2 DGM for elliptic problems

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12 2 Elliptic problems Γ ⃗n Γ K (L) Γ K (R) Γ Fig. 2.2. Interior face Γ, elements K (L) Γ and K (R) Γ and the orientation of n Γ fig:normals For each Γ ∈ Fh I such that Γ ⊂ ∂K (L) Γ normal to the element ∂K (L) Γ Figure fig:normals 2.2. We call elements K (L) Γ introduce the following notation: there exist two neighbouring elements K(L) Γ ,K(R) Γ ∈ T h ∩ ∂K (R) Γ . We use a convention that n Γ is the outer v| (L) Γ v| (R) Γ and the inner normal to the element ∂K (R) Γ , see neighbours. For v ∈ H 1 (Ω, T h ), we , K(R) Γ = the trace of v| K (L) Γ = the trace of v| (R) K Γ ) 〈v〉 Γ = 1 2 [v] Γ = v| (L) Γ ( v| (L) Γ + v|(R) Γ − v|(R) Γ . , on Γ, (2.14) A1.15 on Γ, The value [v] Γ depends on the orientation of n Γ , but the value [v] Γ n Γ is independent of this orientation. For Γ ∈ Fh DN there exists element K (L) Γ ∈ T h such that Γ ⊂ K (L) Γ ∩ ∂Ω. Then for v ∈ H 1 (Ω, T h ), we introduce the following notation: v| (L) Γ = the trace of v| K (L) Γ 〈v〉 Γ = [v] Γ = v| (L) Γ . on Γ, (2.15) A1.17 For Γ ∈ Fh DN by v| (R) Γ we formally denote the exterior trace of v on Γ given either by a boundary condition or by an extrapolation from the interior of Ω. In case that [·] Γ , 〈 · 〉 Γ and n Γ appear in the integrals ∫ Γ ... dS, Γ ∈ F h, we omit the subscript Γ and write simply [·], 〈 · 〉 and n, respectively. sec:2.1.3 2.2.3 Spaces of discontinuous piecewise polynomial functions As we already mentioned in Introduction, DGM is based on the use of discontinuous piecewise polynomial approximations.
2.2 Spaces of discontinuous functions 13 Let T h be a triangulation of Ω introduced in Section sec:2.1.1 2.2.1 and let p ≥ 0 be an integer. We define the space of discontinuous piecewise polynomial functions S hp = {v;v| K ∈ P p (K) ∀K ∈ T h }, (2.16) A1.23 where P p (K) denotes the space of all polynomials on K of degree ≤ p. We call the number p the degree of polynomial approximation. sec:2.1.4 2.2.4 Some auxiliary results Assumptions on the mesh Let us consider a system {T h } h∈(0,h0), h 0 > 0, of partitions of the domain Ω (T h = {K} K∈Th ). In our further considerations we shall meet the following assumptions. (A1) The system {T h } h∈(0,h0) is shape regular: there exists a positive constant C R such that h K ρ K ≤ C R ∀K ∈ T h , ∀h ∈ (0,h 0 ). (2.17) A1.33 (A2) The system {T h } h∈(0,h0) is locally quasi-uniform: there exists a constant C H > 0 such that h K ≤ C H h K ′ ∀K,K ′ ∈ T h , K,K ′ are neighbours, ∀h ∈ (0,h 0 ). (2.18) A1.35 rem:MA Remark 2.2. Sometimes the quasi-uniformity of the mesh is required, i.e. there exists a constant ˜C > 0 such that h ≤ ˜Ch K ∀K ∈ T h . The condition ( 2.18) A1.35 is weaker condition than quasi-uniformity since it only limits the ratio of diameters of neighbouring elements. Let us introduce important tools for the theoretical analysis of the DG method. lem:MTI Lemma 2.3. (Multiplicative trace inequality) Let assumption (A1) be satisfied. Then there exists a constant C M > 0 independent of v, h and K such that ( ) ‖v‖ 2 L 2 (∂K) ≤ C M ‖v‖ L2 (K) |v| H1 (K) + h −1 K ‖v‖2 L 2 (K) , (2.19) eq:MTI K ∈ T h , v ∈ H 1 (K), h ∈ (0,h 0 ). Proof. Let K ∈ T h be arbitrary but fixed. We denote by x K the centre of the largest d−dimensional ball inscribed into the simplex K. (Of course, ρ K is the radius of this ball.) Without the loss of generality we suppose that x K is the origin of the coordinate system. We start from the following relation ∫ ∫ v 2 x · ndS = ∇ · (v 2 x)dx, v ∈ H 1 (K). (2.20) A1.51 ∂K K
Page 1 and 2: 2 DGM for elliptic problems chap:el
Page 3: 2.2 Spaces of discontinuous functio
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
Page 11 and 12: 2.3 DGM based on a primal formulati
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 and 18: 2.4 Properties of the bilinear form
Page 19 and 20: 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 ‖
Page 25 and 26: 2.5 Error estimates 33 The proof of
Page 27 and 28: 2.6 L 2 (Ω)-optimal error estimate
Page 29 and 30: 2.6 L 2 (Ω)-optimal error estimate

2 DGM for elliptic problems

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