2 DGM for elliptic problems

More documents

Recommendations

Info

20 2 Elliptic problems The scheme ( 2.62) A2.22c is called symmetric interior penalty Galerkin (SIPG) method. It was derived by Arnold ([Arn82]) Arnold and Wheeler ([Whe78]) Whe78 by adding penalty terms to the form Bh s . This formulation leads to a symmetric bilinear form which is coercive if the penalty parameter σ is sufficiently large. Moreover, the Aubin-Nitsche trick can be used for obtaining an optimal error estimate in the L 2 (Ω)-norm. The method ( 2.63), A2.22d called nonsymmetric interior penalty Galerkin (NIPG) method, was proposed by Girault, Riviére and Wheeler in [RWG99]. RWG99 In this case the bilinear form B n,σ h is nonsymmetric and does not allow to obtain optimal error estimates in the L 2 (Ω)-norm with the aid of the Aubin-Nitsche trick. However, numerical experiments show that the odd degrees of the polynomial approximation give the optimal order of convergence. On the other hand, a favorable property of NIPG method is the coercivity of B n,σ h (·, ·) for any penalty parameter σ > 0. Finally, the method 2.64), A2.22e called incomplete interior penalty Galerkin (IIPG) method, was studied in [DSW04], Dawson2004 [Sun03], Sun-PhD [SW05]. Sun05 In this case the bilinear form B i,σ h is nonsymmetric and does not allow to obtain an optimal error estimate in the L 2 (Ω)-norm. Moreover, the penalty parameter σ should be chosen sufficiently large in order to guarantee the coercivity of B i,σ h . The advantage of the IIPG method is the simplicity of the discrete diffusion operator. This is particularly advantageous in the case when the diffusion operator is nonlinear. (See, e.g. [Dol08]. iipg07 It would also be possible to define the scheme Bh i (u,v) = li h (v) ∀v ∈ S hp, where Bh i (u,v) = ai h (u,v) and li h (v) = F h i (v), but this method does not make sense, because it does not contain the Dirichlet boundary condition ( 2.2). A0.2a sec:2.4 2.4 Properties of the bilinear forms We are interested in the existence of the numerical solutions of method ( 2.60) A2.22a – ( 2.64) A2.22e and in a priori error estimates of the difference of numerical solution and the exact one. Therefore, we define the following mesh-dependent norm |||u||| Th = ( |u| 2 H 1 (Ω,T h ) + Jσ h(u,u)) 1/2 , (2.68) en In what follows, if there is no danger of misunderstanding, we shall omit the subscript T h . This means that we shall simply write ||| · ||| = ||| · ||| Th . exer1 Exercise 2.10. Prove that ||| · ||| is a norm on the spaces H 1 (Ω, T h ) and S hp . Let Γ ∈ Fh I be an inner face and K(L) Γ and K (R) Γ the elements sharing Γ. There are several possibilities how to define the penalty weight σ. We can set σ| Γ = ( max C W h K (L) Γ , h K (R) Γ ), (2.69) A3.1
where h K (L) Γ and h K (R) Γ 2.4 Properties of the bilinear forms 21 are the diameters of the elements K (L) Γ and K (R) Γ , respectively, adjacent to the face Γ, and C W > 0 is a suitable constant. For boundary faces Γ ∈ F D h (i.e. Γ ⊂ Γ D) we put σ| Γ = C W h K (L) Γ , (2.70) A3.2 where C W > 0 and h KΓ denotes the diameter of the element K (L) Γ adjacent to Γ. If we use the simplified notation ( ) h Γ = max h (L) K , h (R) Γ K for Γ ∈ Fh I and h Γ = h (L) Γ K for Γ ∈ Fh D , (2.71) Γ A3.2a we can write σ| Γ = C W , Γ ∈ F h . (2.72) A3.3 h Γ Let Γ ⊂ ∂K be a face of an element K. Then, in virtue of ( 2.18) A1.35 and ( 2.71), A3.2a we have h K ≤ h Γ ≤ C H h K , K ∈ T h , Γ ∈ F h , Γ ⊂ K. (2.73) A3.5 Another possibility is to replace ( 2.69) A3.1 by σ| Γ = h K (L) Γ 2C W In this case, σ| Γ is defined by ( A3.3 2.72) with h Γ = h K (L) Γ + h K (R) Γ . (2.74) A3.1a + h (R) K Γ . (2.75) A3.1b 2 If the partition T h represents a standard conforming triangulation formed by simplexes, then one ususally defines the weight σ| Γ as σ| Γ = C W , (2.76) A3.1c diam(Γ) i.e. h Γ = diam(Γ). Under the introduced notation, in view of ( 2.40) A2.17 and ( 2.72), A3.3 the interior and boundary penalty forms read Jh σ (u,v) = ∑ ∫ C W [u][v]dS, (2.77) A3.4 h Γ Γ ∈F ID h J σ D(v) = ∑ Γ ∈F D h ∫ Γ Γ C W h Γ u D v dS.
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 and 10: Due to the assumption that u ∈ H
Page 11: 2.3 DGM based on a primal formulati
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?