ ψ ∥ ∥ ∥ L 2 ( T )
估計第一部分……∥ w ^ − w ^ ε ∥ L 2 ( T ^ ) ≲ ε ∣ w ^ ∣ H 1 ( R n ) ≲ ε ∣ w ^ ∣ H 1 ( ω ^ T )
\left\|\widehat{w}-\widehat{w}_{\varepsilon}\right\|_{L_{2}(\widehat{T})} \lesssim \varepsilon|\widehat{w}|_{H^{1}\left(\mathbb{R}^{n}\right)} \lesssim \varepsilon|\widehat{w}|_{H^{1}\left(\widehat{\omega}_{T}\right)}
∥ w − w ε ∥ L 2 ( T ) ≲ ε ∣ w ∣ H 1 ( R n ) ≲ ε ∣ w ∣ H 1 ( ω T )
估計第三部分……∥ ( w ^ ε − w ^ ) ∘ ψ ^ ∥ L 2 ( T ^ ) ≲ ∥ w ^ ε − w ^ ∥ L 2 ( ω ^ T ) ≲ ε ∣ w ^ ∣ H 1 ( ω ^ T )
\left\|\left(\widehat{w}_{\varepsilon}-\widehat{w}\right) \circ \widehat{\psi}\right\|_{L_{2}(\widehat{T})} \lesssim\left\|\widehat{w}_{\varepsilon}-\widehat{w}\right\|_{L_{2}\left(\widehat{\omega}_{T}\right)} \lesssim \varepsilon|\widehat{w}|_{H^{1}\left(\widehat{\omega}_{T}\right)}
∥ ∥ ∥ ( w ε − w ) ∘ ψ ∥ ∥ ∥ L 2 ( T ) ≲ ∥ w ε − w ∥ L 2 ( ω T ) ≲ ε ∣ w ∣ H 1 ( ω T )
估計第二部分……∥ w ^ ε − w ^ ε ∘ ψ ^ ∥ L 2 ( T ^ ) 2 ≲ ε n ∑ i ∥ w ^ ε − w ^ ε ∘ ψ ^ ∥ L ∞ ( B ( y i , ε ) ∩ T ^ ) ≲ ε 2 ∑ i ∣ w ^ ∣ H 1 2 ( B ( y i , 3 ε ) ) ≲ ε 2 ∣ w ^ ∣ H 1 ( R n ) 2 ≲ ε 2 ∣ w ^ ∣ H 1 ( ω ^ T ) 2
\begin{aligned}\left\|\widehat{w}_{\varepsilon}-\widehat{w}_{\varepsilon} \circ \widehat{\psi}\right\|_{L_{2}(\widehat{T})}^{2} & \lesssim \varepsilon^{n} \sum_{i}\left\|\widehat{w}_{\varepsilon}-\widehat{w}_{\varepsilon} \circ \widehat{\psi}\right\|_{L_{\infty}\left(B\left(\mathbf{y}_{i}, \varepsilon\right) \cap \widehat{T}\right)} \\ & \lesssim \varepsilon^{2} \sum_{i}|\widehat{w}|_{H^{1}}^{2}\left(B\left(\mathbf{y}_{i}, 3 \varepsilon\right)\right) \lesssim \varepsilon^{2}|\widehat{w}|_{H^{1}\left(\mathbb{R}^{n}\right)}^{2} \lesssim \varepsilon^{2}|\widehat{w}|_{H^{1}\left(\widehat{\omega}_{T}\right)}^{2} \end{aligned}
∥ ∥ ∥ w ε − w ε ∘ ψ ∥ ∥ ∥ L 2 ( T ) 2 ≲ ε n i ∑ ∥ ∥ ∥ w ε − w ε ∘ ψ ∥ ∥ ∥ L ∞ ( B ( y i , ε ) ∩ T ) ≲ ε 2 i ∑ ∣ w ∣ H 1 2 ( B ( y i , 3 ε ) ) ≲ ε 2 ∣ w ∣ H 1 ( R n ) 2 ≲ ε 2 ∣ w ∣ H 1 ( ω T ) 2
估計ϵ \epsilon ϵ ε ≤ 2 L h T − 1 β T
\varepsilon \leq 2 L h_{T}^{-1} \beta_{T}
ε ≤ 2 L h T − 1 β T
把所有東西合起來，完事。∥ w ~ − w ~ ∘ ψ ∥ L 2 ( T ~ ) 2 ≲ h T n ∥ w ^ − w ^ ∘ ψ ^ ∥ L 2 ( T ^ ) 2 ≲ h T n ε 2 ∣ w ^ ∣ H 1 ( ω ^ T ) 2 ≲ h T n h T − 2 β T 2 h T 2 − n ∣ w ~ ∣ H 1 ( w ~ T ) 2 = β T 2 ∣ w ~ ∣ H 1 ( ω ~ T ) 2
\begin{aligned}\|\widetilde{w}-\widetilde{w} \circ \psi\|_{L_{2}(\widetilde{T})}^{2} & \lesssim h_{T}^{n}\|\widehat{w}-\widehat{w} \circ \widehat{\psi}\|_{L_{2}(\widehat{T})}^{2} \lesssim h_{T}^{n} \varepsilon^{2}|\widehat{w}|_{H^{1}\left(\widehat{\omega}_{T}\right)}^{2} \\ & \lesssim h_{T}^{n} h_{T}^{-2} \beta_{T}^{2} h_{T}^{2-n}|\widetilde{w}|_{H^{1}\left(\widetilde{w}_{T}\right)}^{2}=\beta_{T}^{2}|\widetilde{w}|_{H^{1}\left(\widetilde{\omega}_{T}\right)}^{2} \end{aligned}
∥ w − w ∘ ψ ∥ L 2 ( T ) 2 ≲ h T n ∥ w − w ∘ ψ ∥ L 2 ( T ) 2 ≲ h T n ε 2 ∣ w ∣ H 1 ( ω T ) 2 ≲ h T n h T − 2 β T 2 h T 2 − n ∣ w ∣ H 1 ( w T ) 2 = β T 2 ∣ w ∣ H 1 ( ω T ) 2
下面有另外一個命題， Proposition 35 (Lipschitz perturbation). Let Ω 1 , Ω 2 ⊂ ⊂ Ω ⊂ R n + 1 be Lipschitz bounded domains and L : Ω 1 → Ω 2 be a bi-Lipschitz isomorphism. If r : = max x ∈ Ω 1 ∣ L ( x ) − x ∣ is sufficiently small so that ( Ω 1 ∪ Ω 2 ) + B ( 0 , r ) ⊂ Ω then for all g ∈ H 1 ( Ω ) we have ∥ g − g ∘ L ∥ L 2 ( Ω 1 ) ≲ r ∥ g ∥ H 1 ( Ω )
\begin{array}{l}{\text { Proposition 35 (Lipschitz perturbation). Let } \Omega_{1}, \Omega_{2} \subset \subset \Omega \subset \mathbb{R}^{n+1} \text { be Lipschitz}} \\ {\text { bounded domains and } \mathbf{L}: \Omega_{1} \rightarrow \Omega_{2} \text { be a bi-Lipschitz isomorphism. If }} \\ {\qquad r:=\max _{\mathbf{x} \in \Omega_{1}}|\mathbf{L}(\mathbf{x})-\mathbf{x}|} \\ {\text { is sufficiently small so that }\left(\Omega_{1} \cup \Omega_{2}\right)+B(0, r) \subset \Omega \text { then for all } g \in H^{1}(\Omega) \text { we have }} \\ {\qquad\|g-g \circ \mathbf{L}\|_{L^{2}\left(\Omega_{1}\right)} \lesssim r\|g\|_{H^{1}(\Omega)}}\end{array}
Proposition 35 (Lipschitz perturbation). Let Ω 1 , Ω 2 ⊂ ⊂ Ω ⊂ R n + 1 be Lipschitz bounded domains and L : Ω 1 → Ω 2 be a bi-Lipschitz isomorphism. If r : = max x ∈ Ω 1 ∣ L ( x ) − x ∣ is sufficiently small so that ( Ω 1 ∪ Ω 2 ) + B ( 0 , r ) ⊂ Ω then for all g ∈ H 1 ( Ω ) we have ∥ g − g ∘ L ∥ L 2 ( Ω 1 ) ≲ r ∥ g ∥ H 1 ( Ω )
Lemma 36 (approximability in H 1 ( Γ ) ) . Let γ be a surface of class C 2 and u ~ ∈ H 2 ( γ ) . Let K ∞ be defined in ( 30 ) and β T ( Γ ) be given in ( 86 ) . Then we have (95) inf V ∈ V ( T ) ∥ ∇ Γ ( u ~ ∘ P d − V ) ∥ L 2 ( Γ ) ≲ h T ∣ u ~ ∣ H 2 ( γ ) + β T ( Γ ) K ∞ ∥ ∇ γ u ~ ∥ L 2 ( γ )
\begin{array}{l}{ \text { Lemma }\left.36 \text { (approximability in } H^{1}(\Gamma)\right) . \text { Let } \gamma \text { be a surface of class } C^{2} \text { and } \widetilde{u} \in} \\ {H^{2}(\gamma) . \text { Let } K_{\infty} \text { be defined in }(30) \text { and } \beta_{\mathcal{T}}(\Gamma) \text { be given in }(86) . \text { Then we have }} \\ {\begin{array}{llll}{\text { (95) }} & {\inf _{V \in \mathrm{V}(\mathcal{T})}\left\|\nabla_{\Gamma}\left(\widetilde{u} \circ \mathbf{P}_{d}-V\right)\right\|_{L_{2}(\Gamma)}} & {\lesssim h_{\mathcal{T}}|\widetilde{u}|_{H^{2}(\gamma)}+\beta_{\mathcal{T}}(\Gamma) K_{\infty}\left\|\nabla_{\gamma} \widetilde{u}\right\|_{L_{2}(\gamma)}}\end{array}}\end{array}
Lemma 3 6 (approximability in H 1 ( Γ ) ) . Let γ be a surface of class C 2 and u ∈ H 2 ( γ ) . Let K ∞ be defined in ( 3 0 ) and β T ( Γ ) be given in ( 8 6 ) . Then we have (95) inf V ∈ V ( T ) ∥ ∇ Γ ( u ∘ P d − V ) ∥ L 2 ( Γ ) ≲ h T ∣ u ∣ H 2 ( γ ) + β T ( Γ ) K ∞ ∥ ∇ γ u ∥ L 2 ( γ ) H 1 H^1 H 1 ( 96 ) inf V ∈ V ( T ) ∥ ∇ Γ ( u ~ ∘ P d − V ) ∥ L 2 ( Γ ) 2 ≲ ∑ T ∈ T inf V T ∈ V ( T ) ∥ ∇ Γ ( u ~ ∘ P d − V T ) ∥ L 2 ( T ) 2
(96) \quad \inf _{V \in \mathbf{V}(\mathcal{T})}\left\|\nabla_{\Gamma}\left(\widetilde{u} \circ \mathbf{P}_{d}-V\right)\right\|_{L_{2}(\Gamma)}^{2} \lesssim \sum_{T \in \mathcal{T}} \inf _{V_{T} \in \mathbf{V}(T)}\left\|\nabla_{\Gamma}\left(\widetilde{u} \circ \mathbf{P}_{d}-V_{T}\right)\right\|_{L_{2}(T)}^{2}
( 9 6 ) V ∈ V ( T ) inf ∥ ∇ Γ ( u ∘ P d − V ) ∥ L 2 ( Γ ) 2 ≲ T ∈ T ∑ V T ∈ V ( T ) inf ∥ ∇ Γ ( u ∘ P d − V T ) ∥ L 2 ( T ) 2
下面是H 1 H^1 H 1 Theorem 37 ( H 1 a-priori error estimate for C 2 surfaces). Let γ be of class C 2 f ~ ∈ L 2 , # ( γ ) and u ~ ∈ H 2 ( γ ) be the solution of ( 18 ) . Let U ∈ V # ( T ) be the solution to ( 78 ) with F = f ~ ∘ P q q r defined via the lift P . If the geometric assumptions ( 69 ) , (90), and (91) are valid, then ∥ ∇ Γ ( u ~ ∘ P − U ) ∥ L 2 ( Γ ) ≲ ( h T + λ τ ( Γ ) ) ∥ f ~ ∥ L 2 ( γ ) ≲ h τ ∥ f ~ ∥ L 2 ( γ ) as well as ∥ ∇ Γ ( u ~ ∘ P d − U ) ∥ L 2 ( Γ ) ≲ ( h T + μ T ( Γ ) ) ∥ f ~ ∥ L 2 ( γ ) ≲ h T ∥ f ~ ∥ L 2 ( γ )
\begin{array}{l}{\text { Theorem } 37\left(H^{1} \text { a-priori error estimate for } C^{2} \text { surfaces). Let } \gamma \text { be of class } C^{2}\right.} \\ {\tilde{f} \in L_{2, \#}(\gamma) \text { and } \widetilde{u} \in H^{2}(\gamma) \text { be the solution of }(18) . \text { Let } U \in \mathbb{V}_{\#}(\mathcal{T}) \text { be the solution }} \\ {\text { to }(78) \text { with } F=\widetilde{f} \circ \mathbf{P} \frac{q}{q_{\mathrm{r}}} \text { defined via the lift } \mathbf{P} . \text { If the geometric assumptions }(69),} \\ {\text { (90), and (91) are valid, then }} \\ {\qquad\left\|\nabla_{\Gamma}(\widetilde{u} \circ \mathbf{P}-U)\right\|_{L_{2}(\Gamma)} \lesssim(h _\mathcal{T}+\lambda \tau(\Gamma))\|\widetilde{f}\|_{L_{2}(\gamma)} \lesssim h \tau\|\widetilde{f}\|_{L_{2}(\gamma)}} \\ {\text { as well as }} \\ {\qquad\left\|\nabla_{\Gamma}\left(\widetilde{u} \circ \mathbf{P}_{d}-U\right)\right\|_{L_{2}(\Gamma)} \lesssim\left(h_{\mathcal{T}}+\mu_{\mathcal{T}}(\Gamma)\right)\|\tilde{f}\|_{L_{2}(\gamma)} \lesssim h_{\mathcal{T}}\|\tilde{f}\|_{L_{2(\gamma)}}}\end{array}
Theorem 3 7 ( H 1 a-priori error estimate for C 2 surfaces). Let γ be of class C 2 f ~ ∈ L 2 , # ( γ ) and u ∈ H 2 ( γ ) be the solution of ( 1 8 ) . Let U ∈ V # ( T ) be the solution to ( 7 8 ) with F = f ∘ P q r q defined via the lift P . If the geometric assumptions ( 6 9 ) , (90), and (91) are valid, then ∥ ∇ Γ ( u ∘ P − U ) ∥ L 2 ( Γ ) ≲ ( h T + λ τ ( Γ ) ) ∥ f ∥ L 2 ( γ ) ≲ h τ ∥ f ∥ L 2 ( γ ) as well as ∥ ∇ Γ ( u ∘ P d − U ) ∥ L 2 ( Γ ) ≲ ( h T + μ T ( Γ ) ) ∥ f ~ ∥ L 2 ( γ ) ≲ h T ∥ f ~ ∥ L 2 ( γ ) H 1 H^1 H 1
在L 2 L^2 L 2 Theorem 38 ( L 2 a-priori error estimate for C 2 surfaces). Let γ be of class C 2 and be described by a generic lift P of class C 2 . Let the geometric conditions ( 69 ) , , and ( 91 ) be satisfied. Let u ~ ∈ H # 1 ( γ ) solve ( 19 ) and U ∈ V # ( T ) solve ( 78 ) with F = f ~ ∘ P q q T . Then (97) ∥ u ~ ∘ P − U ∥ L 2 ( Γ ) ≲ h T 2 ∥ f ~ ∥ L 2 ( γ )
\begin{array}{l}{\text { Theorem 38 ( } L_{2} \text { a-priori error estimate for } C^{2} \text { surfaces). Let } \gamma \text { be of class } C^{2} \text { and }} \\ {\text { be described by a generic lift } \mathbf{P} \text { of class } C^{2} . \text { Let the geometric conditions }(69), \text { , }} \\ {\text { and }(91) \text { be satisfied. Let } \widetilde{u} \in H_{\#}^{1}(\gamma) \text { solve }(19) \text { and } U \in \mathbb{V}_{\#}(\mathcal{T}) \text { solve }(78) \text { with }} \\ {F=\widetilde{f} \circ \mathbf{P} \frac{q}{q_{\mathrm{T}}} . \text { Then }} \\ {\begin{array}{ll}{\text { (97) }} & {\|\widetilde{u} \circ \mathbf{P}-U\|_{L_{2}(\Gamma)} \lesssim h_{\mathcal{T}}^{2}\|\widetilde{f}\|_{L_{2}(\gamma)}}\end{array}}\end{array}
Theorem 38 ( L 2 a-priori error estimate for C 2 surfaces). Let γ be of class C 2 and be described by a generic lift P of class C 2 . Let the geometric conditions ( 6 9 ) , , and ( 9 1 ) be satisfied. Let u ∈ H # 1 ( γ ) solve ( 1 9 ) and U ∈ V # ( T ) solve ( 7 8 ) with F = f ∘ P q T q . Then (97) ∥ u ∘ P − U ∥ L 2 ( Γ ) ≲ h T 2 ∥ f ∥ L 2 ( γ ) ∫ γ ∇ γ z ~ ⋅ ∇ γ w ~ = ∫ γ ( u ~ − U ~ # ) w ~ ∀ w ~ ∈ H # 1 ( γ )
\int_{\gamma} \nabla_{\gamma} \widetilde{z} \cdot \nabla_{\gamma} \widetilde{w}=\int_{\gamma}\left(\widetilde{u}-\widetilde{U}_{\#}\right) \widetilde{w} \quad \forall \widetilde{w} \in H_{\#}^{1}(\gamma)
∫ γ ∇ γ z ⋅ ∇ γ w = ∫ γ ( u − U # ) w ∀ w ∈ H # 1 ( γ ) ∫ Γ ∇ Γ Z ⋅ ∇ Γ W = ∫ Γ ( u # − U ) W , ∀ W ∈ V ( T )
\int_{\Gamma} \nabla_{\Gamma} Z \cdot \nabla_{\Gamma} W=\int_{\Gamma}\left(u_{\#}-U\right) W, \quad \forall W \in \mathbb{V}(\mathcal{T})
∫ Γ ∇ Γ Z ⋅ ∇ Γ W = ∫ Γ ( u # − U ) W , ∀ W ∈ V ( T ) ∥ u ~ − U ~ # ∥ L 2 ( γ ) 2 = ∫ γ ∇ γ ( u ~ − U ~ ) ⋅ ∇ γ ( z ~ − Z ~ ) + ∫ γ f ~ ( Z ∘ P d − 1 − Z ∘ P − 1 ) + ∫ γ ∇ γ U ~ ⋅ E ∇ γ Z ~
\begin{aligned}\left\|\widetilde{u}-\widetilde{U}_{\#}\right\|_{L_{2}(\gamma)}^{2} &=\int_{\gamma} \nabla_{\gamma}(\widetilde{u}-\widetilde{U}) \cdot \nabla_{\gamma}(\widetilde{z}-\widetilde{Z}) \\ &+\int_{\gamma} \widetilde{f}\left(Z \circ \mathbf{P}_{d}^{-1}-Z \circ \mathbf{P}^{-1}\right) \\ &+\int_{\gamma} \nabla_{\gamma} \widetilde{U} \cdot \mathbf{E} \nabla_{\gamma} \widetilde{Z} \end{aligned}
∥ ∥ ∥ u − U # ∥ ∥ ∥ L 2 ( γ ) 2 = ∫ γ ∇ γ ( u − U ) ⋅ ∇ γ ( z − Z ) + ∫ γ f ( Z ∘ P d − 1 − Z ∘ P − 1 ) + ∫ γ ∇ γ U ⋅ E ∇ γ Z C 2 C^2 C 2 C 3 C^3 C 3 P = P d \mathbf{P}=\mathbf{P}_{d} P = P d (99) ∥ u ~ ∘ P − U ∥ L 2 ( Γ ) ≲ h T 2 ∣ d ∣ W ∞ 2 ( N ) ∥ f ~ ∥ L 2 ( γ )
\begin{array}{ll}{\text { (99) }} & {\|\widetilde{u} \circ \mathbf{P}-U\|_{L_{2}(\Gamma)} \lesssim h_{\mathcal{T}}^{2}|d|_{W_{\infty}^{2}(\mathcal{N})}\|\widetilde{f}\|_{L_{2}(\gamma)}}\end{array}
(99) ∥ u ∘ P − U ∥ L 2 ( Γ ) ≲ h T 2 ∣ d ∣ W ∞ 2 ( N ) ∥ f ∥ L 2 ( γ )
對於C1曲面，我們同樣有H 1 H^1 H 1 Lemma 39 (approximability in H 1 ( Γ ) ) . Let γ be a surface of class C 1 , α and u ~ ∈ H 1 + s ( γ ) , where 0 < s < α < 1 or 0 < s ≤ α = 1. Then we have (100) inf V ∈ V ( T ) ∥ ∇ Γ ( u ~ ∘ P − V ) ∥ L 2 ( Γ ) ≲ h T s ∣ u ~ ∣ H 1 + s ( γ )
\begin{array}{l}{ \text { Lemma }\left.39 \text { (approximability in } H^{1}(\Gamma)\right) . \text { Let } \gamma \text { be a surface of class } C^{1, \alpha} \text { and }} \\ {\widetilde{u} \in H^{1+s}(\gamma), \text { where } 0<s<\alpha<1 \text { or } 0<s \leq \alpha=1 . \text { Then we have }} \\ {\begin{array}{lll}{\text { (100) }} & {\inf _{V \in \mathbb{V}(\mathcal{T})}\left\|\nabla_{\Gamma}(\widetilde{u} \circ \mathbf{P}-V)\right\|_{L_{2}(\Gamma)} \lesssim h_{\mathcal{T}}^{s}|\widetilde{u}|_{H^{1+s}(\gamma)}}\end{array}}\end{array}
Lemma 3 9 (approximability in H 1 ( Γ ) ) . Let γ be a surface of class C 1 , α and u ∈ H 1 + s ( γ ) , where 0 < s < α < 1 or 0 < s ≤ α = 1 . Then we have (100) inf V ∈ V ( T ) ∥ ∇ Γ ( u ∘ P − V ) ∥ L 2 ( Γ ) ≲ h T s ∣ u ∣ H 1 + s ( γ )
那麼C1曲面的H 1 H^1 H 1 Theorem 40 ( H 1 a-priori error estimate for C 1 , α surfaces). Let γ be of class C 1 , α , 0 < α ≤ 1 , and assume that the geometric assumptions ( 69 ) , ( 90 ) , and ( 91 ) are valid. Let f ^ ∈ L 2 , # ( γ ) and u ~ ∈ H 1 + s ( γ ) be the solution of ( 18 ) and satisfy
\begin{array}{l}{\text { Theorem } 40\left(H^{1} \text { a-priori error estimate for } C^{1, \alpha} \text { surfaces). Let } \gamma \text { be of class } C^{1, \alpha},\right.} \\ {0<\alpha \leq 1, \text { and assume that the geometric assumptions }(69),(90), \text { and }(91) \text { are }} \\ {\text { valid. Let } \widehat{f} \in L_{2, \#}(\gamma) \text { and } \widetilde{u} \in H^{1+s}(\gamma) \text { be the solution of }(18) \text { and satisfy }}\end{array}
Theorem 4 0 ( H 1 a-priori error estimate for C 1 , α surfaces). Let γ be of class C 1 , α , 0 < α ≤ 1 , and assume that the geometric assumptions ( 6 9 ) , ( 9 0 ) , and ( 9 1 ) are valid. Let f ∈ L 2 , # ( γ ) and u ∈ H 1 + s ( γ ) be the solution of ( 1 8 ) and satisfy ∥ u ~ ∥ H 1 + s ( γ ) ≲ ∥ f ~ ∥ L 2 ( γ ) provided 0 < s < α < 1 or 0 < s ≤ α = 1. If U ∈ V # ( T ) is the solution to ( 78 ) with F = f ~ ∘ P q q r defined via the lift P , then ∥ ∇ Γ ( u ~ ∘ P − U ) ∥ L 2 ( Γ ) ≲ h T s ∥ u ~ ∥ H 1 + s ( γ ) + λ T ( Γ ) ∥ f ~ ∥ L 2 ( γ ) ≲ h T s ∥ f ~ ∥ L 2 ( γ )
\begin{array}{l}{\qquad\|\widetilde{u}\|_{H^{1+s}(\gamma)} \lesssim\|\tilde{f}\|_{L_{2}(\gamma)}} \\ {\text { provided } 0<s<\alpha<1 \text { or } 0<s \leq \alpha=1 . \text { If } U \in \mathbb{V}_{\#}(\mathcal{T}) \text { is the solution to }(78)} \\ {\text { with } F=\widetilde{f} \circ \mathbf{P} \frac{q}{q_{\mathrm{r}}} \text { defined via the lift } \mathbf{P}, \text { then }} \\ {\quad\left\|\nabla_{\Gamma}(\widetilde{u} \circ \mathbf{P}-U)\right\|_{L_{2}(\Gamma)} \lesssim h_{\mathcal{T}}^{s}\|\widetilde{u}\|_{H^{1+s}(\gamma)}+\lambda_{\mathcal{T}}(\Gamma)\|\tilde{f}\|_{L_{2}(\gamma)} \lesssim h_{\mathcal{T}}^{s}\|\tilde{f}\|_{L_{2}(\gamma)}}\end{array}
∥ u ∥ H 1 + s ( γ ) ≲ ∥ f ~ ∥ L 2 ( γ ) provided 0 < s < α < 1 or 0 < s ≤ α = 1 . If U ∈ V # ( T ) is the solution to ( 7 8 ) with F = f ∘ P q r q defined via the lift P , then ∥ ∇ Γ ( u ∘ P − U ) ∥ L 2 ( Γ ) ≲ h T s ∥ u ∥ H 1 + s ( γ ) + λ T ( Γ ) ∥ f ~ ∥ L 2 ( γ ) ≲ h T s ∥ f ~ ∥ L 2 ( γ )
後驗誤差估計依賴於數值解U U U u ~ \tilde u u ~
要說明後驗誤差估計，我們可以先了解一下scott-zhang插值：I T s z : H 1 ( Γ ) → V ( T )
\mathcal{I}_{\mathcal{T}}^{\mathrm{sz}}: H^{1}(\Gamma) \rightarrow \mathbb{V}(\mathcal{T})
I T s z : H 1 ( Γ ) → V ( T ) ( 101 ) ∥ v − I T s z v ∥ L 2 ( T ) ≲ h T ∥ ∇ Γ v ∥ L 2 ( ω T ) , ∥ ∇ Γ I T s z v ∥ L 2 ( T ) ≲ ∥ ∇ Γ v ∥ L 2 ( ω T )
(101) \quad\left\|v-\mathcal{I}_{\mathcal{T}}^{\mathrm{sz}} v\right\|_{L^{2}(T)} \lesssim h_{T}\left\|\nabla_{\Gamma} v\right\|_{L^{2}\left(\omega_{T}\right)}, \quad\left\|\nabla_{\Gamma} \mathcal{I}_{\mathcal{T}}^{\mathrm{sz}} v\right\|_{L^{2}(T)} \lesssim\left\|\nabla_{\Gamma} v\right\|_{L^{2}\left(\omega_{T}\right)}
( 1 0 1 ) ∥ v − I T s z v ∥ L 2 ( T ) ≲ h T ∥ ∇ Γ v ∥ L 2 ( ω T ) , ∥ ∇ Γ I T s z v ∥ L 2 ( T ) ≲ ∥ ∇ Γ v ∥ L 2 ( ω T )
我們還需要定義兩個函數量“內部”和“跳躍殘差”：R T ( V ) : = F ∣ T + Δ Γ V ∣ T ∀ T ∈ T J S ( V ) : = ∇ Γ V + ∣ S ⋅ μ S + + ∇ Γ V − ∣ S ⋅ μ S − ∀ S ∈ S T
\begin{aligned} R_{T}(V) &:=\left.F\right|_{T}+\left.\Delta_{\Gamma} V\right|_{T} \quad \forall T \in \mathcal{T} \\ J_{S}(V) &:=\left.\nabla_{\Gamma} V^{+}\right|_{S} \cdot \boldsymbol{\mu}_{S}^{+}+\left.\nabla_{\Gamma} V^{-}\right|_{S} \cdot \boldsymbol{\mu}_{S}^{-} \quad \forall S \in S_{\mathcal{T}} \end{aligned}
R T ( V ) J S ( V ) : = F ∣ T + Δ Γ V ∣ T ∀ T ∈ T : = ∇ Γ V + ∣ ∣ S ⋅ μ S + + ∇ Γ V − ∣ ∣ S ⋅ μ S − ∀ S ∈ S T η T ( V , T ) 2 : = h T 2 ∥ R T ( V ) ∥ L 2 ( T ) 2 + h T ∥ J ∂ T ( V ) ∥ L 2 ( ∂ T ) 2 ∀ T ∈ T
\eta _\mathcal{T}(V, T)^{2}:=h_{T}^{2}\left\|R_{T}(V)\right\|_{L^{2}(T)}^{2}+h_{T}\left\|J_{\partial T}(V)\right\|_{L^{2}(\partial T)}^{2} \quad \forall T \in \mathcal{T}
η T ( V , T ) 2 : = h T 2 ∥ R T ( V ) ∥ L 2 ( T ) 2 + h T ∥ J ∂ T ( V ) ∥ L 2 ( ∂ T ) 2 ∀ T ∈ T η T ( V ) 2 : = ∑ T ∈ T η T ( V , T ) 2
\eta_{\mathcal{T}}(V)^{2}:=\sum_{T \in \mathcal{T}} \eta \mathcal{T}(V, T)^{2}
η T ( V ) 2 : = T ∈ T ∑ η T ( V , T ) 2
那麼，我們首先有C1曲面的後驗誤差上界（H1空間）： Theorem 41 (a-posteriori upper bound for C 1 , α surfaces). Let γ be of class C 1 , α , be parametrized by χ = P ∘ X and satisfy the geometric assumption (69). Let u ~ ∈ H 1 ( γ ) be the solution to ( 18 ) and U ∈ V # ( T ) be the solution to (78) with F = f ~ ∘ P q q r ∈ L 2 , # ( Γ ) . Then, for U ~ : = U ∘ P − 1 : γ → R we have ∥ ∇ γ ( u ~ − U ~ ) ∥ L 2 ( γ ) 2 ≲ η T ( U ) 2 + λ T 2 ( Γ ) ∥ f ~ ∥ L 2 ( γ ) 2
\begin{array}{l}{\text { Theorem } 41 \text { (a-posteriori upper bound for } C^{1, \alpha} \text { surfaces). Let } \gamma \text { be of class } C^{1, \alpha},} \\ {\text { be parametrized by } \chi=\mathbf{P} \circ \mathbf{X} \text { and satisfy the geometric assumption (69). Let }} \\ {\widetilde{u} \in H^{1}(\gamma) \text { be the solution to }(18) \text { and } U \in \mathbb{V}_{\#}(\mathcal{T}) \text { be the solution to (78) with }} \\ {F=\tilde{f} \circ \mathbf{P} \frac{q}{q_{\mathrm{r}}} \in L_{2, \#}(\Gamma) . \text { Then, for } \tilde{U}:=U \circ \mathbf{P}^{-1}: \gamma \rightarrow \mathbb{R} \text { we have }} \\ {\qquad\left\|\nabla_{\gamma}(\widetilde{u}-\widetilde{U})\right\|_{L^{2}(\gamma)}^{2} \lesssim \eta_{\mathcal{T}}(U)^{2}+\lambda_{\mathcal{T}}^{2}(\Gamma)\|\widetilde{f}\|_{L_{2}(\gamma)}^{2}}\end{array}
Theorem 4 1 (a-posteriori upper bound for C 1 , α surfaces). Let γ be of class C 1 , α , be parametrized by χ = P ∘ X and satisfy the geometric assumption (69). Let u ∈ H 1 ( γ ) be the solution to ( 1 8 ) and U ∈ V # ( T ) be the solution to (78) with F = f ~ ∘ P q r q ∈ L 2 , # ( Γ ) . Then, for U ~ : = U ∘ P − 1 : γ → R we have ∥ ∥ ∥ ∇ γ ( u − U ) ∥ ∥ ∥ L 2 ( γ ) 2 ≲ η T ( U ) 2 + λ T 2 ( Γ ) ∥ f ∥ L 2 ( γ ) 2 ( 102 ) ∫ γ ∇ γ ( u ~ − U ~ ) ⋅ ∇ γ v ~ = I 1 + I 2 + I 3
(102) \quad \int_{\gamma} \nabla_{\gamma}(\widetilde{u}-\widetilde{U}) \cdot \nabla_{\gamma} \widetilde{v}=I_{1}+I_{2}+I_{3}
( 1 0 2 ) ∫ γ ∇ γ ( u − U ) ⋅ ∇ γ v = I 1 + I 2 + I 3 I 1 = − ∫ Γ ∇ Γ U ⋅ ∇ Γ ( v − V ) + ∫ Γ F ( v − V ) I 2 = ∫ γ ∇ γ U ~ ⋅ E ∇ γ v ~ I 3 = ∫ γ f ~ v ~ − ∫ Γ F v
\begin{aligned} I_{1} &=-\int_{\Gamma} \nabla_{\Gamma} U \cdot \nabla_{\Gamma}(v-V)+\int_{\Gamma} F(v-V) \\ I_{2} &=\int_{\gamma} \nabla_{\gamma} \widetilde{U} \cdot \mathbf{E} \nabla_{\gamma} \widetilde{v} \\ I_{3} &=\int_{\gamma} \widetilde{f} \widetilde{v}-\int_{\Gamma} F v \end{aligned}
I 1 I 2 I 3 = − ∫ Γ ∇ Γ U ⋅ ∇ Γ ( v − V ) + ∫ Γ F ( v − V ) = ∫ γ ∇ γ U ⋅ E ∇ γ v = ∫ γ f v − ∫ Γ F v ( 103 ) I 1 = ∑ T ∈ T ∫ T R T ( U ) ( v − V ) + ∑ S ∈ S ∫ S J S ( U ) ( v − V )
(103) \quad I_{1}=\sum_{T \in \mathcal{T}} \int_{T} R_{T}(U)(v-V)+\sum_{S \in \mathcal{S}} \int_{S} J_{S}(U)(v-V)
( 1 0 3 ) I 1 = T ∈ T ∑ ∫ T R T ( U ) ( v − V ) + S ∈ S ∑ ∫ S J S ( U ) ( v − V ) ( 104 ) I 1 ≲ η T ( U ) ∥ ∇ Γ v ∥ L 2 ( Γ ) ≲ η T ( U ) ∥ ∇ γ v ~ ∥ L 2 ( γ )
(104) \quad I_{1} \lesssim \eta_{\mathcal{T}}(U)\left\|\nabla_{\Gamma} v\right\|_{L^{2}(\Gamma)} \lesssim \eta_{\mathcal{T}}(U)\left\|\nabla_{\gamma} \tilde{v}\right\|_{L^{2}(\gamma)}
( 1 0 4 ) I 1 ≲ η T ( U ) ∥ ∇ Γ v ∥ L 2 ( Γ ) ≲ η T ( U ) ∥ ∇ γ v ~ ∥ L 2 ( γ )
之後分別估計每一部分的界即可。
對於C1下界，有如下估計： Theorem 42 (a-posteriori lower bound for C 1 , α surfaces). Under the same condi- tions of Theorem 41 (a-posteriori upper bound for C 1 , α surfaces), we have η T ( U , T ) 2 ≲ ∥ ∇ γ ( u ~ − U ~ ) ∥ L 2 ( ω ~ T ) 2 + osc T ( F , ω T ) 2 + λ T 2 ( ω T )
\begin{array}{l}{\text { Theorem } 42 \text { (a-posteriori lower bound for } C^{1, \alpha} \text { surfaces). Under the same condi- }} \\ {\text { tions of Theorem } 41 \text { (a-posteriori upper bound for } C^{1, \alpha} \text { surfaces), we have }} \\ {\qquad \eta_{\mathcal{T}}(U, T)^{2} \lesssim\left\|\nabla_{\gamma}(\widetilde{u}-\widetilde{U})\right\|_{L^{2}\left(\widetilde{\omega}_{T}\right)}^{2}+\operatorname{osc}_{\mathcal{T}}\left(F, \omega_{T}\right)^{2}+\lambda_{\mathcal{T}}^{2}\left(\omega_{T}\right)}\end{array}
Theorem 4 2 (a-posteriori lower bound for C 1 , α surfaces). Under the same condi- tions of Theorem 4 1 (a-posteriori upper bound for C 1 , α surfaces), we have η T ( U , T ) 2 ≲ ∥ ∥ ∥ ∇ γ ( u − U ) ∥ ∥ ∥ L 2 ( ω T ) 2 + o s c T ( F , ω T ) 2 + λ T 2 ( ω T )
下面介紹C2曲面的後驗誤差上界，爲此，介紹常量“數據震盪”如下：osc T ( F , T ) 2 : = h T 2 ∥ F − F ˉ ∥ L 2 ( T ) 2 , osc T ( F ) 2 : = ∑ T ∈ T osc T ( F , T ) 2
\operatorname{osc} \mathcal{T}(F, T)^{2}:=h_{T}^{2}\|F-\bar{F}\|_{L^{2}(T)}^{2}, \quad \operatorname{osc} \mathcal{T}(F)^{2}:=\sum_{T \in \mathcal{T}} \operatorname{osc} \mathcal{T}(F, T)^{2}
o s c T ( F , T ) 2 : = h T 2 ∥ F − F ˉ ∥ L 2 ( T ) 2 , o s c T ( F ) 2 : = T ∈ T ∑ o s c T ( F , T ) 2
那麼，我們有如下定理（C2曲面的後驗誤差上界）： Theorem 43 (a-posteriori upper bound for C 2 surfaces). Let γ be of class C 2 and ( 67 ) , ( 74 ) , ( 90 ) , and (91) hold. Let u ~ be the solution of ( 18 ) with f ~ ∈ L 2 , # ( γ ) and U ∈ V ( T ) be the solution to ( 78 ) with F = f ~ ∘ P q q r , where q corresponds to the parametrization χ = P ∘ X of γ . Then ∥ ∇ γ ( u ~ − U ∘ P d − 1 ) ∥ L 2 ( γ ) 2 ≲ η T ( U ) 2 + μ T 2 ( Γ ) ∥ f ~ ∥ L 2 ( γ ) 2
\begin{array}{l}{\text { Theorem } 43 \text { (a-posteriori upper bound for } C^{2} \text { surfaces). Let } \gamma \text { be of class } C^{2} \text { and }} \\ {(67),(74),(90), \text { and (91) hold. Let } \widetilde{u} \text { be the solution of }(18) \text { with } \widetilde{f} \in L_{2, \#}(\gamma) \text { and }} \\ {U \in \mathbb{V}(\mathcal{T}) \text { be the solution to }(78) \text { with } F=\widetilde{f} \circ \mathrm{P} \frac{q}{q_{\mathrm{r}}}, \text { where } q \text { corresponds to the }} \\ {\text { parametrization } \chi=\mathrm{P} \circ \mathrm{X} \text { of } \gamma \text { . Then }} \\ {\qquad\left\|\nabla_{\gamma}\left(\widetilde{u}-U \circ \mathbf{P}_{d}^{-1}\right)\right\|_{L_{2}(\gamma)}^{2} \lesssim \eta _\mathcal{T}(U)^{2}+\mu_{\mathcal{T}}^{2}(\Gamma)\|\tilde{f}\|_{L_{2}(\gamma)}^{2}}\end{array}
Theorem 4 3 (a-posteriori upper bound for C 2 surfaces). Let γ be of class C 2 and ( 6 7 ) , ( 7 4 ) , ( 9 0 ) , and (91) hold. Let u be the solution of ( 1 8 ) with f ∈ L 2 , # ( γ ) and U ∈ V ( T ) be the solution to ( 7 8 ) with F = f ∘ P q r q , where q corresponds to the parametrization χ = P ∘ X of γ . Then ∥ ∥ ∇ γ ( u − U ∘ P d − 1 ) ∥ ∥ L 2 ( γ ) 2 ≲ η T ( U ) 2 + μ T 2 ( Γ ) ∥ f ~ ∥ L 2 ( γ ) 2 ( 105 ) ∫ γ ∇ γ ( u ~ − U ~ ) ⋅ ∇ γ v ~ = I 1 + I 2 + I 3 with I 1 = − ∫ Γ ∇ Γ U ⋅ ∇ Γ ( v − V ) + ∫ Γ F ( v − V ) I 2 = ∫ γ ∇ γ U ~ ⋅ E ∇ γ v ~ I 3 = ∫ γ f ~ v ~ − ∫ Γ F v
\begin{array}{l}{(105) \quad \quad \int_{\gamma} \nabla_{\gamma}(\widetilde{u}-\widetilde{U}) \cdot \nabla_{\gamma} \widetilde{v}=I_{1}+I_{2}+I_{3}} \\ {\text { with }} \\ {\qquad \begin{aligned} I_{1} &=-\int_{\Gamma} \nabla_{\Gamma} U \cdot \nabla_{\Gamma}(v-V)+\int_{\Gamma} F(v-V) \\ I_{2} &=\int_{\gamma} \nabla_{\gamma} \widetilde{U} \cdot \mathbf{E} \nabla_{\gamma} \widetilde{v} \\ I_{3} &=\int_{\gamma} \widetilde{f} \tilde{v}-\int_{\Gamma} F v \end{aligned}}\end{array}
( 1 0 5 ) ∫ γ ∇ γ ( u − U ) ⋅ ∇ γ v = I 1 + I 2 + I 3 with I 1 I 2 I 3 = − ∫ Γ ∇ Γ U ⋅ ∇ Γ ( v − V ) + ∫ Γ F ( v − V ) = ∫ γ ∇ γ U ⋅ E ∇ γ v = ∫ γ f v ~ − ∫ Γ F v I 3 I_3 I 3 ( 106 ) I 3 = ∫ γ f ~ ( v ~ − v ~ ∘ P d ∘ P − 1 )
(106) \quad I_{3}=\int_{\gamma} \widetilde{f}\left(\widetilde{v}-\widetilde{v} \circ \mathbf{P}_{d} \circ \mathbf{P}^{-1}\right)
( 1 0 6 ) I 3 = ∫ γ f ( v − v ∘ P d ∘ P − 1 ) I 3 ≲ β T ( Γ ) ∥ f ~ ∥ L 2 ( γ ) ∥ ∇ γ v ~ ∥ L 2 ( γ )
I_{3} \lesssim \beta_{\mathcal{T}}(\Gamma)\|\tilde{f}\|_{L_{2}(\gamma)}\left\|\nabla_{\gamma} \tilde{v}\right\|_{L_{2}(\gamma)}
I 3 ≲ β T ( Γ ) ∥ f ~ ∥ L 2 ( γ ) ∥ ∇ γ v ~ ∥ L 2 ( γ )
C2曲面的後驗誤差下界如下所述： Theorem 44 (a-posteriori lower bound for C 2 surfaces). Under the same condi- tions as Theorem 43 (a-posteriori upper bound for C 2 surfaces), we have η T ( U , T ) 2 ≲ ∥ ∇ γ ( u ~ − U ~ ) ∥ L 2 ( ω ~ T ) 2 + osc T ( F , ω T ) 2 + μ T ( ω T ) 2 where μ T ( ω T ) = max T ′ ⊂ ω T μ T ′
\begin{array}{l}{\text { Theorem } 44 \text { (a-posteriori lower bound for } C^{2} \text { surfaces). Under the same condi- }} \\ {\text { tions as Theorem } 43 \text { (a-posteriori upper bound for } C^{2} \text { surfaces), we have }} \\ {\qquad \eta_{\mathcal{T}}(U, T)^{2} \lesssim\left\|\nabla_{\gamma}(\widetilde{u}-\widetilde{U})\right\|_{L^{2}\left(\widetilde{\omega}_{T}\right)}^{2}+\operatorname{osc}_{\mathcal{T}}\left(F, \omega_{T}\right)^{2}+\mu \mathcal{T}\left(\omega_{T}\right)^{2}} \\ {\text { where } \mu_{\mathcal{T}}\left(\omega_{T}\right)=\max _{T^{\prime} \subset \omega_{T}} \mu_{T^{\prime}}}\end{array}
Theorem 4 4 (a-posteriori lower bound for C 2 surfaces). Under the same condi- tions as Theorem 4 3 (a-posteriori upper bound for C 2 surfaces), we have η T ( U , T ) 2 ≲ ∥ ∥ ∥ ∇ γ ( u − U ) ∥ ∥ ∥ L 2 ( ω T ) 2 + o s c T ( F , ω T ) 2 + μ T ( ω T ) 2 where μ T ( ω T ) = max T ′ ⊂ ω T μ T ′
