A smooth fictitious domain/multiresolution method for elliptic equations on general domains

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A smooth fictitious domain/multiresolution method for elliptic equations on general domains

Ping Yin, Jacques Liandrat

To cite this version:

Ping Yin, Jacques Liandrat. A smooth fictitious domain/multiresolution method for elliptic equations

on general domains. Numerical Algorithms, Springer Verlag, 2016, �10.1007/s11075-015-0063-x�. �hal-

01266859�

A smooth fictitious domain/multiresolution method for elliptic equations on general domains

Ping Yin

Science college, Jiangnan University, Wuxi 214122, China

Jacques Liandrat

Centrale Marseille, I2M, UMR 7373, CNRS, Aix-Marseille Univ., Marseille, 13453, France

Abstract

We propose a smooth fictitious domain/multiresolution method for enhanc- ing the accuracy order in solving second order elliptic partial differential equations on general bivariate domains. We prove the existence and unique- ness of the solution of corresponding discrete problem and the interior error estimate which justifies the improved accuracy order. Numerical experiments are conducted on a cassini oval.

Keywords: Elliptic partial differential equations, Smooth fictitious domain methods, Multiresolution discretization, Interior error estimate

1. Introduction

Recently, multiresolution methods have became a new powerful tool in the numerical approximation of partial differential equations due to their high accuracy and related fast algorithms [1, 2]. However, it is difficult to construct scaling and wavelet functions adapted to the geometry of general domains. This is an important issue, especially in the case of moving bound- ary problems.

∗Corresponding author

Email addresses: wanderapple@gmail.com(Ping Yin), jliandrat@centrale-marseille.fr(Jacques Liandrat)

Fictitious domain methods were introduced to encounter this problem.

They consist in solving a problem raised on a general domain ω by merging ω into a larger and simpler domain Ω. Fictitious domain methods differ fun- damentally in the way how the boundary conditions are enforced on γ, the boundary of ω. In penalization methods, additional terms are added in Ω\ ω.

These terms are usually weighted with a coefficient

. To the limit when ² goes to zero, the boundary conditions are enforced [3, 4]. The Lagrange mul- tipliers methods emerge from the optimization techniques under constraints.

A minimization problem on a subset is transformed into a saddle point prob- lem on a larger set. Lagrange multipliers as supplementary unknowns that may live on ω, on its boundary γ or on Ω \ ω [5, 6].

In this paper, the starting point is a fictitious domain method coupled with Lagrange multipliers on the boundary. By shifting the Lagrange mul- tipliers on a control boundary Γ located away from γ in the outer normal direction(as proposed in [7]), this new approach is shown to preserve the good approximation properties of multiresolution methods and to improve the accuracy order of the numerical solution on ω.

This work is a generalization to bivariate framework of the results pre- sented in [8, 9]. The paper is organized as follows. Section 2 presents the for- mulation of our smooth fictitious domain method and gives the approximate controllability result. Section 3 contains the multiresolution discretization and proves the existence of an unique solution of the discrete saddle point problem. Section 4 gives the main theorem of interior error estimate which theoretically justifies the improved accuracy order. We list the important aspects of numerical implementation in Section 5. Some numerical examples are presented in Section 6 to verify the accuracy of the proposed method.

Section 7 concludes this paper.

2. The smooth fictitious domain method

We consider a Dirichlet boundary-value problem on an open, bounded domain ω ⊂ R

with Lipschitz boundary γ,

( u ˆ − ν4ˆ u = f in ω, ˆ

u = g on γ, (1)

where ν > 0, f ∈ L

(ω) and g ∈ H

(γ). In order to separate the difficulty of

approximating the differential operator and the integration of the boundary

condition as well as to overcome the difficulty of constructing basis functions adapted to the complicated domain ω, a fictitious domain/Lagrange mul- tiplier method (FDLM) [10] can be defined. It introduces a rectangular Ω containing ω, ¯ ω ⊂ Ω, and uses the Lagrange multipliers λ on the bound- ary γ to enforce the boundary conditions. Thus the extended problem has a solution u on Ω such that u|

= ˆ u [11]. Since the auxiliary variables λ represent the normal derivative jump of u at γ by green formula [12], the regularity of u is limited in the vicinity of γ such that u ∈ H

(Ω), for any ² > 0. It is well known that the smoothness degree of the solution of a boundary-value problem does affect the order of convergence of a numerical solution [13]. J.Haslinger et al. [7] investigated a new formulation of ficti- tious domain methods connecting in putting the Lagrange multiplier λ on a control boundary Γ located outside of ¯ ω to enforce the boundary condition on γ(see in Figure 1). We will call it the smooth fictitious domain/Lagrange multiplier method (SFDLM). The Dirichlet boundary value problem (1) is

Figure 1: Geometry: ωis the original domain,γis its boundary. Γ is the control boundary and Ξ is the control domain with Ξ⊃ω. Ω represents the fictitious domain with Ω¯ ⊃Ξ.¯

then rewritten as:

 



 

Find (u, λ) ∈ H

(Ω) × H

(Γ) such that, a(u, v) + b

(v, λ) = ( ˜ f , v), ∀v ∈ H

(Ω), b

(u, µ) =< g, µ >

, ∀µ ∈ H

(γ),

(2)

where, if the trace operators B and B are defined, Bu := u|

∈ H

(γ), Bu := u|

∈ H

(Γ), ∀u ∈ H

(Ω) then

b

(u, q) :=< Bu, q >

:=< Bu, q >

, ∀q ∈ H

(γ),

b

(u, p) :=< Bu, p >

:=< Bu, p >

, ∀p ∈ H

(Γ).

The bilinear form a(·, ·) reads a(u, v) =

Z

Ω

(uv + νOuOv)dx, ∀(u, v) ∈ H

(Ω) × H

(Ω),

and ˜ f is an extension of f from ω to Ω. Note that the bilinear forms b

and b

are distinct. This kind of problem can be considered as a generalization of saddle point problems which have been studied in [14, 15]. Since the derivative jump λ lays on Γ at a positive distance from γ, this new method is expected to produce a smoother solution in the control domain Ξ which motivates the proof of interior error estimate depending on the smoothness of u in Ξ. For any λ ∈ H

(Γ)(if H

(Ω) stands for the sobolev space of order 1 of periodic functions on Ω), there exists a unique solution ¯ u of the

problem (2’): (

Find ¯ u ∈ H

(Ω) such that a(¯ u, v) = ( ˜ f , v) − b

(v, λ).

However, the extended problem (2) does not have a solution in general [16].

We will use the following approximate controllability result [7].

Lemma 1. Defining the linear mapping Φ : H

(Γ) 7→ H

(γ) by Φ(λ) =

¯

u|

, where u ¯ is the solution of (2’). Then for any ² > 0, there exists a value of λ ¯ and the corresponding solution u ¯ associated to ¯ g = Φ(¯ λ) in (2’) such that

kˆ u − uk ¯

≤ ². (3) 3. The multiresolution discretization

The finite dimensional subspaces used in the numerical approximation of problem (2) include the multiresolution analysis on Ω, Γ, γ denoted by U

⊂ H

(Ω), Q

⊂ H

(Γ) ,Q

⊂ H

(γ), and the subspace V

which is the image of U

by the operator (I −ν4)

, i.e., V

= (I −ν4)

U

. This space is spanned by biorthogonal vaguelettes [17]. We call it the Petrov-Galerkin and wavelet-vaguelette method [18]. Then the discretization formulation of (2) reads 

 

 

Find (u

, λ

) ∈ U

× Q

, such that a(u

, v

) + b

(v

, λ

) = ( ˜ f , v

), ∀v

∈ V

, b

(u

, µ

) =< g, µ

>

, ∀µ

∈ Q

.

(4)

We assume that Γ and γ are piecewise regular in the sequel. The definition of the various spaces reads as follows:

U

= span{Φ

, α ∈ K

:= {(j, k

, k

), 0 ≤ k

, k

≤ 2

− 1, (j, k

, k

) ∈ N

}}, where ∀f ∈ L

(R

), f

(x, y) = 2

f(2

x − k

, 2

y − k

). It is an orthogonal m-order(m ∈ N) multiresolution

on Ω with h = 2

(see [17]).

The basis function Φ

is the tensor product of φ

(x) and φ

(y) with Ω = [0, L

] × [0, L

], where φ

(x)(φ

(y)) is the scaling function on the interval [0, L

](resp [0, L

]) in the x(resp y) direction.

V

= span{θ

, α ∈ K

}. θ

= (I − ν4)

Φ

. For any ², there exists an integer r > 0 and g ∈ U

, ˜ h = 2

such that kθ

− gk ≤ ² [19]. Thus we can approximate θ

by an element in U

up to arbitrary precision.

Q

= span{φ

, α

∈ K

:= {(j

, k

), (j

, k

) ∈ N

}}. It is an orthogo- nal m’-order(m

∈ N) multiresolution on Γ constructed by a map from the multiresolution on [0, 1] with h

= 2

[20].

Q

= span{φ

, α

∈ K

:= {(j

, k

), (j

, k

) ∈ N

}}}. It is an orthog- onal m”-order(m

∈ N) multiresolution on γ with h

= 2

.

The matrix form of (4) is

 



 

Find U

∈ R

, Λ

∈ R

, such that µ I C

D 0

¶ µ U

Λ

¶

= µ F

G

¶

, (5)

where the off-diagonal matrices C

=

Z

Γ

θ

φ

ds, D

= Z

γ

Φ

φ

ds, and the vectors at the right side

(F )

= Z

Ω

f θ ˜

dx, (G)

= Z

γ

gφ

ds.

Assuming the discretization level j

and j

are equal, the matrix involved in (5)

A :=

µ I C D 0

¶ .

1The multiresolution is of order mif it produces polynomials of degree up tom−1.

is square. To prove that there exists a unique solution of (4), we only need to show that A is invertible. We will use the following lemma [21].

Lemma 2. A is invertible if and only if C has a full-column rank and Ker(D) ∩ R(C) = {0},

where Ker(D) is the null space of matrix D and R(C) is the range space of matrix C.

Therefore, if the conditions in Lemma 2 are satisfied, (4) has a unique solu- tion.

Theorem 3. Let U

, V

, Q

and Q

be constructed as previously. If Q

∈ H

(Γ) with s

> 2, Q

∈ H

(γ) with s

> 1 such that j(s

− 2) − j

(s

− 1) > B where B is a positive constant, then there exists a unique solution of (4).

Proof. We mimic the proof of Theorem 3.1 in [22]. The condition j(s

− 2) − j

(s

− 1) > B ensures that the following LBB condition holds [23], i.e., there exist constants β

, β

such that

sup

vh∈VΩ vh6=0h

b

(v

, p

)

kv

k

≥ β

kp

k

2(Γ)

, ∀p

∈ Q

, (6)

sup

uh∈UΩ uh6=0h

b

(u

, q ˜

)

ku

k

≥ β

k˜ q

k

H⁻¹²(γ)

, ∀˜ q

∈ Q

. (7) We then estimate the condition number of C

C, D

D and (DC)

DC as follows.

-estimation of C

C

According to the definition of the matrix C, it follows

∀w ∈ R

, (C

Cw, w) = X

α∈Kj

( X

α⁰∈K⁰_j

C

w

)

= X

α∈Kj

| Z

Γ

θ

X

α⁰∈K_j⁰

w

φ

|

.

Substituting v

= P

α∈Kj

z

θ

, p

= P

α⁰∈K_j⁰

w

φ

in (6), we have sup

z∈R^4j

R

Γ

P

α∈Kj

z

θ

P

α⁰∈K_j⁰

w

φ

k P

α∈Kj

z

θ

k

≥ β

k X

α⁰∈K_j⁰

w

φ

k

, ∀w

∈ R

.

By the discrete Cauchy-Schwartz inequality, it yields Z

Γ

X

α∈Kj

z

θ

X

α⁰∈K_j⁰

w

φ

≤ (z, z)

(C

Cw, w)

.

Using this inequality, it leads to β

k X

α⁰∈K_j⁰

w

φ

k

≤ sup

z∈R^4j

R

Γ

P

α∈Kj

z

θ

P

α⁰∈K_j⁰

w

φ

k P

α∈Kj

z

θ

k

≤ (z, z)

k P

α∈Kj

z

θ

k

(C

Cw, w)

.

(8)

Due to θ

= (I − ν4)

Φ

and the orthonormality of the bases {Φ

}

in the norm k · k

, it follows

k X

α∈Kj

z

θ

k

∼ k X

α∈Kj

z

Φ

k

= (z, z)

. (9)

Applying the Bernstein inequality[24] on the left term, we have k X

α⁰∈K_j⁰

w

φ

k

≥ c2

(w, w)

, (10)

where the constant C > 0 depends on the property of the multiresolution on the boundary Γ.

Combing the (8),(9) and (9) together, it follows

(C

Cw, w) ≥ K

2

(w, w), ∀w ∈ R

.

On the other hand, we use the continuous Cauchy-Schwartz inequality to

get

Z

Γ

X

α∈Kj

z

θ

X

α⁰∈K_j⁰

w

φ

≤ k X

α∈Kj

z

θ

k

k X

α⁰∈K_j⁰

w

φ

k

≤ Ck X

α∈Kj

z

θ

k

(w, w)

≤ C(z, z)

(w, w)

,

where C > 0 only denotes a constant, does not mean the same value.

Choosing z

= R

Γ

θ

P

α⁰∈K_j⁰

w

φ

, then we have (C

Cw, w) ≤ K

(w, w).

Therefore,

K

2

(w, w) ≤ (C

Cw, w) ≤ K

(w, w). (11) -estimation of D

D

We use the definition of the matrix D to get

∀z ∈ R

, (D

Dz, z) = X

α⁰⁰∈K_j⁰⁰

( X

α∈Kj

D

z

)

= X

α⁰⁰∈K_j⁰⁰

| Z

γ

φ

X

α∈Kj

z

Φ

|

. Taking u

= P

α∈Kj

z

Φ

, ˜ q

= P

α⁰⁰∈K_j⁰⁰

w

φ

in (7), it follows sup

z∈R^4j

R

γ

P

α∈Kj

z

Φ

P

α⁰⁰∈K_j⁰⁰

w

φ

k P

α∈Kj

z

Φ

k

≥ β

k X

α⁰⁰∈K_j⁰⁰

w

φ

k

. (12) By the discrete Cauchy-Schartz inequality, it follows

Z

γ

X

α∈Kj

z

Φ

X

α⁰⁰∈K⁰⁰_j

w

φ

≤ (w, w)

(D

Dz, z)

.

Using (12) and the Bernstein inequality of the multiresolution Q

, we have (D

Dz, z)

≥ β

2

k X

α∈Kj

z

Φ

k

≥ β

2

k X

α∈Kj

z

Φ

k

= β

2

(z, z)

.

The inequality above equals to

(D

Dz, z) ≥ K

2

(z, z).

By the continuous Cauchy-Schwartz inequality, it follows Z

γ

X

α∈Kj

z

Φ

X

α⁰⁰∈K_j⁰⁰

w

φ

≤ k X

α∈Kj

z

Φ

k

k X

α⁰⁰∈K_j⁰⁰

w

φ

k

≤ Ck X

α∈Kj

z

Φ

k

(w, w)

≤ C2

(z, z)

(w, w)

,

where we use the Bernstein inequality of the multiresolution U

and the trace theorem.

Choosing w

= R

γ

φ

P

α∈Kj

z

Φ

, we get (D

Dz, z) ≤ K

2

(z, z).

Therefore,

K

2

(z, z) ≤ (D

Dz, z) ≤ K

2

(z, z). (13) -estimation of (DC)

DC

Taking z = cw in (13), we have

2

K

(Cw, Cw) ≤ (D

DCw, Cw) ≤ 2

K

(Cw, Cw).

By (11), it yields

2

K

K

(w, w) ≤ (D

DCw, Cw) ≤ 2

K

K

(w, w).

Denote K

= K

K

, k

= K

K

, and here j

= j

, we have K

2

(w, w) ≤ (DCw, DCw) ≤ K

2

(w, w), ∀w ∈ R

,

Thus it yields that C has a full-column rank and Ker(D) ∩ R(C) = {0}. By

Lemma 2, the proof concludes.

4. Interior error estimate

We will use the notation A . B which means that there exits a constant c > 0 such that A ≤ cB, similarly for &. Local interior error estimate deals with the error between the computed solution u

and ¯ u, a good approxima- tion of u on ω, on ¤ a subdomain of Ξ including ¯ ω. This estimate depends on the regularity of ¯ u. We first recall the results for the bilinear form a(·, ·) [22].

Lemma 4. For U

, V

constructed as previously, there exist sup

vh∈VΩ vh6=0h

a(u

, v

)

kv

k

& ku

k

, ∀u

∈ U

, (14)

sup

uh∈UΩ uh6=0h

a(u

, v

)

ku

k

& kv

k

, ∀v

∈ V

. (15) Then, we derive the interior error estimate as follows.

Theorem 5. Let ω ⊂⊂ ¤ ⊂⊂ Ξ ⊂⊂ Ω, u ˆ (resp u

) be the solution of problem of (1) (resp problem (4)). Moreover, (¯ u, ¯ λ) is the solution of problem (2’) associated to g ¯ = Φ(¯ λ) such that u ¯ satisfies (3). Then if u ¯ ∈ H

(¤), 1 < s ≤ m, λ ¯ ∈ H

(Γ), 0 < s

≤ m

, and j ≥ j

(j

depending on ω and Ξ), we have

kˆ u − u

k

. 2

k¯ uk

+ 2

k λk ¯

+ ², ∀² > 0.

Proof. Let ω ⊂⊂ ¤

⊂⊂ ¤ ⊂⊂ Ξ ⊂⊂ Ω and δ = 1 on ω, δ ∈ C

(¤

) and set ˜ u = δ u. Let ¯ T u ˜ ∈ U

be the unique solution of

a(˜ u − T u, v ˜

) = −b

(v

, λ ¯ − λ), ˜ ∀v

∈ V

, (16) where ˜ λ ∈ Q

solves k ˜ λ − λk ¯

= inf

h0

kλ

− ¯ λk

. The existence of T u ˜ is guaranteed by Lemma 4 and Lax-Milgram lemma.

Now

k˜ u − T uk ˜

. sup

v∈H¹(Ω)

a(˜ u − T u, v) ˜ kv k

. sup

v∈H¹(Ω)

( a(˜ u − T u, v ˜ − P

v)

kvk

− b

(P

v, λ ¯ − ˜ λ) kvk

),

(17)

where P

v ∈ V

satisfies

a(u

, v − P

v ) = 0, ∀u

∈ U

. (18) Furthermore, it follows from (15) that

kP

vk

. sup

uh∈U_h^Ω

a(u

, P

v)

ku

k

= sup

uh∈U_h^Ω

a(u

, v)

ku

k

. kvk

. Hence, for any η ∈ U

,

a(˜ u − T u, v ˜ − P

v) = a(˜ u − η, v − P

v) . k˜ u − ηk

kvk

, (19) where we have used (18). Also

|b

(P

v, λ ¯ − λ)| ˜ = |(BP

v, λ ¯ − λ) ˜

| . kP

vk

k λ ¯ − λk ˜

. kvk

2

k λk ¯

, (20) where the trace theorem and Jackson estimate are used.

It follows from (17), (19) and (20) that k˜ u − T uk ˜

. inf

η∈U_h^Ω

k˜ u − ηk

+ 2

k ¯ λk

. 2

k˜ uk

+ 2

k λk ¯

(Γ)

. 2

k¯ uk

+ 2

k λk ¯

. Hence,

k¯ u − T uk ˜

= k˜ u − T uk ˜

≤ k˜ u − T uk ˜

. 2

k¯ uk

+ 2

k ¯ λk

(Γ)

. (21)

Let us now estimate kˆ u − u

k

.

kˆ u − u

k

≤ kˆ u − uk ¯

+ k¯ u − u

k

≤ ² + k¯ u − T uk ˜

+ kT u ˜ − u

k

. (22) Note (¯ u, λ) (resp (u ¯

, λ

)) is the solution of (2’) (resp (4)), we subtract these two equations to have

a(¯ u − u

, v

) + b

(v

, ¯ λ − λ

) = 0, ∀v

∈ V

.

It follows then from (16) that

a(T u ˜ − u

, v

) = a(˜ u − u, v ¯

) + b

(v

, λ

− λ), ˜ ∀v

∈ V

. (23) Let P

h

be the biorthogonal projector on V

[22], we have a(u

, P

h

v) = a(u

, v), ∀(u

, v) ∈ U

× H

(Ω), and

kP

h

vk

. kvk

, ∀v ∈ H

(¤).

Hence,

kT u ˜ − u

k

≤ kT u ˜ − u

k

. sup

v∈H₀¹(¤)

a(T u ˜ − u

, v) kvk

. sup

v∈H¹₀(¤)

a(T u ˜ − u

, P

h

v) kP

h

vk

.

(24)

If the scale level j is sufficiently large, then P

h

v ∈ ˚ V

(¤) := {v

|v

∈ V

, suppv

⊂ ¤}, ∀v ∈ H

(¤). It follows from (23) and (24) that

kT u ˜ − u

k

. sup

vh∈˚V_h^Ω(¤)

a(T u ˜ − u

, v

)

kv

k

= sup

vh∈˚V_h^Ω(¤)

a(˜ u − u, v ¯

) kv

k

. inf

η∈U_h^Ω

k˜ u − ηk

+ inf

η∈U_h^Ω

k¯ u − ηk

. 2

k¯ uk

.

(25)

By (21), (22) and (25), we have

kˆ u − u

k

. 2

k¯ uk

+ 2

k λk ¯

(Γ)

+ ², which completes the proof.

Here the second term in the right hand side of the estimate of Theorem 5

illustrates the influence of the Lagrange multiplier, i.e., the amplitude of the

normal gradient jump at the control boundary.