FOR THE FINITE-DIFFERENCE SCHEMES

INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES

WARSZAWA 1994

MULTIPLIERS IN SOBOLEV SPACES AND EXACT CONVERGENCE RATE ESTIMATES

FOR THE FINITE-DIFFERENCE SCHEMES

B O ˇ S K O J O V A N O V I ´ C

University of Belgrade, Faculty of Sciences Studentski trg 16, POB 550, 11000 Belgrade, Yugoslavia

Abstract. In this paper we present some recent results concerning convergence rate esti- mates for finite-difference schemes approximating boundary-value problems. Special attention is given to the problem of minimal smoothness of coefficients in partial differential equations necessary for obtaining the results.

1. Introduction. Recently, increased attention is given to approximation of generalized solutions of partial differential equations with finite-difference methods.

For a problem with the solution belonging to the Sobolev space W

(Ω), the convergence estimate

(1) ku − vk

p(ω)

≤ Ch

kuk

p(Ω)

, s > k ,

is said to be compatible with the smoothness of the solution [12]. Here u ∈ W

(Ω) denotes the solution of the original boundary-value problem, v denotes the so- lution of the corresponding finite-difference scheme, h is the discretization pa- rameter, W

(ω) denotes the discrete Sobolev space, and C is a positive generic constant, independent of h and u.

Estimates of this type have been obtained for a broad class of elliptic problems (see [6, 10, 11, 13, 16]). Analogous results have also been obtained for parabolic and hyperbolic problems (see [5, 7, 8, 9]). As a rule, the Bramble–Hilbert lemma [2, 4] is used in their proofs.

1991 Mathematics Subject Classification: 46E35, 65N15.

Key words and phrases: Sobolev spaces, multipliers, boundary-value problems, finite diffe- rences.

This work was supported by SF of Serbia, grant number 0401F.

The paper is in final form and no version of it will be published elsewhere.

[165]

For equations with variable coefficients, the natural problem arises of estab- lishing the minimal smoothness properties of coefficients for obtaining the same type (1) of estimate. Such coefficients belong to classes of multipliers in Sobolev spaces.

2. Multipliers in Sobolev spaces. Let Ω be a domain in R

. By D(Ω) = C ˙

(Ω) we denote the space of infinitely smooth functions with compact support in Ω, and by D

(Ω) the space of distributions. Moreover, x = (x

, x

, . . . , x

) denote vectors from R

, and α = (α

, α

, . . . , α

) are multi-indices. Let |α| = α

+ α

+ . . . + α

. Partial derivatives are denoted by

D

u = ∂u/∂x

and D

u = D

D

. . . D

u .

Suppose V and W are two function spaces contained in D

(Ω). A function a defined on Ω is called a pointwise multiplier , or simply a multiplier, from V to W if, for every v in V , the product a · v belongs to W . The set of all multipliers from V to W is denoted by M (V → W ). In particular, when V = W we put M (V ) = M (V → V ).

In this section, we shall be concerned with multipliers in Sobolev spaces which belong to M (W

(Ω) → W

(Ω)), 1 < p < ∞. Naturally, we assume that t ≥ s.

To begin, we consider multipliers in pairs of Sobolev spaces on R

. Motivated by the definition of multiplication of a distribution with a smooth function, for a ∈ M (W

(R

) → W

(R

)) and u ∈ W

(R

), 1/p + 1/p

= 1, we define the product a · u ∈ W

(R

) by

ha · u, ϕi

p0 ×W_p^t

= hu, a · ϕi

p0 ×W_p^s

, ∀ϕ ∈ W

(R

) .

This definition implies that M (W

(R

) → W

(R

))=M (W

(R

) → W

(R

)), and therefore it suffices to explore the properties of the sets M (W

(R

) → W

(R

)) and M (W

(R

) → W

(R

)) for t ≥ s ≥ 0.

We recall a collection of fundamental results on multipliers in Sobolev spaces (see [14]).

Lemma 1. If a ∈ M (W

(R

) → W

(R

)), t ≥ s ≥ 0, then:

a ∈ M (W

(R

) → L

(R

)) ,

a ∈ M (W

(R

) → W

(R

)), 0 < σ < s ,

D

a ∈ M (W

(R

) → W

(R

)), |α| ≤ s ,

D

a ∈ M (W

(R

) → L

(R

)), |α| ≤ s .

Lemma 2. For t ≥ s ≥ 0, M (W

(R

) → W

(R

)) ⊆ W

, where W

= {f | sup

z∈Rⁿ

kη(x − z) · f (x)k

p

< ∞, ∀η ∈ D(R

), η ≡ 1 on B

} , and B

is the unit ball with center 0. If tp > n, then M (W

(R

) → W

(R

)) = W

.

Lemma 3. For s ≥ 0, M (W

(R

)) ⊆ L

(R

).

Lemma 4. Suppose 1 < p < ∞, and let s and t be nonnegative integers such that t ≥ s. If

a = X

|α|≤t

D

a

and a

∈ M (W

(R

) → W

(R

)) ∩ M (W

(R

) → L

(R

)), 1/p + 1/p

= 1, then a ∈ M (W

(R

) → W

(R

)).

Lemma 5. Let p > 1, t > s > 0, and suppose that either q ∈ [n/t, ∞] and tp < n, or q ∈ (p, ∞) and tp = n. If

a ∈ B

= {f | sup

z∈Rⁿ

kη(x − z) · f (x)k

q,p

< ∞, ∀η ∈ D(R

), η ≡ 1 on B

} , where B

is the Besov space, then a ∈ M (W

(R

) → W

(R

)). The result is also true for t = s, provided a ∈ B

∩ L

(R

).

Lemma 6. If a

∈ M (W

(R

) → W

(R

)), s ≥ k, for every multi-index α then the differential operator

(2) Lu = X

|α|≤k

a

(x)D

u, x ∈ R

, defines a continuous mapping from W

(R

) to W

(R

).

The analogous result holds true for s < 0. If p = 2 then the result holds true for every s. Under certain conditions we have the converse result:

Lemma 7. Let the operator (2) define a continuous mapping from W

(R

) to W

(R

), and p(s − k) > n, p > 1. Then a

∈ M (W

(R

) → W

(R

)), for every multi-index α.

All of these results can be transfered to Sobolev spaces in an open subset of R

. More precisely, if Ω is an open set in R

with a Lipschitz continuous boundary and a ∈ M (W

(Ω) → W

(Ω)), then a can be extended to a function e a, defined on the whole of R

, such that e a ∈ M (W

(R

) → W

(R

)). The converse is also true: the restriction to Ω of a multiplier a ∈ M (W

(R

) → W

(R

)) is an element of M (W

(Ω) → W

(Ω)).

For bounded domains, W

and B

are replaced by standard Sobolev

and Besov spaces, respectively. Employing Lemmas 2, 3, 5, imbedding theorems

for Besov spaces [1, 17] and the representation of distributions from negative Sobolev spaces [18], we obtain the following results:

Lemma 8. Suppose that Ω is a bounded open subset of R

with a Lipschitz continuous boundary , s > 0 and p > 1. If a ∈ W

(Ω) where

q = p, t = s when sp > n, and q ≥ n/s, t = s + ε, ε > 0 when sp ≤ n , then a ∈ M (W

(Ω)).

Lemma 9. Let Ω be a bounded open set in R

with a Lipschitz continuous boundary, s > 0 and p > 1. If a ∈ L

(Ω) where

q = p when sp > n , q > p when sp = n, and q ≥ n/s when sp < n , then a ∈ M (W

(Ω) → L

(Ω)).

Lemma 10. Let Ω be a bounded open set in R

with a Lipschitz continuous boundary and

a(x) = a

(x) +

n

X

i=1

D

a

(x) .

If a

∈ M (W

(Ω) → L

(Ω)), and a

∈ M (W

(Ω) → W

(Ω))∩M (W

(Ω) → L

(Ω)), i = 1, 2, . . . , n, where 0 < s ≤ 1 ≤ t < 2, s 6= 1/2, then a ∈ M (W

(Ω) → W

(Ω)).

3. Boundary-value problem and its approximation. As a model problem let us consider the first boundary-value problem for a second-order linear elliptic equation with variable coefficients, in the square Ω = (0, 1)

:

(3) −

2

X

i,j=1

D

(a

D

u) + au = f in Ω, u = 0 on Γ = ∂Ω .

We assume that the generalized solution of the problem (3) belongs to the Sobolev space W

(Ω), 1 < s ≤ 3, with the right-hand side f (x) belonging to W

(Ω).

Consequently, the coefficients a

(x) and a(x) belong to the following classes of multipliers: a

∈ M (W

(Ω)), a ∈ M (W

(Ω) → W

(Ω)). According to Lemmas 8–10 sufficient conditions are the following:

a

∈ W

(Ω) , a ∈ W

(Ω) , for |s − 1| > 1 , a

∈ W

(Ω) , a = a

+

2

X

i=1

D

a

,

a

∈ L

(Ω) , a

∈ W

(Ω) ,

where ε > 0,

δ > 0 , p > 2/|s − 1| for 0 < |s − 1| ≤ 1 , and δ = 0 , p = ∞ for s = 1 .

The following estimates do not depend on δ in any way, so we can put δ = 0.

We also assume that the following conditions hold:

a

= a

,

2

X

i,j=1

a

y

y

≤ c

2

X

i=1

y

, x ∈ Ω, c

= const > 0, a(x) ≥ 0 in the sense of distributions, i.e.

ha · ϕ, ϕi

≥ 0, ∀ϕ ∈ D(Ω) .

Let ω be the uniform mesh in Ω with step h, ω = ω ∩ Ω, γ = ω ∩ Γ , γ

= {x ∈ γ | x

= k, 0 < x

< 1}, k = 0, 1, and ω

= ω ∪ γ

. We define finite differences as usual:

v

= (v

− v)/h, v

= (v − v

)/h , where v

(x) = v(x ± hr

), and r

is the unit vector on the x

axis.

We also define the Steklov smoothing operators:

T

f (x) =

1

R

0

f (x + htr

) dt = T

f (x + hr

) = T

f (x + 0.5hr

) . These operators commute and transform derivatives to differences:

T

D

u = u

, T

D

u = u

.

We approximate the problem (3) with the following finite-difference scheme:

(4) L

v = T

T

f in ω, v = 0 on γ where L

v = −0.5 P

i,j=1

[(a

v

)

+ (a

v

)

] + (T

T

a)v and T

= T

T

. The difference scheme (4) is a standard symmetric difference scheme (see [15]) with the right-hand side and coefficient a(x) averaged. For 1 < s ≤ 3 these coefficients may not be continuous, so the difference scheme with non-averaged data is not well defined.

4. Convergence of the finite-difference scheme. Let u denote the solution to the boundary value problem (3) and v the solution to the difference scheme (4). The error z = u − v satisfies the conditions

(5) L

z =

2

X

i,j=1

η

+ ζ in ω, z = 0 on γ where η

= T

T

(a

D

u)−0.5(a

u

+a

u

j

) and ζ = (T

T

a)u−T

T

(au).

For θ ⊆ ω let (·, ·)

= (·, ·)

and k · k

= k · k

denote the discrete inner product and the discrete L

-norm on θ. We also define the discrete W

-norm on ω:

kvk

2(ω)

= kvk

+ kv

k

+ kv

k

.

Using the energy method [15] it is easy to prove the next lemma.

Lemma 11. The finite-difference scheme (5) is stable in the sense of the a priori estimate

(6) kzk

2(ω)

≤ C  X

i,j=1

kη

k

+ kζk

 .

The problem of deriving the convergence rate estimate for the finite-difference scheme (4) is now reduced to estimating the right-hand side terms in (6). Esti- mates which follow are based on the following bilinear version of the Bramble–

Hilbert lemma [3, 10]:

Lemma 12. Let E be a bounded open set in R

with a Lipschitz continuous boundary and let η(u, v) be a bounded bilinear functional on W

(E) × W

(E), 1 ≤ p, q ≤ ∞, s, t > 0, such that

η(u, v) = 0

if either u is a polynomial of degree < s and v ∈ W

(E), or v is a poly- nomial of degree < t and u ∈ W

(E). Then there exists a positive constant C = C(E, p, s, q, t) such that

|η(u, v)| ≤ |u|

|v|

, ∀(u, v) ∈ W

(E) × W

(E), with the seminorms of the corresponding spaces at the right-hand side.

First, we decompose η

in the following way:

η

= η

+ η

+ η

+ η

, where η

= T

T

(a

D

u) − (T

T

a

) · (T

T

D

u) , η

= [T

T

a

− 0.5(a

+ a

)] · (T

T

D

u) , η

= 0.5(a

+ a

) · [T

T

D

u − 0.5(u

+ u

j

)], and η

= −0.25(a

− a

) · (u

− u

j

) . For 1 < s ≤ 2 we set ζ = ζ

+ ζ

+ ζ

, where

ζ

= (T

T

a

)u − T

T

(a

u), and

ζ

= (T

T

D

a

)u − T

T

(D

a

· u), i = 1, 2 . For 2 < s ≤ 3 we set ζ = ζ

+ ζ

, where

ζ

= (T

T

a) · (u − T

T

u), and

ζ

= (T

T

a) · (T

T

u) − T

T

(a · u) .

Let us introduce the elementary rectangles e = e(x) = {y | |y

− x

| ≤ h, j = 1, 2} and e

= e

(x) = {y | x

≤ y

≤ x

+ h, |y

− x

| ≤ h}, i = 1, 2.

The value η

at the node x ∈ ω

is a bounded bilinear functional on W

(e

)×

W

(e

) where λ ≥ 0, µ ≥ 1 and q > 2. Moreover, η

= 0 if either a

is a constant or u is a first-degree polynomial. Using Lemma 12 and a procedure proposed in [11], developed in [10], we obtain

|η

| ≤ C(h)|a

|

q(ei)

|u|

2q/(q−2)(ei)

, 0 ≤ λ ≤ 1, 1 ≤ µ ≤ 2 ,

where C(h) = Ch

. Summation with the use of the H¨ older inequality yields (7) kη

k

≤ Ch

|a

|

q(Ω)

|u|

2q/(q−2)(Ω)

, 0 ≤ λ ≤ 1, 1 ≤ µ ≤ 2 . Set λ = s − 1, µ = 1 and q = p. By the imbedding theorem [17], W

⊆ W

for 1 < s ≤ 2. Therefore, from (7) we obtain

(8) kη

k

≤ Ch

ka

k

p (Ω)

kuk

2(Ω)

, 1 < s ≤ 2 . Similar estimates hold for η

, η

, ζ

and ζ

.

Let now q > 2 be a constant. The following imbeddings hold: W

⊆ W

for µ > 2 − 2/q, and W

⊆ W

for λ > 2/q. Setting λ + µ = s we obtain from (7),

(9) kη

k

≤ Ch

ka

k

2 (Ω)

kuk

2(Ω)

, 2 < s ≤ 3 . In the same manner we can estimate η

.

For s > 2, η

(x) is a bounded bilinear functional on W

(e

) × W

(e

) which vanishes if either a

is a first-degree polynomial or u is a constant. Using the same lemma and the imbedding W

⊆ W

we obtain for η

an estimate of the form (9).

Similarly, η

(x) is a bounded bilinear functional on L

(e

) × W

(e

), s > 1, which vanishes if u is a second-degree polynomial. In the same way, using the imbeddings W

⊆ L

(for 1 < s ≤ 2) and W

⊆ L

(for s > 2) we again obtain estimates of the forms (8) and (9).

Let 2 < q < 2/(3 − s). For 2 < s ≤ 3, ζ

(x) is a bounded bilinear functional on L

(e)×W

(e). Moreover, ζ

= 0 if u is a first-degree polynomial. Using the Bramble–Hilbert lemma and the imbeddings W

⊆ L

and W

⊆ W

we obtain the estimate

(10) kζ

k

≤ Ch

kak

2 (Ω)

kuk

2(Ω)

, 2 < s ≤ 3 .

For 2 < s ≤ 3, ζ

(x) is a bounded bilinear functional on W

(e) × W

(e).

Using the same methodology and the imbedding W

⊆ W

, we obtain for ζ

an estimate of the form (10).

Finally, let 2 < q < min{2 + ε, 2/(2 − s)}. For 1 < s ≤ 2, ζ

(x) is a bounded

bilinear functional on L

(e)×W

(e) which vanishes if u is a constant. Using

the imbeddings L

⊆ L

and W

⊆ W

, we obtain the estimate (11) kζ

k

≤ Ch

ka

k

kuk

2(Ω)

, 1 < s ≤ 2 . Combining (6) with (8)–(11) we obtain the final result:

Theorem. The finite-difference scheme (4) converges and the following esti- mates hold :

(12) ku − vk

2(ω)

≤ Ch

(max

i,j

ka

k

2 (Ω)

+ kak

2 (Ω)

)kuk

, for 2 < s ≤ 3 , and

ku − vk

2(ω)

≤ Ch

(max

i,j

ka

k

p (Ω)

+ max

i

ka

k

p (Ω)

(13)

+ ka

k

)kuk

, for 1 < s ≤ 2 .

The obtained convergence-rate estimates (12) and (13) are compatible with the smoothness of data. An analogous estimate in L

-norm is obtained in [6].

Non-stationary problems were considered in [7, 8].

References

[1] R. A. A d a m s, Sobolev Spaces, Academic Press, New York 1975.

[2] J. H. B r a m b l e and S. R. H i l b e r t, Bounds for a class of linear functionals with appli- cation to Hermite interpolation, Numer. Math. 16 (1971), 362–369.

[3] P. G. C i a r l e t, The Finite Element Method for Elliptic Problems, North-Holland, Ams- terdam 1978.

[4] T. D u p o n t and R. S c o t t, Polynomial approximation of functions in Sobolev spaces, Math. Comp. 34 (1980), 441–463.

[5] B. S. J o v a n o v i ´ c, On the convergence of finite-difference schemes for parabolic equations with variable coefficients, Numer. Math. 54 (1989), 395–404.

[6] —, Optimal error estimates for finite-difference schemes with variable coefficients, Z.

Angew. Math. Mech. 70 (1990), 640–642.

[7] —, Convergence of finite-difference schemes for parabolic equations with variable coeffi- cients, ibid. 71 (1991), 647–650.

[8] —, Convergence of finite-difference schemes for hyperbolic equations with variable coeffi- cients, ibid. 72 (1992), to appear.

[9] B. S. J o v a n o v i ´ c, L. D. I v a n o v i ´ c and E. E. S ¨ u l i, Convergence of a finite-difference scheme for second-order hyperbolic equations with variable coefficients, IMA J. Numer.

Anal. 7 (1987), 39–45.

[10] —, —, —, Convergence of finite-difference schemes for elliptic equations with variable coefficients, ibid., 301–305.

[11] R. D. L a z a r o v, On the question of convergence of finite-difference schemes for generalized solutions of the Poisson equation, Differentsial’nye Uravneniya 17 (1981), 1285–1294 (in Russian).

[12] R. D. L a z a r o v, V. L. M a k a r o v and A. A. S a m a r s k i˘ı, Application of exact dif-

ference schemes for construction and investigation of difference schemes for generalized

solutions, Mat. Sb. 117 (1982), 469–480 (in Russian).

[13] R. D. L a z a r o v, V. L. M a k a r o v and W. W e i n e l t, On the convergence of difference schemes for the approximation of solutions u ∈ W

(m > 0.5) of elliptic equations with mixed derivatives, Numer. Math. 44 (1984), 223–232.

[14] V. G. M a z ’ y a and T. O. S h a p o s h n i k o v a, Theory of Multipliers in Spaces of Differ- entiable Functions, Monographs Stud. Math. 23, Pitman, Boston 1985.

[15] A. A. S a m a r s k i˘ı, Theory of Difference Schemes, Nauka, Moscow 1983 (in Russian).

[16] E. S ¨ u l i, B. J o v a n o v i ´ c and L. I v a n o v i ´ c, Finite difference approximations of gener- alized solutions, Math. Comp. 45 (1985), 319–327.

[17] H. T r i e b e l, Interpolation Theory , Function Spaces, Differential Operators, Deutscher Verlag der Wissenschaften, Berlin 1978.

[18] J. W l o k a, Partial Differential Equations, Cambridge Univ. Press, 1987.