PEERS + Raviart-Thomas in 2D

5.3. PARTICULAR CHOICES OF FINITE ELEMENT SUBSPACES 41

5.3. PARTICULAR CHOICES OF FINITE ELEMENT SUBSPACES 42 Note here that the product space Hh(Ω_S)×L_h(Ω_S)×L²_h(Ω_S), with Hh(Ω_S) and L_h(Ω_S) defined according to (5.2), constitutes the classical PEERS originally introduced in [1] for a mixed finite element aproximation of the linear elasticity problem with Dirichlet boundary conditions (see also [33]). In turn,H_h(Ω_D)×L_h(Ω_D) is the Raviart-Thomas stable element of lowest order for the mixed formulation of the Poisson problem (see, e.g. [7], [36]). These facts are particularly important for the rest of the analysis since, as we will make it clear below, all the discrete inf-sup conditions that are required in the hypotheses indicated in Section 5.2, either are already available in the literature or can be derived from related results provided there.

Next, in order to define the spaces on the interface Σ, thus completing the list in (5.1), we follow the simplest approach suggested in [25] and [35]. To this end, we assume, without loss of generality, that the number of edgeseof Σhis even. Then, we let Σ2h be the partition of Σ arising by joining pairs of adjacent edges of Σ_h, and denote the resulting edges still by e. Since Σh is inherited from the interior triangulations, it is automatically of bounded variation (that is, the ratio of lengths of adjacent edges is bounded) and, therefore, so is Σ_2h. Certainly, if the number of edges of Σ_h were odd, we simply reduce it to the even case by joining any pair of two adjacent elements, and then construct Σ_2h from this partition.

Hence, denoting byx₀ and x₁ the extreme points of Σ, we define Λ^S_h(Σ) := n

ψ ∈ C(Σ) : ψ|_e ∈ P₁(e) ∀e∈Σ_2h, ψ(x₀) = ψ(x₁) = 0o

, and Λ^D_h(Σ) =

n

ξ ∈ C(Σ) : ξ|_e ∈ P1(e) ∀e ∈ Σ_2h o

.

(5.24) Our analysis below will also utilize the finite element subspaces of H₀₀^−1/2(Σ) andH^−1/2₀₀ (Σ) given by

Φ_h(Σ) :=

n

φ_h∈L²(Σ) : φ_h|_e∈P0(e) ∀edgee∈Σ_h o

, and Φ_h(Σ) := n

φ_h ∈L²(Σ) : φ_h|_e ∈P₀(e) ∀edgee∈Σ_ho .

In what follows we establish from (5.23), (5.24), and the accompanying definitions (5.2) and (5.4), that the hypotheses(H.0) - (H.3)are satisfied. In fact, the verification of(H.0) and (H.2) is quite straightforward from the definitions given in (5.23). Now, the discrete inf-sup conditions (5.10) and (5.11) are proved in [33, Theorem 4.5] and [7, Chapter IV, Section IV.1.2], respectively. Alternatively, one can also look at [1, Lemma 4.4] and [36, Chapter 7, Section 7.2.2]. In turn, the existence ofψ₀ ∈H^1/2₀₀ (Σ) verifying (5.12) follows as

5.3. PARTICULAR CHOICES OF FINITE ELEMENT SUBSPACES 43 in [25, Section 3.2] (see also [27, Section 3.2]). In fact, we pick one interior corner point of Σ and define a function v that is continuous, linear on each side of Σ, equal to one in the chosen vertex, and zero on all other ones. If n1 and n2 are the normal vectors on the two sides of Σ that meet at the corner point, then ψ₀ := v(n₁+n₂) satisfies that property. If the interface Σ were a line segment (without interior corners), we pickv as the continuous linear function on Σ, equal to one in any interior point and zero in the extreme points, and define ψ₀ := vn. We have thus verified the assumptions required by(H.1).

On the other hand, concerning the discrete inf-sup conditions yielding (H.3), we first recall from the analyses in [25] and [35], that the existence of a stable discrete lifting of the normal traces of He_0,h(ΩD) implies that a sufficient condition for (5.17) is the existence of βb_D>0, independent ofh, such that

sup

φh∈Φh(Σ) φh6=0

hφ_h, ξhi_Σ

kφ_hk_−1/2,Σ ≥βbDkξ_hk_1/2,Σ ∀ξ_h ∈Λ^D_h(Σ). (5.25)

In fact, a detailed proof of (5.17), whose main ingredients were the explicit construction of such a lifting and then the demonstration of (5.25), was first provided in [25, Lemmas 4.2, 5.1 and 5.2] under the assumption of quasi-uniformity around the interface Σ. This result was improved recently in [35, Sections 4 and 5] where it was shown for the 2D case without any requirement on the meshes. In turn, in order to proceed similarly with (5.16), we need to introduce suitable changes into the arguments from [25] and [35]. The reason for it is rather technical and has to do with the fact that the tensors τ_S,h ∈ Heh(Ω_S) (cf. (5.15)), space where the supremum in (5.16) is taken, must also satisfy the discrete symmetry condition (τ_S,h,η_S,h)S = 0 ∀η_S,h ∈ L²h(ΩS). More precisely, since the Raviart-Thomas or related projection operators do not preserve any kind of symmetry, the way in which the lifting was built in [25] is not applicable to construct a stable discrete lifting of the normal traces of Heh(Ω_S). Instead of it, we now proceed a bit differently and still show, using results from [20], [25], and [35], that a sufficient condition for (5.16) is the analogue of (5.25), that is the existence of βb_S>0, independent of h, such that

sup

φh∈Φh(Σ) φh6=0

hφ_h,ψ_hi_Σ

kφ_hk_−1/2,Σ ≥βb_Skψ_hk_1/2,Σ ∀ψ_h ∈Λ^S_0,h(Σ). (5.26)

In fact, givenφ_h ∈Φh(Σ), we let σeh,(ueh,γe_h,ϕe_h)

∈Hh(ΩS)× Lh(ΩS)×L²_h(ΩS)×Λ^S_h(Σ)

5.3. PARTICULAR CHOICES OF FINITE ELEMENT SUBSPACES 44 be the unique solution of the Galerkin scheme:

(σeh,τh)S + (divτh,uh)S + (τh,γe_h)S + hτ_hn,ϕe_hi_Σ = 0,

(divσe_h,v_h)_S + (σe_h,η_h)_S + hσe_hn,ψ_hi_Σ = hφ_h,ψ_hi_Σ,

(5.27)

for all τh,(vh,η_h,ψ_h)

∈Hh(ΩS)× Lh(ΩS)×L²_h(ΩS)×Λ^S_h(Σ)

. Note that (5.27) actually corresponds to the PEERS-based mixed finite element approximation of a particular linear elasticity problem in Ω_S (see, e.g. (4.10)) with homogeneous Dirichlet boundary condition on ΓS and Neumann boundary condition given byφ_hon Σ. Moreover, the well-posedness of (5.27) is proved, modulus minor changes, by combining [20, Section 4.3] with [35, Theorem 5.1] and [25, Lemma 5.2]. In particular, the associated stability result insures the existence of C >e 0, independent ofh, such that

k σe_h,(eu_h,eγ_h,ϕe_h)

k ≤ Cekφ_hk_−1/2,Σ. (5.28) Therefore, since the second equation in (5.27) establishes that σeh belongs to Heh(ΩS) and that hσe_hn,ψ_hi_Σ = hφ_h,ψ_hi_Σ ∀ψ_h∈Λ^S_h(Σ), we deduce, using also (5.28), that

| hφ_h,ψ_hi_Σ|

kφ_hk_−1/2,Σ = | hσehn,ψ_hi_Σ| kφ_hk_−1/2,Σ ≤ 1

Ce

| hσehn,ψ_hi_Σ| kσe_hk_div;ΩS , which implies that

sup

φh∈Φh(Σ) φh6=0

hφ_h,ψ_hi_Σ kφ_hk_−1/2,Σ ≤ 1

Ce sup

τh∈eHh(ΩS) τh6=0

hτ_hn,ψ_hi_Σ

kτ_hk_div;ΩS ∀ψ_h ∈Λ^S_h(Σ). (5.29)

Thus, it is quite clear from (5.29) that the discrete inf-sup condition (5.16) is a straight- forward consequence of (5.26). Moreover, since the latter has already been proved in [25, Lemma 5.2], we conclude in this way the full verification of the hypothesis (H.3).

Thanks to the previous results and analyses, we can establish the following theorems.

Theorem 5.3 Assume that the stabilization parameterρlives in

0,^α_γ⁰2 0

, and let((t,ϕ), p)∈ X×M be the unique solution of the continuous formulation (2.34). In addition, let Xh :=

X1,h×M1,h and Mh be the finite element subspaces defined by (5.4)in terms of the specific discrete spaces given by (5.23) and (5.24). Then, the Galerkin scheme (5.5) has a unique solution ((t_h,ϕ_h), p_h)∈Xh×Mh and there exist C1, C2>0, independent of h, such that

k((t_h,ϕ

h), p

h)k_X×M ≤ C₁n kF|_Xhk

X⁰h + kG|_Mhk

M⁰h

o

, (5.30)

5.3. PARTICULAR CHOICES OF FINITE ELEMENT SUBSPACES 45 and

k((t,ϕ), p)−((t_h,ϕ_h), p_h)k_X×M ≤ C2 inf

((r_h,ψ_h),q_h)∈Xh×Mh

k((t,ϕ), p)−((r_h,ψ_h), q_h)k_X×M. (5.31) Proof. Having verified the hypotheses (H.0), (H.1), (H.2) and (H.3), the proof is a straightforward application of Theorems 5.1 and 5.2.

The following theorem provides the theoretical rate of convergence of the Galerkin scheme (5.5), under suitable regularity assumptions on the exact solution.

Theorem 5.4 Let((t,ϕ), p)∈X×Mand((t_h,ϕ_h), p_h)∈Xh×Mhbe the unique solutions of the continuous and discrete formulations (2.34) and (5.5), respectively. Assume that there exists δ ∈ (0,1] such that t_S ∈ H^δ(Ω_S), u_D ∈ H^δ(Ω_D), divu_D ∈ H^δ(Ω_D), σ_S∈H^δ(Ω_S), divσ_S∈H^δ(Ω_S), and γ_S ∈H^δ(Ω_S). Then,u_S∈H^1+δ(Ω_S),p_D∈H^1+δ(Ω_D),ϕ∈H^1/2+δ(Σ), λ∈ H^1/2+δ(Σ), and there exists C > 0, independent of h and the continuous and discrete solutions, such that

k((t,ϕ), p) − ((t_h,ϕ_h), p_h)k_X×M ≤ C h^δn

kt_Sk_δ,ΩS + ku_Dk_δ,ΩD + kdivu_Dk_δ,ΩD + kσ_Sk_δ,ΩS + kdivσ_Sk_δ,ΩS + kγ_Sk_δ,ΩS + ku_Sk_1+δ,ΩS + kp_Dk_1+δ,ΩDo

.

(5.32)

Proof. We first recall from Theorem 4.2 that∇u_S=t_S+γ_S and ∇p_D = −K⁻¹u_D, which implies that u_S ∈ H^1+δ(Ω_S), and p_D ∈H^1+δ(Ω_D), whence ϕ = −u_S|_Σ ∈ H^1/2+δ(Σ) and λ = p_D|_Σ ∈ H^1/2+δ(Σ). The rest of the proof follows from the Cea estimate (5.31), the approximation properties of the subspaces involved (see, e.g. [4], [7] and [29]), and the fact that, thanks to the trace theorems in ΩS and ΩD, respectively, there holds

kϕk_1/2+δ,Σ ≤ cku_Sk_1+δ,ΩD and kλk_1/2+δ,Σ ≤ ckp_Dk_1+δ,ΩD.

5.3. PARTICULAR CHOICES OF FINITE ELEMENT SUBSPACES 46