2 DGM for elliptic problems

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26 2 Elliptic problems |Jh σ (u h ,v h )| ≤ ∑ ∫ Γ ∈F ID h ⎛ ≤ ⎝ ∑ Γ ∈F ID h Γ ∫ σ|[u h ][v h ]|dS (2.98) A2.43 Γ ⎞ σ[u h ] 2 dS⎠ 1/2 ⎛ = J σ h(u h ,u h ) 1/2 J σ h(v h ,v h ) 1/2 , which together with ( en 2.68) and ( A2.42 2.97) gives ⎝ ∑ Γ ∈F ID h ∫ Γ ⎞ σ[v h ] 2 dS⎠ |B h (u h ,v h )| ≤ c a |||u||| |||v||| + J σ h (u h ,u h ) 1/2 J σ h(v h ,v h ) 1/2 ≤ (c a + 1)|||u h ||| |||v h |||, which gives ( A2.41 2.96) with C B := c a + 1. ⊓⊔ Remark 2.20. In the same way as in ( A2.43 2.98) it is possible to show that |J σ h (u,v)| = J σ h(u,u) 1/2 J σ h(v,v) 1/2 ∀u,v ∈ H 1 (Ω, T h ). (2.99) A2.J 1/2 coercofB lem:2.4 2.4.2 Coercivity of the bilinear forms Lemma 2.21. (NIPG coercivity) For any C W > 0 the bilinear form B n,σ h defined by ( 2.53) A2.20d satisfies the coercivity condition B n,σ h (v,v) ≥ |||v|||2 ∀v ∈ H 2 (Ω, T h ). (2.100) A2.50 Proof. From ( ah_N 2.45) and ( A2.20d 2.53) it follows immediately B n,σ h (v,v) = an h(v,v) + J σ h(v,v) = |v| 2 H 1 (Ω,T h ) + Jσ h (v,v) = |||v||| 2 .⊓⊔ (2.101) A2.51 The proof of the coercivity of the symmetric bilinear form B s,σ h complicated. is more Lemma 2.22. (SIPG coercivity) Let assumptions (A1) and (A2) be satis- fied, let C W ≥ 4C H C M (1 + C I ), (2.102) A3.40 where C M , C I and C H are constants from ( 2.19), eq:MTI ( 2.26) eq:InvI and ( 2.18) A1.35 respectively, and let the penalty parametr σ be given by ( 2.69) A3.1 for all Γ ∈ Fh ID. Then B s,σ h (v h,v h ) ≥ 1 2 |||v h||| 2 ∀v h ∈ S hp , ∀h ∈ (0,h 0 ). (2.103) A3.41 lem:2.8 Proof. Let δ > 0, then from ( ah_S 2.44) and the Cauchy and Young’s inequalities it follows that
2.4 Properties of the bilinear forms 27 a s h(v h ,v h ) (2.104) A3.42 = |v h | 2 H 1 (Ω,T h ) − 2 ∑ ∫ n · 〈∇v h 〉[v h ]dS Γ ∈F ID h Γ ≥ |v h | 2 H 1 (Ω,T h ) ⎧ ⎨ 1 ∑ −2 ⎩δ Γ ∈F ID h ∫ Γ ≥ |v h | 2 H 1 (Ω,T h ) − ω − ⎫1/2 ⎧ ⎬ ⎨ h Γ (n · 〈∇v h 〉) 2 dS ⎭ ⎩ δ ∑ δ C W J σ h (v h ,v h ), Γ ∈F ID h ∫ Γ ⎫ 1 ⎬ [v h ] 2 dS h Γ ⎭ 1/2 where ω = 1 δ ∑ Γ ∈F ID Γ h Further, from ( 2.69), A3.1 ( 2.81),( A3.7 2.19) eq:MTI and ( 2.26), eq:InvI we get ω ≤ C H δ ≤ C HC M δ ∑ ∫ h Γ |〈∇v h 〉| 2 dS. (2.105) A3.43 h K ‖v h ‖ 2 L 2 (∂K) (2.106) A3.44 K∈T h ∑ ( ) h K |v h | H1 (K)|∇v h | H1 (K) + h −1 K |v h| 2 H 1 (K) K∈T h ≤ C HC M (1 + C I ) |v h | 2 H δ 1 (Ω,T h ) . Now let us choose δ = 2C H C M (1 + C I ). (2.107) A3.45 then it follows from ( 2.102) A3.40 and ( 2.104) A3.42 – ( 2.107) A3.45 that a s h(v h ,v h ) (2.108) A3.46 ≥ 1 ( |v h | 2 H 2 1 (Ω,T h ) − 4C ) HC M (1 + C I ) J σ C h(v h ,v h ) W ≥ 1 ( ) |v h | 2 H 2 1 (Ω,T h ) − Jσ h(v h ,v h ) . Finally, from ( A2.20c 2.52) and ( A3.46 2.108) we have B s,σ h (v h,v h ) = a s h(v h ,v h ) + Jh σ (v h ,v h ) (2.109) A3.47 ≥ 1 ( ) |v h | 2 H 2 1 (Ω,T h ) + Jσ h(v h ,v h ) = 1 2 |||v h||| 2 , which we wanted to prove. ⊓⊔ Lemma 2.23. (IIPG coercivity) Let assumptions (A1) and (A2) be satis- fied, let lem:2.8I
Page 1 and 2: 2 DGM for elliptic problems chap:el
Page 3 and 4: 2.2 Spaces of discontinuous functio
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Page 7 and 8: lem:II lem:III def:Pi_h 2.2 Spaces
Page 9 and 10: Due to the assumption that u ∈ H
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Page 13 and 14: where h K (L) Γ and h K (R) Γ 2.4
Page 15 and 16: lem:A11a 2.4 Properties of the bili
Page 17: 2.4 Properties of the bilinear form
Page 21 and 22: 2.5 Error estimates 29 lem:2.7 wher
Page 23 and 24: ∑ 2.5 Error estimates 31 h K ‖
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Page 29 and 30: 2.6 L 2 (Ω)-optimal error estimate

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