Finite element approximations in projection methods for solution of

odeling

MATHEMATICAL MODELING AND COMPUTING, Vol. 5, No. 1, pp. 74–87 (2018)

MMC

omputing

athematical

Finite element approximations in projection methods for solution of some Fredholm integral equation of the first kind Polishchuk O. Pidstryhach Institute for Applied Problems of Mechanics and Mathematics National Academy of Sciences of Ukraine 3-b Naukova str., 79060, Lviv, Ukraine (Received 5 April 2018) Approximation properties of B-splines and Lagrangian finite elements in Hilbert spaces of functions defined on surfaces in three-dimensional space are investigated. The conditions for the convergence of Galerkin and collocation methods for solution of the Fredholm integral equation of the first kind for the simple layer potential that is equivalent to the Dirichlet problem for Laplace equation in R3 are established. The estimation of the error of approximate solution of this problem, obtained by means of the potential theory methods, is determined. Keywords: potential, integral equation, well-posed solvability, B-spline, Lagrange interpolation, Galerkin method, collocation method, convergence. 2000 MSC: 65E05, 35J05 UDC: 519.6

DOI: 10.23939/mmc2018.01.074

1. Introduction Many physical processes (e.g. diffusion, heat flux, electrostatic field, perfect fluid flow, elastic motion of solid bodies, groundwater flow, etc.) are modeled using boundary value problems for Laplace equation [1–3]. The powerful tools for solving such problems are potential theory methods, especially in the case of tired boundary surface or complex shape surface [4–6]. These methods are a convenient for calculating desired solution in small domains [7, 8]. In number of cases, application of potential theory methods requires solving Fredholm integral equation of the first kind. In particular, one of the cases is solving Dirichlet problem in the space of functions with normal derivative jump on crossing boundary surface using simple layer potential [9, 10]. When solving Neumann problem in the space of functions with jump on crossing boundary surface using double layer potential, we also proceed to integral equation of the first kind [11, 12]. The need to solve integral equations of the first kind also arises when the sum of simple and double layer potentials is used to solve the double-sided Dirichlet or Neumann problem [13] or double-sided Dirichlet-Neumann problem [14] in the space of functions that, same as their normal derivatives, have jump on crossing boundary surface. Many systems of integral equations for the simple and double layer potentials that are equivalent to mixed boundary value problems for Laplace equation, also contain integral equations of the first kind [15, 16]. In general, researches of projection methods convergence mainly focus on solving integral equations of the second kind [4,6,17]. Defining well-posed solvability conditions for integral equations of the first kind that are equivalent to boundary value problems for Laplace’s equation in Hilbert spaces [18–20] allows us to use projection methods for numerical solution of such equations, thus avoiding resourceconsuming regularization procedures [21–23]. For detailed review of numerical methods for solving integral equations, please see [2–4, 6]. In [24, 25], convergence conditions are defined for the series of projection methods for solving Fredholm integral equation of the first kind for simple layer potential that is equivalent to three-dimensional Dirichlet problem for Laplace equation while approximating desired potential density with complete systems of orthonormal functions. However, if boundary 74

c 2018 Lviv Polytechnic National University

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Finite element approximations in projection methods for solution . . .

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surface has a complex shape usage of such approximations poses considerable difficulties for practical implementation of numerical methods [7]. In this case, finite elements of different types should be used for approximation of desired potential densities [26, 27]. Derived approximations, among other things, allow us to create effective algorithms for singularities removal in kernels and desired integral equation densities [28]. The purpose of the paper is to define convergence conditions of projection methods for approximate solution of Fredholm integral equations of the first kind by the example of integral equation for the simple layer potential that is equivalent to Dirichlet problem for Laplace equation using approximation of desired potential density with systems of finite elements of different types and orders.

2. Approximations of Hilbert spaces and the basic convergence theorem Consider the operator equation Au = f,

u ∈ U,

(1)

f ∈ F,

where U and F are the Hilbert spaces, A ∈ L(U, F ). To solve equation (1) we apply the projection method QN APN u = QN f. (2) In formula (2) PN and QN are projection operators from U and F onto closed finite dimensional subspaces UN ∈ U and FN ∈ F accordingly. Define operators PN and QN in the following way. Denote by rN the restriction operator from the space U to the finite dimensional subspace VN ⊂ RN and introduce in VN the extension operator pN as an isomorphism from VN onto subspace UN ∈ U . The norm in VN is determined by the relation kuN kVN = kpN uN kU , uN ∈ VN . Then PN = pN rN and we can determine the triple (VN , pN , rN ) as approximation of the space U . Such approximation are called convergent if lim ku − pN rN ukU = 0. N →∞

Denote by sN the restriction operator from the space F to the finite dimensional subspace ΦN ⊂ RN . The extension operator qN from ΦN onto subspace FN = AUN ⊂ F introduce by the formula qN f N = ApN uN ,

f N ∈ ΦN .

The norm in ΦN is determined by the relation kf N kΦN = kqN f N kF , f N ∈ ΦN . Then QN = qN sN and we can determine the triple (ΦN , qN , sN ) as approximation of the space F . Thus, the solution of problem (2) is reduced to solution of the system of linear algebraic equations AN uN = f N ,

AN = sN ApN ,

AN ∈ L(VN , ΦN ),

f N = sN f.

(3)

Operators AN are called stable if there is independent of N constant µ > 0 such that for arbitrary uN ∈ VN is performed inequality µ kuN kVN 6 kAN uN kFN . (4) Let us the pairs of operators (rN , pN ) and (sN , qN ) are selected. Assume as an approximate solution of equation (1) the function pN uN ∈ UN where uN is the solution of problem (3). Then we have the next basic theorem of convergence [29]. Theorem 1. Let us the operator A is an isomorphism from U into F . Then the sequence pN uN converges to solution u of equation (1) if and only if the approximations (VN , pN , rN ) of the space U are convergent and operators AN are stable. In addition, error estimation of the approximate solution is given by the ratio ku − pN uN kU 6 (1 + kAk /µ) ku − pN rN ukU . The choice of triples (VN , pN , rN ) and (ΦN , qN , sN ) defines one or another projection method of solving the equation (1) (Galerkin, smallest squares, smallest mismatch, collocation, etc.). Mathematical Modeling and Computing, Vol. 5, No. 1, pp. 74–87 (2018)

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3. B-splines approximations Let us S = [0, a]×[0, b] ⊂ R2 . Construct in the domain S a rectangular grid Sh with the steps h1 = a/n and h2 = b/k, n, k = 1, 2, . . .. Introduce in the space H m (S), m = 0, 1, 2, . . ., the system of B-splines of m-th degree n−1 k−1  (5) Bij (ξ) i=−m j=−m , n, k > m + 1. N : H m (S) → V N ⊂ RN Denote by UBN the linear shell of system (5). Select restriction operator rB B N N N m and extension operator pB : VB → UB ⊂ H (S) in the form Z (i,j) N (rB v)i,j ≡ vN = Bij (ξ)v(ξ) dSξ , i = −m(1)(n − 1), j = −m(1)(k − 1), (6) S

N N rB v = vB ∈ VBN , N pN B vB =

n−1 X

k−1 X

v ∈ H m (S),

(i,j)

vN Bij (ξ),

(7)

N = (n + m)(k + m).

i=−m j=−m

The next result is in order [29]. N m Lemma 2. Approximations (VBN , pN B , rB ) of the space H (S) are convergent and for arbitrary v ∈ H m (S) are performed the estimates

N σ−t

v − pN kvkH σ (S) , 0 6 t 6 σ 6 m, (8) B rB v H t (S) 6 Ch

where constant C > 0 does not depend on v.

Denote by G a bounded open domain in R3 with boundary Γ. Suppose that exists M open balls Bl ⊂ R 3 ,

Γ⊂

M [

Bl ,

Bl ∩ Γ = Γl 6= 0,

l = 1, M ,

l=1

¯l m times differentiated real vector-function f (l) (x) = such that for each ball Bl there is defined on B (l) (l) (l) (f1 , f2 , f3 ) such that yl = f (l) (x) carries a mutually unambiguous mapping of the ball Bl onto some open bounded set in R3 where Γl is mapped on an open set Sl ⊂ R2 . In addition, the Jacobian (l) (l) (l) ∂(f1 ,f2 ,f3 ) ¯l , l = 1, M . Then the surface Γ is called m-smooth Jl = is positive and continuous if x ∈ B ∂(x1 ,x2 ,x3 )

surface in R3 [30]. M We associate to the partition Γ = ∪M l=1 Γl the partition of one {ψl (x)}l=1 , x ∈ Γ, with the following properties: ψl (x) ∈ C ∞ (Γl ),

supp{ψl } ⊂ Γl ,

0 6 ψl (x) 6 1,

l = 1, M ,

M X

ψl (x) = 1,

l=1

and exists m times continuously differentiated mapping τl : Γl → Sl = [0, al ] × [0, bl ],

τl−1 : Sl → Γl ,

l = 1, M .

Then for arbitrary function u(x) defined on Γ we can put into a mutually unambiguous correspondance the set of defined in R2 functions {vl (ξ)}M l=1 ,

vl (ξ) = u(τl−1 (ξ)),

ξ ∈ Sl ,

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which have the compact support on Sl and ul (x) = ψl (x)u(x),

x ∈ Γ,

u(x) ∈ H m (Γ),

and kukH m (Γ) =

M X

if vl (ξ) ∈ H m (Sl ),

l = 1, M ,

kvl kH m (Sl ) .

l=1

(l)

(l)

Construct in each domain Sl a rectangular grid Slh with the steps h1 = al /nl and h2 = bl /kl , and define in each grid domain Slh a system of functions  (l) nl −1 kl −1 Bij (ξ) i=−m j=−m ,

nl , kl > m + 1, l = 1, M .

−1 h Assign to them the grid Γh = ∪M l=1 τl (Sl ) on the surface Γ and system of functions M [   (l) nl −1 kl −1 ˜k (x) NB = B Bij (τl (x)) i=−m , k=1 j=−m

NB =

l=1

M X (nl + m)(kl + m).

(9)

l=1

Denote Γl1 ,l2 = Γl1 ∩ Γl2 , l1 , l2 = 1, M , and suppose that the grid on surface Γ satisfies condition  (l) supp Bij (τl (x)) 6⊂ Γl,k ,

i = −m(1)(nl − 1),

j = −m(1)(kl − 1),

k 6= l,

k, l = 1, M .

˜i (x)} = ˜i (x)} for i1 6= i2 , the functions of system {B ˜k (x)}NB are linearly Since supp{B 6 supp{B 1 2 k=1 independent. NB Nl Denote by r˜B the restriction operator from H m (Γ) onto finite dimensional space VBNB , and r˜B is m its restriction to H (Γl ), that is,  Nl M NB = r˜B , r˜B l=1

Nl Nl r˜B ul (x) = rB vl (ξ),

Nl = (nl + m)(kl + m),

(10)

Nl where rB is the similar to (6) restriction operator from H m (Sl ) onto finite dimensional space VBNl , l = 1, M , and VBNB = VBN1 × VBN2 × . . . × VBNM . NB B The extension operator p˜N onto UBNB ⊂ H m (Γ) is introduced by the formula B from VB NB X
B ˜i (x), p˜N u (x) = ui B NB B

uNB ∈ VBNB .

(11)

i=1

From lemma 2 follows that

B NB lim u − p˜N r ˜ u

B B

NB →∞

H m (Γ)

=

M X l=1



l Nl lim vl − pN r v

l B B

Nl →∞

H m (Sl )

= 0,

NB B i.e. approximations (VBNB , p˜N ˜B ) of the space H m (Γ) are convergent. Further, from estimates (8) B ,r we obtain

2

N N

u − p˜B B r˜B B u

H t (Γ)

=

M

2 X

N N

vl − pB l rB l vl

H t (Sl )

l=1

6 C 2 h2(σ−t)

M X

kvl k2H σ (Sl ) = C 2 h2(σ−t) kuk2H σ (Γ) ,

l=1

0 6 t 6 σ 6 m + 1,

t 6 m,

Nl m l where pN B is the similar to (7) extension operator from VB onto the spaces H (Sl ), constant C > 0 (l) (l) does not depend on u, and h = max {h1 h2 }. 16l6M

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Polishchuk O. Thus, it is proved

NB B Lemma 3. Approximations (VBNB , p˜N ˜B ) of the space H m (Γ) are convergent and for arbitrary B ,r u ∈ H m (Γ) are valid the estimates

NB NB − p ˜ r ˜ u (12)

u

t 6 C hσ−t kukH σ (Γ) , 0 6 t 6 σ 6 m, B B H (Γ)

in which constant C > 0 does not depend on u.

4. Lagrangian approximations Assign to each element of the grid Sh of domain S Pij = [h1 i, h1 (i + 1)] × [h2 j, h2 (j + 1)],

i = 0(1)(n − 1),

j = 0(1)(k − 1),

a smaller rectangular grid Pijε with the steps ε1 = h1 /m and ε2 = h2 /m. Denote Sh,ε = ∪i,j Pijε and associate with the set of nodes Sh,ε a system of piecewise polynomial functions  mn mk Lpt (ξ) p=0 t=0 ,

satisfying conditions

 supp Lpt (ξ) = P˜pt ,

Lpt (ξls ) = δpl δts ,

P˜pt =

(13)

n[ i,j

o Pij : ξpt ∈ Pij ,

ξls ∈ P˜pt ,

(14)

where δpl is the Kronecker symbol. Functions (13)–(14) form a system of Lagrangian finite elements of m-th degree in H m (S). Denote by ULN1 the linear shell of this system, N1 = (1 + mn)(1 + mk). It is obvious that the restriction of system (13)–(14) onto an arbitrary rectangle Pij of the grid Sh is a basis in the space of polynomials Pm (Pij ) of degree not higher than m, defined on Pij . Then UBN ⊂ ULN1 .

(15)

N1 N1 N1 m N1 1 Choose the extension operator pN L : VL → UL ⊂ H (S), where VL ⊂ R , in the form 1 N1 pN L vL

=

mn X mk X

(i,j)

vN1 Lij (ξ),

(0,0)

(0,1)

(mn,mk)


vLN1 = vN1 , vN1 , . . . , vN1

i=0 j=0

.

(16)

Then, by virtue of the embedding (15), there exists a restriction operator rLN1 : H m (S) → VLN1 such N1 m 1 that approximations (VLN1 , pN L , rL ) of the space H (S) are convergent and valid the estimates

1 N1 ˜ σ−t kvk σ , 0 6 t 6 σ 6 m, r v 6 Ch (17)

v − pN H (S) L L t H (S)

in which constant C˜ > 0 does not depend on v. Thus, it is proved

Lemma 4. There is a restriction operator rLN1 : H m (S) → VLN1 such that approximations N1 m 1 (VLN1 , pN L , rL ) of the space H (S) are convergent and valid the estimates (17). Assume that surface Γ satisfy the conditions of p. 3. Construct in each domain Sl the rectangular (l) (l) grid Slh with the steps h1 = al /nl and h2 = bl /kl and set on each element Pijl of the grid Slh a smaller (l)

(l)

(l)

(l)

grid with the steps ε1 = h1 /m and ε2 = h2 /m, l = 1, M . Define analogously to (13), (14) in each Mathematical Modeling and Computing, Vol. 5, No. 1, pp. 74–87 (2018)

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grid domain Slh,ε = ∪i,j Pijl,ε the system of Lagrangian finite elements  (l) (l) nl m Lij (ξ ) i=0

kl m , j=0

ξ (l) ∈ Sl ,

l = 1, M .

h,ε l,ε −1 −1 Assign to the family Slh,ε the grid Γh,ε = ∪M l=1 τl (Sl ) on the surface Γ, where τl (Pij ) are

the elements of the grid Γh,ε, l = 1, M . Denote by Tl the set of nodes of the grid Slh,ε , l = 1, M , PM T = ∪M l=1 Kl , l=1 Tl . We number all elements of the set T with the cross-cutting index t = 1, K, K = Kl = (1+ nl m)(1+ kl m), and put in correspondence to each node xp of the grid Γh,ε the set of elements Pp∗ element

=

P˜p =

n

Pijl,ε

l=1

n[ i,j

the set of indexes

o Slh,ε : xp ∈ τl−1 (Pijl,ε ) ,

o τl−1 (Pijl,ε ), Pijl,ε ∈ Pp∗ , l = 1, M ,

n o (l) Tp∗ = t ∈ T : τl−1 (Pijl,ε ) = xp , ξt ∈ Plh,ε l = 1, M ,

and function ˜ p (x) = L

⊂

M [

X

Llt (τl (x)),

x ∈ Γl ,

t∈Tp∗

 ˜ p (x) = P˜p , supp L

 (l) nl m (l) Lt (ξ (l) ) ∈ Lij (ξ (l) ) i=0

kl m . j=0

Denote by r˜LNL the restriction operator from H m (Γ) into the finite dimensional space VLNL and by r˜LNl — its restriction to H m (Γl ), i.e. r˜LNL = {˜ rLNl }M l=1 ,

r˜LNl ul (x) = rLKl vl (ξ),

(18)

where rLKl is the restriction operator from H m (Sl ) into the corresponding finite dimensional space VLKl , l = 1, M , and NL is the number of nodes in the grid Γh,ε . NL L ˜ p (x)}NL , U NL ⊂ The extraction operator p˜N into the linear shell ULNL of the system {L p=1 L from VL L H m (Γ), introduce by formula NL X
(i) ˜ L NL p˜N u (x) = uN L i (x), L L

NL L uN L ∈ VL .

(19)

i=1

From Lemma 4 follows that

L NL lim u − p˜N r ˜ u

L L

NL →∞

H m (Γ)

=

M X l=1



l Nl lim vl − pN r v

l L L

H m (Sl )

Nl →∞

= 0,

L i.e. approximations (VLNL , p˜N ˜LNL ) of the space H m (Γ) are convergent. Further from estimate (17) L ,r we obtain

2

N N

u − p˜L L r˜L L u

H t (Γ)

M

2 X

N N =

vl − pL l rL l vl l=1

H t (Sl )

6 C˜ 2 h2(σ−t)

M X

kvl k2H σ (Sl ) = C˜ 2 h2(σ−t) kuk2H σ (Γ) ,

l=1

0 6 t 6 σ 6 m,

Nl m l ˜ where pN L is a similar to (16) extension operator from VL into H (Sl ), constant C > 0 does not (l) (l) depend on u and h = max {h1 h2 }. 16l6M

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Thus, it is proved Lemma 5. There is a restriction operator r˜LNL : H m (Γ) → VLNL such that approximations L (VLNL , p˜N ˜LNL ) of the space H m (Γ) are convergent and valid the estimates L ,r



N N ˜ σ−t kuk σ , 0 6 t 6 σ 6 m, (20)

u − p˜L L r˜L L u t 6 Ch H (Γ) H (Γ)

in which constant C˜ > 0 does not depend on u.

5. Galerkin method ¯ and introduce in G and G′ the Sobolev spaces [30] Let us denote G′ = R3 \G H m (G) = {v ∈ L2 (G) : ∂ α v ∈ L2 (G), |α| 6 m}, 2

W m (G′ ) = {v ∈ D ′ (G′ ) : (1 + r 2 )(|α|−1) ∂ α v ∈ L2 (G′ ), |α| 6 m},
P3 2 1/2 , x = (x , x , x ) ∈ R3 . where m > 0, and r = 1 2 3 i=1 xi Consider the next boundary value problem: to find function n o [ m+1 = v ∈ H m+1 (G) W m+1 (G′ ) : v|Γint = v|Γext , ∆v(x) = 0, x ∈ G, G′ v ∈ HΓ,∆=0

(21)

satisfying condition

v|Γ = f,

f ∈ H m+1/2 (Γ).

(22)

In [9] was proved the next Theorem 6. Problem (21)–(22) has one and only one solution. We will search a solution of the problem (21)–(22) in the form of simple layer potential Z 1 u(y) v(x) = dΓy , x ∈ G, G′ . 4π Γ |x − y| The unknown potential density is determined from the equation Z 1 u(y) (Au)(x) ≡ dΓy = f (x), 4π Γ |x − y|

x ∈ Γ.

(23)

The next result is in order [9]. Theorem 7. Operator A is an isomorphism of H s (Γ) onto H s+1 (Γ), s > −1/2. From the last statement and the Banach theorem follows the validity of inequalities αs kukH s (Γ) 6 kAukH s+1 (Γ) 6 βs kukH s (Γ) ,

(24)

in which constants αs and βs , 0 < αs 6 βs , does not depend on u ∈ H s (Γ). Suppose that for approximation of unknown potential density u ∈ H m (Γ) uses the system of Bsplines of the form (9), and UNB is its linear shell. We choose the operators r˜NB : H m (Γ) → VNB and p˜NB : VNB → UNB in the form (10) and (11) respectively and determine the restriction operator sNB : H m+1 (Γ) → ΦNB in the form sNB = r˜NB . In this case, the system AG NB u NB = f NB ,

AG ˜NB A˜ p NB , NB = r

f NB = rNB f,

implements Galerkin method of solving the equation (23). From Lax-Milgram lemma [31] follows that matrix AG NB is nondegenerate and, accordingly, the definition of operator qNB in the form qNB f NB = Mathematical Modeling and Computing, Vol. 5, No. 1, pp. 74–87 (2018)

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A˜ pNB uNB is correct. Taking into account the left side of inequalities (24), the bijectivity of mapping p˜NB : VNB → UNB , the expressions for the norms in the spaces VNB and ΦNB in the case U = H m (Γ), F = H m+1 (Γ), and equality QNB APNB u = APNB u, we obtain the following inequalities

αm kuNB kVN 6 AG (25) NB u NB Φ NB

B

for arbitrary uNB ∈ VNB and αm does not depend on uNB . Then from the inequalities (24) and (25), Lemma 3 and Theorem 1 we obtain the validity of following statement.

Theorem 8. For arbitrary f ∈ H m+1 (Γ), m = 0, 1, . . . , the approximate solution uB NB of equation (23) obtained by the Galerkin method under approximation of unknown potential density by the system of functions constructed on the basis of B-splines of m-th degree converges to its exact solution, and there is an estimate

u − uB t 6 C(1 + βt /αt ) hσ−t kf k σ+1 , NB H (Γ) H (Γ) ασ

0 6 t 6 σ 6 m,

(26)

where h is the maximum area of the grid element on Γ.

Similarly, from the inequalities (24) and (25), Lemma 5 and Theorem 1, we obtain the validity of following statement. Theorem 9. For arbitrary f ∈ H m+1 (Γ), m = 0, 1, . . . , the approximate solution uL NL of equation (23) obtained by the Galerkin method under approximation of unknown potential density by the system of functions constructed on the basis of Lagrangian finite elements of m-th degree converges to its exact solution, and there is an estimate ˜ + βt /αt )

C(1

u − uL

hσ−t kf kH σ+1 (Γ) , NL H t (Γ) 6 ασ

0 6 t 6 σ 6 m,

(27)

where h is the maximum area of the grid element on Γ.

6. Collocation method To simplify the presentation, we assume that for approximation of unknown potential density u ∈ H m (Γ), m > 0, of equation (23) a system of linearly independent functions {ϕi }∞ i=1 is chosen, UN is a m (Γ) → V , p : V → U are the similar to described in p. linear shell of the system {ϕi }N , r : H N N N N N i=1 2 restriction and extraction operators. Denote by XN the set of pairwise different points belonging to the surface Γ XN = {xj }N xj ∈ Γ, j = 1, N , j=1 , and introduce in H m+1 (Γ) restriction operator sN : H m+1 (Γ) → ΦN by formula (28)

(sN f )j = f (˜ yj ) in which  y˜j ∈ y˜ ∈ δ(yj ) : |f (˜ y)| = min |f (y)| , yj ∈ XN , y∈δ(yj )

δ(yj ) = {y ∈ Γ : |y − yj | < δ},

(29)

in particular ρ(y ∗ , δ(yj )) > 0 for arbitrary y ∗ ∈ XN , y ∗ 6= yj , j = 1, N . If f ∈ C(Γ), then operator sN can be defined as usual (sN f )j = f (yj ),

yj ∈ XN ,

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(30)

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Polishchuk O.

i.e. y˜j = yj , j = 1, N . It is easy to see that, with this choice of operator sN , a system of linear algebraic equations AcN uN = sN f, AcN = sN ApN , uN ∈ VN , (31) implements the collocation method of solving the equation (23). The set XN is called a set of collocation points. Denote YN = {˜ yj }N j=1 and consider the system of functions rj (x) =

1 , |x − y˜j |

y˜j ∈ YN ,

j = 1, N .

From the choice of the set XN and conditions (29) follow that the functions of system {rj (x)}N j=1 are linearly independent [32]. Define in L∞ (Γ) the family of linear continuous functionals Z ϕ(x)rj (x)dΓx , ϕ ∈ L∞ (Γ), j = 1, N . lj (ϕ) = Γ

Denote by Ker(lj ) the zero subspace of functional lj in L∞ (Γ) Ker(lj ) = {ϕ ∈ L∞ (Γ) : lj (ϕ) = 0} c and suppose that KN = ∩N j=1 Ker(lj ). The degeneracy of matrix AN is equivalent to the linear N or β dependence of its that is, the existence of such sets αN = {αi }N N = i=1 ∈ R PNrows2 or columns, PN N N 2 {βj }j=1 ∈ R , i=1 αi > 0, j=1 βj > 0, that

Z X N Γ

or Z

Γ

i=1

 αi ϕi (x) rj (x)dΓx = 0,

N X  ϕi (x) βi rj (x) dΓx = 0,

j = 1, N ,

(32)

i = 1, N .

(33)

j=1

Implementation of equations (32), (33) is only possible if KN ∩ UN 6= 0. From this follows sufficient conditions for the invertibility of matrix AcN , which we formulate in the next statement. Lemma 10. Let us the system of linearly independent functions {ϕi }N i=1 is chosen for the approximate solution of equality (23) and determined the set of collocation points XN (and, consequently, the set KN is defined). Then, if KN ∩ UN = 0, (34) then the matrix AcN of the system of collocation equations (31) is non-degenerate for arbitrary N . A similar result is obtained if the restriction operator sN is chosen in the form Z 1 (sN f )j = f (y)dΓy mes δ(yj ) δ(yj ) and rj (x) =

1 mes δ(yj )

Z

δ(yj )

dΓy , |x − y|

(35)

j = 1, N .

It is obvious that under conditions of Lemma 10 the operator AcN , where sN is defined according to (27)–(28) or (33), or in the case of f ∈ C(Γ) according to (30), is stable in sense (4). Mathematical Modeling and Computing, Vol. 5, No. 1, pp. 74–87 (2018)

Finite element approximations in projection methods for solution . . .

83

Consider a discrete analog of condition (34). Let us the quadrature formula Z

ϕ(x)ri (x)dΓx ≈ Γ

N X

Aj ϕ(xj )ri (xj ),

xj ∈ Γ,

xj 6= xi ,

if j 6= i,

(36)

j=1

is used to calculate the integrals Z

ϕ(x)ri (x)dΓx ,

ϕ(x) ∈ UN ,

i = 1, N ,

Γ

which is exact for integrals

Z

ϕ(x)ψ(x)dΓx ,

ϕ(x), ψ(x) ∈ UN .

Γ

Consider the system of functions ψi (x) =

N X

(i)

(37)

αk ϕk (x),

k=1

(i)

the coefficients αk , k, i = 1, N , of which we define from N systems of linear algebraic equations N X

(i)

αk ϕk (xj ) = ri (xj ),

(38)

i, j = 1, N .

k=1

Define the conditions under which the functions ψi (x), i = 1, N , are linearly independent. From (37) we obtain that N N X N  X X (i) ci ψi (x) = ci αk ϕk (x) = 0 i=1

k=1

i=1

if and only if

N X

(i)

ci αk = 0,

(39)

k = 1, N .

i=1

Let us the set of colocation points XN = {yj }N j=1 ⊂ Γ is chosen in such a way that 0 < |xi − yi | < ε,

d < |xi − yj | ,

i, j = 1, N ,

j 6= i,

0