The Continuous time Galerkin Approximation with Curved Triangles

problem (4.1.1)-(4.1.3) and optimal order of convergence for the semidiscrete solution in L²(H¹) norm is obtained.

4.2 The Continuous time Galerkin Approximation

which, combine with (4.2.3), leads to

A(Phv, φ) = (f^∗, φ) = A(v, φ) ∀φ∈Vh, v ∈X∩H_o¹(Ω). (4.2.4) Below, we present a proof that shows optimal error bounds for the projection operator Ph. This lemma is very crucial for our later analysis.

Lemma 4.2.1 Let P_h be defined by (4.2.4). Then there exists a positive constant C independent of h such that

kv−Phvk+hkv −Phvk1 ≤Ch²kvkX ∀v ∈X∩H₀¹(Ω).

Proof. By the definition of f^∗, we obtain

A(v, φ) = (f^∗, φ) ∀φ∈H₀¹(Ω). (4.2.5) For k = 1,2,define fk: Ω→R by

fk=

( f^∗|_Ωk in Ωk, 0 otherwise.

Clearly, fk ∈L²(Ω) and f^∗ =f1+f2 a.e. in Ω. We now consider the following interface problems: Let w_k ∈H₀¹(Ω) be the solution of the interface problem

A(wk, φ) = (fk, φ) ∀φ∈H₀¹(Ω). (4.2.6) Then by the coercivity of the operator A it follows immediately that v =w₁+w₂.

Let w_h^k ∈Vh be the finite element approximation towk defined by

A(w^k_h, φ) = (fk, φ) ∀φ∈Vh. (4.2.7) Again by the coercivity of the operator A and definition (4.2.4) of the Ph operator, it follows that Phv =w¹_h+w_h².

Since fk|_Ωs = 0, s = 1(2) if k = 2(1), we have for the elliptic interface problem (4.2.6)-(4.2.7) (cf. Theorem 2.3.1)

kw_k−w_h^kk_H¹_(Ω) ≤ Ch(kw_kk_H²_(Ω1)+kw_kk_H²_(Ω2))

≤ ChkwkkX. (4.2.8)

Then by the elliptic regularity (cf. Theorem 1.2.1) we have kwkkX ≤ Ckfkk_L²_(Ω)

≤ Ckf^∗k_L²_(Ωk) ≤Ckvk_H²_(Ωk). TH-261_BDEKA

This, together with (4.2.8), yields

kwk−w^k_hkH¹(Ω) ≤ChkvkH²(Ω_k). (4.2.9) Then

kv −Phvk_H¹_(Ω) ≤ kw₁−w_h¹k_H¹_(Ω)+kw₂−w_h²k_H¹_(Ω)

≤ C(hkw1kX +hkw2kX)

≤ Ch kvk_H²_(Ω1)+kvk_H²_(Ω2)

≤ Chkvk_X ∀v ∈X∩H₀¹(Ω). (4.2.10) For the L² norm estimate, let us consider the following problem: Find w∈H₀¹(Ω) such that

A(w, φ) = (v−Phv, φ) ∀φ ∈H₀¹(Ω). (4.2.11) By setting φ = v − Phv ∈ H₀¹(Ω) in (4.2.11) and using definition (4.2.4) of the Ph

operator, we have

kv−Phvk²_L2(Ω) = A(w, v−Phv)

= A(w−Phw, v−Phv) +A(Phw, v−Phv)

= A(w−Phw, v−Phv)

≤ Ckw−Phwk_H¹_(Ω)kv−Phvk_H¹_(Ω)

≤ Ch²kvkXkwkX,

in the last inequality, we used (4.2.10). Then applying the elliptic regularity estimate (cf. Theorem 1.2.1) for the interface problem (4.2.11), we get

kv−Phvk²_L2(Ω) ≤ Ch²kvk_Xkwk_X

≤ Ch²kv−Phvk_L²_(Ω)kvkX.

This completes the proof of Lemma 4.2.1.

Let Lh :L²(Ω)→Vh be the standard L² projection defined by

(Lhv, φ) = (v, φ) ∀v ∈L²(Ω), φ∈Vh. (4.2.12) It is well known that Lhv ∈Vh is the best approximation in the L² norm tov ∈L²(Ω).

The following lemma shows that L_hv is a quasi-best approximation to v ∈ X∩H₀¹(Ω) in the H¹ norm.

Lemma 4.2.2 Let P_h and L_h be defined by (4.2.4) and (4.2.12), respectively. Then we have

kL_hv−vk_H¹_(Ω) ≤CkP_hv−vk_H¹_(Ω) ∀v ∈X∩H₀¹(Ω).

TH-261_BDEKA

Proof. For any v ∈X∩H₀¹(Ω), we know that there exists a unique solution w∈H₀¹(Ω) for the elliptic interface problem

A(w, φ) = (Phv−v, φ) ∀φ∈H₀¹(Ω). (4.2.13) Equation (4.2.13), together with (4.2.4) and Lemma 4.2.1, leads to

kPhv−vk²_L2(Ω) = A(w−Phw, Phv−v) +A(Phw, Phv−v)

≤ Ckw−Phwk_H¹_(Ω)kv−Phvk_H¹_(Ω)

≤ Chkwk_Xkv−Phvk_H¹_(Ω)

≤ Chkv−Phvk_L²_(Ω)kv−Phvk_H¹_(Ω). (4.2.14) Here, we used the fact that kwkX ≤Ckv−Phvk_L²_(Ω). Use of the triangle inequality and (4.2.2) yields

kL_hv−vk_H¹_(Ω) ≤ kP_hv −vk_H¹_(Ω)+kL_hv−Phvk_H¹_(Ω)

≤ kPhv −vk_H¹_(Ω)+Ch⁻¹kLhv −Phvk_L²_(Ω)

≤ kP_hv −vk_H¹_(Ω)+Ch⁻¹{kv −Phvk_L²_(Ω)

+kL_hv−vk_L²_(Ω)}. (4.2.15)

We knowLhv is the best approximation ofv ∈L²(Ω) with respect to theL² norm. Since v ∈X∩H₀¹(Ω),Phv ∈Vh. Thus, kv−Lhvk_L²_(Ω) ≤Ckv−Phvk_L²_(Ω). Therefore, (4.2.15) implies

kLhv−vk_H¹_(Ω) ≤ kPhv−vk_H¹_(Ω)+Ch⁻¹kv−Phvk_L²_(Ω). (4.2.16) The desired estimate now follows from (4.2.14) and (4.2.16).

The following Lemma is crucial for our subsequent error analysis.

Lemma 4.2.3 Let L_h be defined by (4.2.12). Then we have kL_hvk_H¹_(Ω) ≤Ckvk_H¹_(Ω) ∀v ∈H₀¹(Ω).

Proof. We define a projection Rh :H₀¹(Ω)→Vh by

A(Rhv, vh) = A(v, vh) ∀v_h ∈Vh, v ∈H₀¹(Ω). (4.2.17) For anyv ∈H₀¹(Ω), we know that there exists a unique solutionw∈X∩H₀¹(Ω) for the elliptic interface problem

A(w, φ) = (Rhv−v, φ) ∀φ∈H₀¹(Ω). (4.2.18) TH-261_BDEKA

Setting φ=Rhv −v in (4.2.18) and using Lemma 4.2.1, we obtain

kR_hv −vk²_L2(Ω) = A(w−Phw, Rhv−v) +A(Phw, Rhv −v)

≤ Ckw−P_hwk_H¹_(Ω)kv−R_hvk_H¹_(Ω)

≤ ChkwkXkv−RhvkH¹(Ω)

≤ Chkv−Rhvk_L²_(Ω)kv −Rhvk_H¹_(Ω). (4.2.19) Here, we used the fact that kwk_X ≤Ckv−Rhvk_L²_(Ω). This, together with (4.2.12), we have

kR_hv−Lhvk²_L2(Ω) = (Rhv−Lhv, Rhv−Lhv)

= (R_hv−v, R_hv−L_hv) + (v−L_hv, R_hv−L_hv)

≤ Ckv−RhvkL²(Ω)kRhv−LhvkL²(Ω)

≤ Chkv−Rhvk_H¹_(Ω)kL_hv−Rhvk_L²_(Ω). (4.2.20) Then, inverse inequality and (4.2.20), leads to

kL_hvk_H¹_(Ω) ≤ kL_hv−Rhvk_H¹_(Ω)+kR_hvk_H¹_(Ω)

≤ C{h⁻¹kLhv−Rhvk_L²_(Ω)+kvk_H¹_(Ω)}

≤ Ckv −Rhvk_H¹_(Ω)+Ckvk_H¹_(Ω)≤Ckvk_H¹_(Ω) ∀v ∈H₀¹(Ω)

where, we used the fact that kR_hvk_H¹_(Ω) ≤ Ckvk_H¹_(Ω) ∀v ∈ H₀¹(Ω), and this completes

the rest of the proof.

Now, we are in a position to discuss the continuous time Galerkin finite element approximation to (4.2.1) which may be stated as follows: Find u_h(t)∈V_h such that

(uht, vh) +A(uh, vh) = (f, vh) +hg, vhiΓ ∀vh ∈Vh, t∈(0, T], (4.2.21) with uh(0) = Lhu0.

Subtracting (4.2.21) from (4.2.1) we have

(u_t−u_ht, v_h) +A(u−u_h, v_h) = 0 ∀v_h ∈V_h. (4.2.22) Define the error e(t) as e(t) = u(t)−uh(t). Setting vh = Lhu in (4.2.22) and using (4.2.12), we obtain

1 2

d

dtkek²_L2(Ω) + A(e, e) =A(u−uh, u−Lhu) + (ut−uht, u−Lhu)

= A(u−u_h, u−L_hu) + (u_t−L_hu_t, u−L_hu) +(Lhut−uht, u−Lhu)

= A(u−uh, u−Lhu) + (ut−(Lhu)t, u−Lhu) +(L_hu_t−u_ht, u−L_hu)

= A(u−u_h, u−L_hu) + 1 2

d

dt(u−L_hu, u−L_hu). (4.2.23) TH-261_BDEKA

In the last equality we used the fact thatLhut−u_ht ∈Vhand the definition (4.2.12) of the Lh operator. Integrate the above equation from 0 tot. Then apply the Cauchy-Schwarz inequality and Young’s inequality to obtain

kek²_L2(Ω) + Z t

0

kek²_H1(Ω)ds

≤C Z t

0

ku−Lhuk²_H1(Ω)ds+ku−Lhuk²_L2(Ω)+ku0−Lhu0k²_L2(Ω)

.

An application of Lemma 4.2.2 leads to kek²_L2(Ω) +

Z t 0

kek²_H1(Ω)ds

≤C Z t

0

ku−Phuk²_H1(Ω)ds+ku−Lhuk²_L2(Ω)+ku0−Lhu0k²_L2(Ω)

,

which, together with Lemma 4.2.1 and the fact kLhv −vk_L²_(Ω) ≤ Chkvk_H¹_(Ω) ∀v ∈ H₀¹(Ω), yields

Z t 0

kek²_H1(Ω)ds≤Ch²n

ku0k²_H1(Ω)+kuk²_X +kuk²_L2(0,T;X)

o.

Thus, we have proved the following optimal H¹ norm estimate.

Theorem 4.2.1 Let u and uh be the solutions of (4.1.1)-(4.1.3) and (4.2.21), respec- tively. Then, for u₀ ∈H₀¹(Ω), f ∈H¹(0, T;L²(Ω)), and g ∈H¹(0, T;H¹²(Γ)), we have

ku−uhk_L²_(0,T;H¹_(Ω)) ≤ Ch

ku₀k_H¹_(Ω)+kuk_X +kuk_L²_(0,T;X) .

Next, for theL² norm error estimate, we shall use the duality argument. For this purpose, we now consider the following auxiliary problem: Find z ∈H₀¹(Ω) such that

A(z, v) = (u−uh, v) ∀v ∈H₀¹(Ω), t ∈(0, T], (4.2.24) with

A_∂^∂z

n

= 0 across the interface Γ. Then its finite element approximation is defined to be a function zh ∈Vh satisfying

A(zh, vh) = (u−uh, vh) ∀v_h ∈Vh, t∈(0, T]. (4.2.25) Setting v =u−uh in (4.2.24) and using (4.2.22), we obtain

ku−uhk²_L2(Ω) = A(z, u−uh)

= A(z−zh, u−uh) +A(zh, u−uh)

= A(z−z_h, u−u_h)−(u_t−u_ht, z_h). (4.2.26) TH-261_BDEKA

Differentiating (4.2.25) with respect to t, we obtain A(zht, vh) = (ut−uht, vh).

Thus, we have

1 2

d

dtA(z_h, z_h) =A(z_ht, z_h) = (u_t−u_ht, z_h), and hence, integrating (4.2.26) from 0 toT we obtain

ku−uhk²_L2(0,T;L²(Ω))+1

2A(zh, zh)

≤C Z T

0

kz−zhk_H¹_(Ω)ku−uhk_H¹_(Ω)ds+1

2A(zh(0), zh(0)).

Further, using Theorem 1.2.1 for the elliptic interface problem (4.2.24) and Lemma 4.2.1, we obtain

kz−zhk_H¹_(Ω) ≤ Ckz−Phzk_H¹_(Ω)

≤ Chkzk_X

≤ Chku−uhkL²(Ω), and hence

ku−uhk²_L2(0,T;L²(Ω))

≤C Z T

0

hku−uhk_L²_(Ω)ku−uhk_H¹_(Ω)ds+ 1

2A(zh(0), zh(0)). (4.2.27) Taking t→0, it now follows from (4.2.25) that

A(zh(0), zh(0)) = (u0−Lhu0, zh(0)) = 0.

This, together with (4.2.27) and Theorem 4.2.1, leads to ku−uhk²_L2(0,T;L²(Ω))

≤Ch Z T

0

ku−uhk²_L2(Ω)

¹2 Z T 0

ku−uhk²_H1(Ω)

¹2

≤Ch²ku−u_hk_L²_(0,T;L²_(Ω))

ku₀k_H¹_(Ω)+kuk_X +kuk_L²_(0,T;X) . Thus, we have proved the following optimal L² norm estimate.

Theorem 4.2.2 Let u and u_h be the solutions of (4.1.1)-(4.1.3) and (4.2.21), respec- tively. Then, for u0 ∈H₀¹(Ω), f ∈H¹(0, T;L²(Ω)), and g ∈H¹(0, T;H¹²(Γ)), we have

ku−uhk_L²_(0,T;L²_(Ω))≤Ch²

ku₀k_H¹_(Ω)+kuk_X +kuk_L²_(0,T;X) . TH-261_BDEKA

4.3 The Continuous time Galerkin Approximation