2 DGM for elliptic problems

More documents

Recommendations

Info

18 2 Elliptic problems a n h(u,v) = ∑ K∈T h ∫K − a i h(u,v) = ∑ ∑ Γ ∈F ID h K∈T h ∫K ∇u · ∇v dx (2.45) ah_N ∫ Γ (n · 〈∇u〉 [v] − n · 〈∇u〉 [v]) dS, ∇u · ∇v dx − ∑ Γ ∈F ID h ∫ Γ n · 〈∇u〉 [v]dS (2.46) ah_I and the linear forms ∫ Fh(v) s = f v dx + ∑ ∫ Fh n (v) = ∫ Fh(v) i = Ω Ω Γ ∈F N h f v dx + ∑ Γ ∈F N h f v dx + ∑ Ω Γ ∈F N h ∫ g N v dS + ∑ ∫ n · ∇v u D dS, (2.47) Fh_S ∫ ∫ Γ Γ Γ Γ ∈F D h g N v dS − ∑ ∫ Γ Γ ∈F D Γ h n · ∇v u D dS, (2.48) Fh_N g N v dS. (2.49) Fh_I Moreover, for u,v ∈ H 2 (Ω, T h ) let us define the bilinear forms and the linear forms B s h(u,v) = a s h(u,v), (2.50) A2.20a Bh(u,v) n = a n h(u,v), (2.51) A2.20b B s,σ h (u,v) = as h(u,v) + Jh σ (u,v), (2.52) A2.20c B n,σ h (u,v) = an h(u,v) + Jh σ (u,v), (2.53) A2.20d B i,σ h (u,v) = ai h(u,v) + Jh σ (u,v) (2.54) A2.20e def:2.1 l s h(v) = F s h(v), (2.55) A2.21a l n h(v) = Fh n (v), (2.56) A2.21b l s,σ h (v) = F h(v) s + JD(v), σ (2.57) A2.21c l n,σ h (v) = F h n (v) + JD(v), σ (2.58) A2.21d l i,σ h (v) = F h(v) i + JD(v). σ (2.59) A2.21e Since S hp ⊂ H 2 (Ω, T h ), the forms ( 2.50) A2.20a – ( 2.59) A2.21e make sense for u h ,v h ∈ S hp . Consequently, we define five numerical schemes. Definition 2.8. A function u h ∈ S hp is called a DG approximate solution of problem ( 2.1) A0.1 – ( 2.3), A0.2b if it satisfies one of the following identities: i) B s h(u h ,v h ) = l s h(v h ) ∀v h ∈ S hp (2.60) A2.22a ii) B n h(u h ,v h ) = l n h(v h ) ∀v h ∈ S hp (2.61) A2.22b
2.3 DGM based on a primal formulation 19 iii) B s,σ h (u h,v h ) = l s,σ h (v h) ∀v h ∈ S hp (2.62) A2.22c iv) B n,σ h (u h,v h ) = l n,σ h (v h) ∀v h ∈ S hp (2.63) A2.22d v) B i,σ h (u h,v h ) = l i,σ h (v h) ∀v h ∈ S hp (2.64) A2.22e where the forms Bh s, Bn h ,... and ls h ,ln h ,... are defined by (A2.20a 2.50) – ( 2.54) A2.20e and ( 2.55) A2.21a – ( 2.59), A2.21e respectively. If we denote by B h any form defined by ( 2.50) A2.20a – ( 2.54) A2.20e and by l h we denote the corresponding form given by ( 2.55) A2.21a – ( 2.59), A2.21e the discrete problem can be formulated to find u h ∈ S hp satisfying the identity B h (u h ,v h ) = l h (v h ) ∀v h ∈ S hp . (2.65) dispr From the construction of the forms B and l one can see that the strong solution u ∈ H 2 (Ω) of problem ( 2.1) A0.1 – ( 2.3) A0.2b satisfies the identity terminology B h (u,v) = l h (v) ∀v ∈ H 2 (Ω, T h ) (2.66) A2.23 which represents the consistency of the method. The expressions ( 2.65) dispr and ( 2.66) A2.23 imply the so-called Galerkin orthogonality of the error e h = u h − u of the method: B h (e h ,v h ) = 0 ∀v h ∈ S hp , (2.67) A2.23a which will be used in the analysis of error estimates. In contrast to standard conforming finite element techniques, both Dirichlet and Neumann boundary conditions are included automatically in formulation ( 2.65) dispr of the discrete problem. This is an advantage particularly in the case of nonhomogeneous Dirichlet boundary conditions, because it is not necessary to construct subsets of finite element spaces formed by functions approximating the Dirichlet boundary condition in a suitable way. Remark 2.9. The method ( 2.60) A2.22a was introduced by Delves et al. ([DH79], Delves1 Delves2 [DP80], [HD79], Delves3 [HDP79]), Delves4 who call it global element method. Its advantage is the symmetry of the discrete problem. On the other hand, a significant disadvantage is that the billinear form Bh s (·, ·) is indefinite. This causes troubles when dealing with time-dependent problem since some eigenvalues can have negative real parts and then the resulting formulations becames unconditionally unstable. Therefore we prove in Lemma 2.18 lem:A11 the continuity of the bilinear form Bh s , but further we shall not be concerned with this method any more. The scheme ( 2.61) A2.22b was introduced by Baumann and Oden in [BBO99], bbo obb [OBB98]. Therefore, we shall call it the Baumann-Oden method. It is straightforward to show that the corresponding billinear form Bh n (·, ·) is positive semidefinite. An interesting property of this method is that it is unstable for piecewise linear approximations, i.e. for p = 1.
Page 1 and 2: 2 DGM for elliptic problems chap:el
Page 3 and 4: 2.2 Spaces of discontinuous functio
Page 5 and 6: 2.2 Spaces of discontinuous functio
Page 7 and 8: lem:II lem:III def:Pi_h 2.2 Spaces
Page 9: Due to the assumption that u ∈ H
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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?