Convergence Analysis

Then, Lemma 3.2.3 immediately implies

|(III)3| ≤Ch³²kgk_H²_(Γ)kuh−Qhuk_H¹_(Ω). (3.3.6) Thus, it follows from (3.3.2)-(3.3.3) and (3.3.5)-(3.3.6) that

kuh−Qhuk_H¹_(Ω) ≤C(u, g)h. (3.3.7) Next, we shall estimate the termku_h−Qhuk_L²_(Ω). For this purpose, we use duality trick.

Consider the following interface problem : Find ˜w∈H₀¹(Ω) such that

A_h( ˜w, v) = (u_h−Q_hu, v) ∀v ∈H₀¹(Ω), (3.3.8) with the jump conditions

[ ˜w] = 0,

"

βh(x)∂w˜

∂n˜

#

= 0 along Γh.

The solution ˜w satisfies the following regularity estimate (cf. [22, 40])

kwk˜ _H¹_(Ω)+kwk˜ _H1+s(Ω^h₁)+kwk˜ _H1+s(Ω^h₂) ≤Cku_h−Qhuk_L²_(Ω). (3.3.9) Before proceeding further, we define the projection Rh :H₀¹(Ω)→Vh by

Ah(Rhw, vh) =Ah(w, vh) ∀v_h ∈Vh, w ∈H₀¹(Ω). (3.3.10) Now, setting v =uh−Qhuin (3.3.8) and using (3.3.10), we have

ku_h−Qhuk²_L2(Ω) = Ah( ˜w, uh−Qhu)

= A_h(R_hw, u˜ _h−Q_hu)

= Ah(Rhw, u˜ h)−A(Rhw, u),˜ (3.3.11) in the last equality, we used (3.2.11). Further, using (3.3.1), we have

Ah(uh, Rhw)˜ −A(u, Rhw) =˜ hgh, Rhwi˜ Γ_h− hg, Rhwi˜ Γ, which, together with (3.3.11), we have

kuh−Qhuk²_L2(Ω) =hgh, Rhwi˜ Γ_h− hg, Rhwi˜ Γ=: (IV). (3.3.12) Using the fact kR_hwk˜ _H¹_(Ω)≤ kwk˜ _H¹_(Ω) and Lemma 3.2.3, we obtain

|(IV)| ≤ Ch³²kgk_H²_(Γ)kRhwk˜ _H¹_(Ω)

≤ Ch³²kgk_H²_(Γ)kwk˜ _H¹_(Ω). TH-261_BDEKA

Further, applying the regularity estimate (3.3.9), we have

|(IV)| ≤Ch³²kgk_H²_(Γ)ku_h−Qhuk_L²_(Ω). (3.3.13) Thus, it follows from (3.3.12)-(3.3.13) that

kuh−Qhuk_L²_(Ω)≤Ch³²kgk_H²_(Γ)kuh−Qhuk_L²_(Ω), (3.3.14) which, together with (3.3.7) and Lemma 3.2.4, yields the desired result. This completes

the proof.

Remark 3.3.1 In this chapter, we have proved a new interface approximation result (Lemma 3.2.2) which plays a crucial role in studying the error analysis. For the L²- norm error estimate, we have considered the dual problem over Ω^h₁ ∪Ω^h₂ ∪Γh. Since the interface Γ is of arbitrary shape,Ω^h_i (i= 1,2)is no more convex, and hence one can not have full regularity in each individual subdomains Ω^h_i (see, [22]). The existing regularity result for non-convex domain is then used to derive sub-optimal L²-norm error estimate.

Therefore, the proposed technique will be useful to deal with the interface problems on nonsmooth domain. Further, the proposed technique can easily be extended to treat more general interface problem of non-selfadjoint type (cf. [55]).

TH-261_BDEKA

Chapter 4

Error Estimates for Spatially Discrete Schemes for Parabolic Interface Problems

In this chapter, we extend the finite element analysis of elliptic interface problems dis- cussed in Chapters 2 and 3 to parabolic interface problems. For the spatially discrete scheme, we first establish optimal order error estimates in L²(L²) and L²(H¹) norms when the grid lines follow the actual interface.² Secondly, when the grid lines form an approximation to the interface, the semidiscrte solution is shown to converge to the exact solution at an optimal rate in L²(H¹) norm.

4.1 Introduction

Let Ω be a bounded domain in R² with smooth boundary ∂Ω and Ω₁ ⊂ Ω is an open domain with C² boundary Γ. Let Ω2 = Ω\Ω₁. We shall recall the following parabolic interface problems of the form

ut+Lu=f(x, t) in Ω×(0, T] (4.1.1) with initial and boundary conditions

u(x,0) =u0(x) in Ω; u(x, t) = 0 on∂Ω×(0, T] (4.1.2) and interface conditions

[u] = 0,

"

A∂u

∂n

#

=g(x, t) along Γ (4.1.3)

2SIAM J. Numer. Anal., 43 (2005), pp. 733-749

TH-261_BDEKA

where f = f(x, t) and g =g(x, t) are real valued functions in Ω×(0, T], and ut = ^∂u_∂t. Further, u0 =u0(x) is real valued function in Ω. For the ease of exposition, we assume Lv =−∇ ·(A∇v). The symbols [v] and n are defined as in Chapter 1, and T <∞.

In the theorem below, we prove the a priori estimate for the solution u of the interface problem (4.1.1)-(4.1.3) under appropriate regularity conditions on f and g.

Theorem 4.1.1 Let f ∈ H¹(0, T;L²(Ω)), g ∈ H¹(0, T;H¹²(Γ)) and u0 ∈H₀¹(Ω). Then the problem (4.1.1)-(4.1.3) has a unique solution u∈L²(0, T;X)∩H¹(0, T;Y)∩H₀¹(Ω).

Further, u satisfies the following a priori estimate kuk_L²_(0,T;X) ≤ Cn

kfk_L²_(0,T;L²_(Ω))+ku0k_H¹_(Ω)+kg(0)k

H¹²(Γ)

+kg(T)k

H¹²(Γ)+kgk

H¹(0,T;H¹²(Γ))

o

. (4.1.4)

Proof. The proof of the existence of a unique solution is in [30]. Next, to obtain the a priori estimate (4.1.4), we first transform the problem (4.1.1)-(4.1.3) into the following equivalent problem.

For a.e. t∈(0, T], find u=u(x, t)∈H₀¹(Ω)∩X satisfying

Lu = f(x, t)−ut in Ω, (4.1.5) u = 0 on∂Ω,

[u] = 0,

"

A∂u

∂n

#

= g(x, t) along Γ.

From the elliptic regularity estimate for the elliptic interface problem (cf. Theorem 1.2.1), it follows that

kukX ≤C

kf −utk_L²_(Ω)+kgk

H¹²(Γ)

. (4.1.6)

Multiply both sides of (4.1.5) by ut and then integrate over Ω to obtain

kutk²_L2(Ω)+ (Lu, ut) = (f, ut). (4.1.7) Note that u∈H¹(0, T;X) and [u] = 0 on Γ imply [ut] = 0 on Γ. Hence, an integration by parts leads to

(Lu, u_t) = Z

Ω1

A₁∇u· ∇u_tdx+ Z

Ω2

A₂∇u· ∇u_tdx+ Z

Γ

"

A∂u

∂n

# utds

= A¹(u, ut) +A²(u, ut) +hg, utiΓ, (4.1.8) where A^l(., .) :H¹(Ωl)×H¹(Ωl)→R is given by

A^l(w, v) = Z

Ωl

A_l∇w· ∇vdx, l = 1,2.

TH-261_BDEKA

Equation (4.1.7), together with (4.1.8), yields ku_tk²_L2(Ω)+1

2 d dt

2

X

i=1

Aⁱ(u, u)

!

= (f, u_t)− d

dthg, ui_Γ+hg_t, ui_Γ.

Integrate the above equation from 0 to T. Then apply the Cauchy-Schwarz inequality and the trace theorem (cf. [1]) to obtain

Z T 0

ku_tk²_L2(Ω)ds+ku(T)k²_H1(Ω1)+ku(T)k²_H1(Ω2)

≤C Z T

0

kfk_L²_(Ω)ku_tk_L²_(Ω)ds

+kg(T)k_L²_(Γ)ku(T)k_L²_(Γ)+kg(x,0)k_L²_(Γ)ku0k_L²_(Γ) +

Z T 0

kgtkL²(Γ)kukL²(Γ)ds+ku0k²_H1(Ω1)+ku0k²_H1(Ω2)

≤C Z T

0

kfk_L²_(Ω)ku_tk_L²_(Ω)ds+kg(T)k

H¹²(Γ)ku(T)k_H¹_(Ω) +kg(x,0)k

H¹²(Γ)ku₀k_H¹_(Ω)+ Z T

0

kg_tk

H¹²(Γ)kuk_H¹_(Ω)ds+ku₀k²_H1(Ω)

.

Use a standard kickback argument to obtain ku_tk²_L2(0,T;L²(Ω))+ku(T)k²_H1(Ω)

≤C Z T

0

kfk²_L2(Ω)ds+kg(T)k²

H¹²(Γ)

+kg(0)k²

H¹²(Γ)+ Z T

0

kg_tk²

H¹²(Γ)ds+ku₀k²_H1(Ω)

+C

Z T 0

ku(s)k²_H1(Ω)ds.

Finally, an application of Gronwall’s lemma completes the proof.

The purpose of the present chapter is to extend the convergence analysis of fitted finite element method for elliptic interface problems to parabolic interface problems.

Due to low global regularity of the true solution, pointwise-in-time error estimates in L² and H¹ norms are difficult and hence, we study the convergence analysis in terms of L²(H¹) and L²(L²) norms. The previous work on finite element analysis for parabolic problems without interface can be found in [29], [38], [58], and references therein. The key to the present analysis is the introduction of some auxiliary projections and duality arguments.

The outline of this chapter is as follows: Section 4.2 is devoted to the error esti- mates for the spatially discrete scheme for a finite element discretization where interface triangles follow exactly the actual interface. Optimal order error estimates inL²(L²) and L²(H¹) norms are shown to hold even if the global regularity of the solution is low. In section 4.3, a finite element discretization based on straight triangles is discussed for the TH-261_BDEKA

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