Spline Approximation Methods for the Biharmonic

Spline Approximation Methods for the Biharmonic Dirichlet Problem on Non-Smooth Domains Victor D. Didenko and Bernd Silbermann Abstract. For the approximate solution of the biharmonic Dirichlet problem we propose and study a boundary element method based on the integral equation of Muskhelishvili. Such an approach has a number of advantages, for instance, this equation does not have any critical geometry and in the case of smooth boundaries the method always converges. If the boundary has corner points, then the convergence of the method depends on the invertibility of some operators from a Toeplitz algebra.

1. Introduction Let D be a two-dimensional domain bounded by a simple closed piecewise smooth ¯ := D ∪ Γ. In the present work we analyse spline Galerkin contour Γ and let D approximation methods for the biharmonic Dirichlet problem  2   ∆ U|D = 0,    U|Γ = f1 , (1)      ∂U = f2 . ∂n Γ Here ∂U (t) is the outward normal derivative to Γ at the point t ∈ Γ, which is ∂n defined everywhere except the corner points of Γ. We assume that the function U is sought in Wp1 (D) ∩ Wp4 (D). The notation Wpk (X) is used for the Sobolev space of k-times differentiable functions on X the derivatives of which belong to the corresponding space Lp (X). The biharmonic problem (1) has been intensively studied by different authors. The reader can consult [17] for results concerning its solvability and solution properties. Problem (1) arises in different branches of applied mathematics. For example, the behaviour of plane “slow” viscous flows, deflection of plates, elastic equilibrium of solids as well as a number of other problems can be modelled by means of the biharmonic equation [8, 9, 22, 23, 24, 25]. It is therefore no wonder that this problem has been attracting great attention by numerical analysts. It suffices to say that

2

V.D. Didenko and B. Silbermann

one of the most powerful approximation procedures, namely, the Galerkin method was discovered while considering a special case of problem (1). Among the variety of approaches to the numerical treatment of problem (1) one can distinguish the so-called boundary element methods [2, 4, 11, 12, 19]. They allow us to reduce the dimension of the initial problem and, as a result, to reduce the computation cost drastically. The authors of the afore-mentioned papers usually use different modifications of the integral equation proposed by C. Christiansen and P. Hougaard [5, 6, 16] or an integral equation of first kind [18]. Although such approaches are widely used, they have some drawbacks. Thus, for some boundaries called “critical” the corresponding integral operators become non-invertible and corrections are to be done before one can start with the construction of approximation methods. In addition, the analysis of the stability of the approximation methods proposed has been accomplished for smooth boundaries only, though conditions for the invertibility of the corresponding integral operators are available for piecewise smooth contours as well [10]. On the other hand, there is a very nice “complex” approach to the problem (1). It takes its beginning from the works of N. I. Muskhelishvili [23] and leads to integral equations without critical geometry. However, in a strange way the Muskhelishvili equation remains an almost unknown quantity in numerical analysis. The most common approach to the approximate solution of this equation is based on trigonometric Fourier expansions and was proposed by N. I. Muskhelishvili himself in the middle of thirties (cf. [3, 7, 21]). Since then in numerical analysis there has been developed a lot of new powerful approximation methods, but they have not been implemented and studied in the case of the Muskhelishvili equation. We mention the papers [26, 27], which deal with spline approximations of solutions of the Muskhelishvili equation but do not contain any stability analysis. It is notable that the integral operators in the Muskhelishvili equation can be “locally” represented as elements of an algebra of Mellin operators with conjugation. The stability of approximation methods for Mellin convolutions was studied in [20], so we could apply some of those results to the operators appearing in the case of the Muskhelishvili equation. It should be also mentioned that from the practical point of view the convergence of the methods considered has to be proved in spaces of differentiable functions. However, the technique used here is well adapted to the norms of Lp spaces. Therefore, we first show the stability in the spaces Lp and then, using [28], we obtain some results for Sobolev norms. This paper is organized as follows. First, we reduce the initial problem (1) to a boundary value problem for two analytic functions in the domain D. One of the functions is a solution of the Muskhelishvili equation. However, the corresponding integral operator is not invertible on the space we work with. Nevertheless, it can be corrected in such a way that a newly obtained operator is invertible and the solution of the associated integral equation is simultaneously a solution of the Muskhelishvili equation. Afterwards we study the stability of spline Galerkin method for the auxiliary integral equation and obtain an approximate solution for the equation of Muskhelishvili.

Spline Approximation Methods

3

2. Auxiliary results This section contains results which are needed to construct an approximate solution to problem (1). Some of these results are known. In slightly modified form, they can be found in the literature dealing with problems of elasticity theory. However, we include them to make the paper self-contained. On the other hand, below we will mention auxiliary results concerning the invertibility of the integral operators related to our problem. Their proofs will be published in another paper. Let z denote the complex coordinate of a point of D ∪ Γ, so z = x + iy, and let c1 , c2 , . . . , cl be the corner points of Γ. By ωj we denote the angle between the corresponding semi-tangents at the point cj . Hence, ωj ∈ (0, 2π),

ωj = π,

j = 1, 2, . . . , l.

For real numbers p and αj (j = 1, 2, . . . , l) satisfying p > 1 and 0 < αj +

1 < 1, p

j = 1, 2, . . . , l,

(2)

we introduce the weight function ρ = ρ(t) =

l

|t − cj |αj ,

t ∈ Γ,

j=1

and by Lp (Γ, ρ) we denote the set of all Lebesgue measurable functions f such that 1/p p ||f || := |f (t)ρ(t)| |dt| < ∞. Γ

Wp1 (Γ, ρ)

Then, refers to the Sobolev space of all functions the derivatives of which belong to Lp (Γ, ρ). Let α = α(t), t ∈ Γ be the angle between the real axis and the outward normal n to Γ at the point t. By s we denote the unit vector defined by the requirement that the angle between s and the real axis is α − π/2. Proposition 2.1. Let (f1 , f2 ) ∈ Wp1 (Γ, ρ) × Lp (Γ, ρ). If U = U(x, y) is a solution of problem (1), then it can be represented in the form U(x, y) = Re [zψ(z) + χ(z)],

z = x + iy,

(3)

where ψ and χ are the functions analytic in D that satisfy the boundary condition ∂f1 , t ∈ Γ. (4) ψ(t) + tψ (t) + χ (t) = e−iα f2 + i ∂s Proof. It is well known [23] that any biharmonic function can be represented in the form (3) with some analytic functions ψ and χ. So our task now is to find these functions in such a way that they will satisfy the boundary conditions (1).

4


Since f1 belongs to the Sobolev space Wp1 (Γ, ρ) and since U was supposed to be differentiable on D, one can write ∂U ∂U ∂f1 = +i ∂s ∂n ∂s ∂U ∂U = eiα (Ux + iUy ). = (cos α + i sin α) −i ∂x ∂y

f2 + i

(5)

On the other hand, if we represent the function ψ of (3) in the form ψ = u + iv, then U(x, y) = xu(x, y) + yv(x, y) + Re χ(x, y). Immediate calculations and the Cauchy-Riemann equations lead to the formula Ux (z) + iUy (z) = u(x, y) + xux (x, y) + yvx (x, y) + Re χx (x, y) + +i(xuy (x, y) + v(x, y) + yvy (x, y) + Re χy (x, y)) = ψ(z) + zψ (z) + χ (z).

(6)

Comparing (5) and (6) we obtain that the functions ψ and χ of (3) satisfy the boundary condition (4). Assume for a moment that the boundary values ψ = ψ(t), χ = χ(t), t ∈ Γ of the functions ψ and χ have been found. Then by the Cauchy’s integral formula, 1 ψ(t)dt ψ(z) = , z ∈ D, 2πi Γ t − z (7) χ(t)dt 1 , z ∈ D, χ(z) = 2πi Γ t − z one will be able to represent the biharmonic function U inside of the domain D with the help of (3). Later on we will see that the function ψ = ψ(t), t ∈ Γ is the solution of an integral equation, so if ψ is known, then χ (t) = f (t) − ψ(t) − tψ (t),

t ∈ Γ,

(8)

where f denotes the right-hand side of (4). It is clear that equation (8) allows us to restore the function χ. Really, let t = t(s) be a 1-periodic parameterization of Γ, and let s1 , s2 , . . . , sl be those points on [0,1) which correspond to the corner points of Γ, i.e., t(sj ) = cj , j = 1, 2, . . . , l. The function t = t(s) is continuously differentiable on (sj , sj+1 ) because Γ was supposed to be a piecewise smooth curve. We set sl+1 = s1 + 1, let Γk be the


5

subarc of Γ which joins the points tk and tk+1 , and let s ∈ [sk , sk+1 ]. Using (8) we can write χ (t(s))t (s) = f (t(s))t (s) − ψ(t(s))t (s) − t(s)ψ (t(s))t (s) for any s ∈ (sk , sk+1 ) and hence introduce the functions s s f (t(s))t (s)ds − ψ(t(s))t (s)ds ζk (t(s)) = sk s

+

sk

dt(s) ψ(t(s))ds − t(s)ψ(t(s)), k = 1, 2, . . . , l. ds

(9)

sk

Then the function χ may be represented in the form  ζ1 (t) + C        ζ2 (t) + (ζ1 (c2 ) − ζ2 (c2 )) + C χ(t) =   ...............      ζl (t) + (ζl−1 (cl ) − ζl (cl )) + . . . + (ζ1 (c2 ) − ζ2 (c2 )) + C

if t ∈ Γ1 if t ∈ Γ2

.

... if t ∈ Γl (10)

The indefinite constant C must be chosen to satisfy the first of the boundary conditions (1). Therefore, if we would have found some approximations {ψn } of the function ψ such that ||ψn − ψ||Lp (Γ,ρ) → 0 as n → ∞, then we might construct “good” approximations χn of χ replacing ψ by ψn in (9)-(10) and obtain ||χn − χ||Lp (Γ,ρ) → 0

as n → ∞,

with the same rate of convergence. So, our task now is to find approximations of the functions ψ. To achieve the goal we will use the Muskhelishvili equation. Let us recall [14, 15] that if ψ is a solution of (4) and belongs to Wp1 (Γ, ρ), then it satisfies the equation 1 τ −t τ −t 1 − = f0 (t), t ∈ Γ, Rψ(t) ≡ −ψ(t) − ψ(τ )d log ψ(τ )d 2πi Γ τ − t 2πi Γ τ −t (11) 1 f (τ )dτ 1 f0 (t) = − f (t) + 2 2πi Γ τ − t and f is the function of the right-hand side of (4). Here we can mention the two properties of the integral operator R which probably prevent us from applying approximation methods to the equation (11). First, the operator R has been investigated in the spaces Wp1 (Γ, ρ) (see [14, 15]), but at present there are no efficient

where

6


tools for studying the stability of approximation methods in spaces Wp1 (Γ, ρ) when Γ possesses corner points. Secondly, R is not invertible, neither as an operator acting on Wp1 (Γ, ρ) nor as one on Lp (Γ, ρ). Fortunately, things can be corrected in an appropriate way. We consider the following finite-dimensional operator T :

ψ(τ ) ψ(τ )dτ ψ(τ )dτ 1 1 1 . dτ + (T ψ)(t) = + 2πi Γ τ t 2πi Γ τ2 τ2 Theorem 2.2. Let Γ be a piecewise smooth curve. There exist δ < 1/2 and δ > 1/2 such that if

1 1 + αj ≤ max + αj < δ , (12) δ < min 1≤j≤l p 1≤j≤l p then the operator R1 = R + T : Lp (Γ, ρ) → Lp (Γ, ρ) is invertible. If the weight ρ satisfies inequality (12), f ∈ Wp1 (Γ, ρ), and Re f (t)dt = 0, (13) Γ

then the solution of the equation R1 ψ = f0

(14)

is simultaneously a solution of the Muskhelishvili equation (11) and of the boundary problem (4). For a proof of this result see [13]. Corollary 2.3. The operator R1 : L2 (Γ) → L2 (Γ) is invertible.

3. Approximate solution of the Muskhelishvili equation on special contours In this section we consider the stability of a spline Galerkin method for the Muskhelishvili operator on special curves. The corresponding results concern the so-called local models. In the sequel they will be used to obtain necessary and sufficient conditions for the stability of the spline Galerkin method on Γ. Let β ∈ [0, 2π) and ω ∈ (0, π) ∪ (π, 2π) be real numbers. We denote by Γβ,ω the infinite angle Γβ,ω := Γ1 ∪ Γ2 , where the ray Γ1 := ei(β+ω) R+ is directed to 0 but Γ2 := eiβ R+ is directed away from 0. Then Lp (Γβ,ω , α), 0 < 1/p + α, is the space of all Lebesgue measurable functions f equipped with the norm

1/p ||f ||p,α,ω = . |f (t)|p |t|αp |dt| Γβ,ω


7

Note that in the case ω = π all the following computations are trivial and will be omitted. Let Rω denote the Muskhelishvili operator on Lp (Γβ,ω , α) : Rω x(t) ≡ −x(t) −

1 2πi

x(τ )d log Γβ,ω

1 τ −t − τ − t 2πi

x(τ )d Γβ,ω

τ −t . τ −t

(15)

First of all we construct spline spaces on Γβ,ω . Let χ[0,1) = χ[0,1) (s), s ∈ R, denote the characteristic function of the interval [0,1). For any natural number m, one may introduce the function ϕm (s) := (ϕ0 ∗ ϕm−1 )(s),

s ∈ R,

with ϕ0 (s) = χ[0,1) (s),

s ∈ R,

and with (f ∗ g) denoting the convolution of the functions f and g, i.e., (f ∗ g)(s) :=

R

f (s − x)g(x)dx.

From now on we fix m ∈ N and set ϕ(s) = ϕm (s),

s ∈ R.

As a next step we fix a natural number n and, for each k ∈ N, we define the function ϕkn = ϕkn (s) by ϕkn (s) := ϕ(ns − k),

s ∈ R.

(16)

Lemma 3.1. Let ϕ denote the function defined by (16). Then 1. supp {ϕ} ⊂ [0, m + 1]; 2. for every s ≥ 0 one has ϕ(−s + m + 1) = ϕ(s).

(17)

Proof. Both assertions of this lemma can be proved by induction. We show the second one only. Thus, if m = 1, then    ϕ(s) =

 

s

if 0 ≤ s < 1,

2 − s if 1 ≤ s < 2, 0

otherwise,

8


and (17) is obvious. Suppose that equality (17) is satisfied for k = m and consider the case k = m + 1. One has χ[0,1) (−s + m + 2 − x)ϕm (x)dx ϕm+1 (−s + m + 2) = R

=

R

χ[0,1) (−s + 1 + u)ϕm (−u + m + 1)du

=

R

χ[0,1) (−s + 1 + u)ϕm (u)du

=

R

χ[0,1) (s − u)ϕm (u)du = ϕm+1 (s),

and the proof is complete.

Now we are able to introduce spline spaces on Γβ,ω . Namely, we denote by Snβ,ω the smallest closed subspace of Lp (Γβ,ω , α) which contains all the functions  ϕkn (s) if t = eiβ s   k ≥ 0,   0 otherwise  (n) φk (t) := (18)   ϕk−m,n (s) if t = ei(β+ω) s   k < 0.  0 otherwise Let us consider the semi-linear form f, g :=

f (t)g(t)|dt|, Γβ,ω

where f ∈ Lp (Γβ,ω , α), g ∈ Lq (Γβ,ω , −α) and 1/p + 1/q = 1. The Galerkin projec n from Lp (Γβ,ω , α) onto S β,ω can be defined by tion operators L n n f, φkn := f, φkn for f ∈ Lp (Γβ,ω , α) L

and φkn ∈ Snβ,ω .

(19)

n , (n = 1, 2, . . . ) are well-defined and that It is known [20] that the operators L n } converges strongly to the identity operator I as n tends to ∞. the sequence {L n → Im L n , is said to be Definition 3.2. An operator sequence {An } , An : Im L n → stable if there exists an n0 such that for all n ≥ n0 the operators An : Im L n are invertible and Im L sup ||A−1 n Ln || < ∞. n≥n0

Our task now is to establish stability conditions for the sequence of the n Rω L n } in the case where Rω is the Muskhelishvili operator Galerkin operators {L (15). To proceed with this problem we have to recall some notions. We denote by M and M −1 the direct and inverse Mellin transforms, respectively: +∞ x1/p+α−zi−1 f (x)dx, z ∈ R, (M f )(z) = 0


(M −1 f )(x) =

1 2π

+∞

xzi−1/p−α f (z)dz,

−∞

9

x ∈ R+ .

Let Lp (R+ , α), p > 1, 0 < 1/p+α < 1, refer to the space of all Lebesgue measurable functions on the half-line R+ equipped with the norm 1/p p αp ||f ||p,α := |f (t)| t dt . R+

We will also consider the set L2p (R+ , α) of all pairs (f1 , f2 )T , f1 , f2 ∈ Lp (R+ , α). The norm in L2p (R+ , α) is defined by 1/p ||(f1 , f2 )T || = ||f1 ||pp,α + ||f2 ||pp,α . It is known (see, e.g., [20]) that under some restrictions on the function b the rule (M(b)f )(σ) = ((M −1 bM )f )(σ) defines a bounded linear operator on Lp (R+ , α). The operator M is called the Mellin operator with the symbol b, and it can be represented in the integral form σ ds f (s) , k = M −1 b. (M(b)f )(σ) = k s s R+ For any Banach space X, we denote by L(X) (Ladd (X)) the space of all bounded linear (additive) operators on X. Following [13] we introduce a mapping η : Lp (Γβ,ω , α) → L2p (R+ , α) by η(f ) = (η1 (f ), η2 (f ))T

(20)

where η1 (f )(s)

= f (sei(β+ω) ),

η2 (f )(s)

= f (seiβ ),

s ∈ R+ ,

s ∈ R+ .

Obviously, the mapping η : Lp (Γβ,ω , α) → L2p (R+ , α) is invertible and A → ηAη −1 is an isometric algebra isomorphism of L(Lp (Γβ,ω , α)) onto L(L2p (R+ , α)). Lemma 3.3. Let Sn be the smallest closed subspace of Lp (R+ , α) which contains all functions ϕkn = ϕkn (s), s ∈ R+ of (16) with k ≥ 0 and let Ln : Lp (R+ , α) → Sn denote the Galerkin projection onto Sn . Then for every n ∈ N the operator n ∈ L(Lp (Γβ,ω , α)) is isometrically isomorphic to the operator diag (Ln , Ln ) ∈ L L(L2p (R+ , α)). Proof. Immediate calculations and Lemma 3.1 show that n )η −1 = diag (Ln , Ln ). η(L

10


Note that in the sequel any diagonal operator of the form diag (T, T, . . . , T ) will be written as T, so we write Ln instead of diag (Ln , Ln ). Lemma 3.4. The operator Rω ∈ Ladd (Lp (Γβ,ω , α)) is isometrically isomorphic to the operator Aω ∈ Ladd (L2p (R+ , α)) defined by Aω = A + B V , where the operators A, B and V are given by 0 A = −e−i2(β+ω) M2π−ω B

=

e−i2β Mω 0

1 2

−I 1 [N − N2π−ω ] ω 2

(21) ,

[Nω − N2π−ω ] −I

,

and V (f1 , f2 ) = (f1 , f2 ), and where Mν , Nν , ν ∈ (0, 2π) are the Mellin operators sin ν ϕ(s) 1 +∞ σ (Mν (ϕ))(σ) = ds π 0 s (1 − (σ/s)eiν )2 s and (Nν (ϕ))(σ) =

1 2πi

+∞

0

ϕ(s)ds . s − σeiν

A proof of the last lemma can be found in [13]. The next result immediately follows from the Lemmas 3.3 and 3.4. n Rω L n } is stable if and only if so is the sequence Proposition 3.5. The sequence {L Ln Aω Ln . Now we can make further simplification. For, we consider the space lp,α of all sequences {ξj }+∞ j=0 of complex numbers ξj such that ||{ξj }||pp,α =

+∞

(1 + j)αp |ξj |p < ∞,

j=0

and we define the operators En : lp,α → Sn and E−n : Sn → lp,α by En : {ξj } →

+∞

ξj ϕjn (t),

j=0

E−n :

+∞ j=0

ξj ϕjn (t) → {ξj }.


11

Proposition 3.6. ([1]) The mappings En : lp,α → Sn and E−n : Sn → lp,α are bounded linear operators, and there are constants C1 > 0 and C2 > 0 such that

+∞

ξj ϕjn Lp (R+ ,α) ≤ C1 n−(1/p+α) {ξj }lp,α ,

(22)

j=0

{ξj } ≤ C2 n(1/p+α)

+∞

ξj ϕjn Lp (R+ ,α) .

(23)

j=0 2 Let lp,α denote the cartesian product of two copies of lp,α .

n Rω L n } is stable if and only if the operator Bω1 := Lemma 3.7. The sequence {L 2 2 E−1 L1 Aω L1 E1 : lp,α → lp,α is invertible. Proof. Let M(b) be a Mellin convolution operator and k = M −1 (b) and let G(b) = −1 (alq )∞ bM )Ln En : lp,α → lp,α . Then l,q=0 be the matrix of the operator E−n Ln (M alq = (M(b)ϕqn )(s)ϕln (σ)dσ R

=

k R

σ s

R+

=

k R

R+

ϕ(ns − q)

u+l t+q

ϕ(t)

ds ϕ(σn − l)du s

dt ϕ(u)du. t

Thus, the entries of the matrix G(b) are independent of n. Taking into account that the operator Bωn := E−n Ln Aω Ln En admits the representation −I G(b3 ) V 0 0 G(b1 ) Bωn = + (24) G(b2 ) 0 −I G(b3 ) 0 V where V ({ξj }) = {ξ j } and b1 = e−2iβ mω (y), b2 = e−2i(β+ω) m2π−ω (y), b3 = knω (y), y = z + i(1/p + α), z ∈ R, one obtains the claim. It is worth mentioning that studying invertibility of the operator Bω1 is a very difficult problem. However, the conditions for the invertibility of the operator Aω are simultaneously the conditions for the Fredholmness of the operator Bω1 . Corollary 3.8. For every ω ∈ (0, 2π) there exists a real number δω > 1/2 such that for every p > 1 and every α satisfying the inequality 0
0 such that ||g − Ln g||L2 (Γ) < c1 n−1 ||g||W21 (Γ) ,

g ∈ W21 (Γ).

The latter inequality yields the estimate ||Ag − Ln ALn g||L2 (Γ) < c2 n−1 ||g||W21 (Γ) ,

g ∈ W21 (Γ).

(29)

Note that the positive constants c1 , c2 are independent of g ∈ W21 (Γ) and n. Taking into account the inverse properties of the splines ψn (recall that n = lr, r ∈ N and that the support of ϕ kn is entirely contained in one of the arcs (γ(j/l), γ((j + 1)/l)), j = 0, 1, . . . , l − 1) and applying Theorem 1.37 of [28] one obtains the stability of the Galerkin method (28). This implies the convergence of {ψn } in W21 (Γ). Using Theorem 2.2 once more we get the result. Thus, the Galerkin method (28) can be used to find approximate solutions of the Muskhelishvili equation (11). This allows us to construct approximate solutions of the biharmonic problem (1) (see Section 2 for details). Acknowledgment. The work of V. D. Didenko was supported in part by Universiti Brunei Darussalam under the Research Grant UBD/PNC2/2/RG/1(12).

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Department of Mathematics, Universiti Brunei Darussalam, Bandar Seri Begawan, BE1410 Brunei E-mail address: [email protected] Faculty of Mathematics, University of Technology Chemnitz, 09107 Chemnitz, Germany E-mail address: [email protected]