Boundary Layer Approximate Approximations and Cubature of. Potentials in Domains

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1 Boundary Layer Aroximate Aroximations and Cubature of Potentials in Domains Tjavdar Ivanov 1, Vladimir Maz'ya 1, Gunther Schmidt Mathematics Subject Classication. 65D32, 65D15, 65N38. Keywords. Volume otentials, semi-analytic cubature formulae, aroximate aroximations, aroximate multi-resolution 1 Link ing University, Deartment of Mathematics, Link ing, Sweden 2 Weierstra Institute of Alied Analysis and Stochastics, Mohrenstr. 39, Berlin, Germany

2 Abstract In this article we resent a new aroach to the comutation of volume otentials over bounded domains, which is based on the quasi-interolation of the density by smooth, almost locally suorted basis functions for which the corresonding volume otentials are known. The quasi-interolant is a linear combination of the basis function with shifted and scaled arguments and with coecients exlicitly given by the oint values of the density. Thus, the aroach results in semi-analytic cubature formulae for volume otentials, which rove to be high order aroximations of the integrals. It is based on multi-resolution schemes for accurate aroximations u to the boundary by alying aroximate renement equations of the basis functions and iterative aroximations on ner grids. We obtain asymtotic error estimates for the quasi-interolation and corresonding cubature formulae and rovide some numerical examles. 1 Introduction In recent years the boundary element method (BEM) has been used extensively to solve boundary value roblems for artial dierential equations with constant coecients which occur in mechanics, electromagnetics and other elds of mathematical hysics. Let, for examle, L be a artial dierential oerator with known fundamental solution E and consider the equation Lf = u in ; comlemented with some boundary condition. The simlest way to aly BEM for solving this roblem is to reresent the solution u as the sum where P u is the volume otential dened by f (x) = f 0 (x) + P u(x); P u(x) = and f 0 satises the homogeneous equation u(y) E(x; y) dy Lf 0 = 0 in ; with boundary conditions adjusted such that the total solution f satises the boundary condition of the original roblem. The remainder f 0 is obtained by solving the corresonding boundary integral equations, involving now the new boundary data for f 0. In order to nd these data suciently recise, one must be able to comute the volume otential (and, very often, its derivatives) very accurately. Even more imortant alications of the volume otentials aear when one combines the BEM with iteration rocedures for linear roblems with variable coecients or for non-linear roblems. Essentially, the aroach for solving boundary roblems for nonlinear equations lums the nonlinearity into body forces and then solves the roblem iteratively. This introduces domain integral contributions or volume otentials to the corresonding boundary integral equations. The construction of closedform articular solutions is ossible only for some secial inhomogeneities. Thus the articular solutions must be aroximated. However, the direct comutation of the otential P u leads to evaluation of a tyically singular integral, which is both numerically exensive and inaccurate if conventional cubature formulae are used. Therefore, starting with the aer of Nardini/Brebbia [11] it has become increasingly oular to reresent the densities u of the volume otentials in terms of simler functions for which articular solutions are known (see, e.g., [12] and the references therein). Thus, the singularity 1

3 is removed and one obtains an aroximation for the otential P u. Tyically, in the case of volume otentials for isotroic dierential oerators the most widely used class of aroximating functions are secial radial basis functions and the aroximant interolates u at certain nodes. Thus, the aroximation of the volume otentials turns to the aroximation-theoretic roblem of the construction of aroximants to given functions u by secial basis functions and the corresonding error estimates. However, the construction of the interolant may be rather involved; see for examle [13], where the case of Gaussian radial basis functions is studied. Let us note that another oular method of transforming domain integrals to boundary integrals relies also on the interolation of the density by linear combinations of certain radial functions (cf. [14] and the references therein). The aim of this article is to resent a new aroach to the comutation of volume otentials over bounded domains, which is based on the quasi-interolation of the density u by smooth, almost locally suorted basis functions for which articular solutions are known. Since the quasi-interolant is a linear combination of the basis function with shifted and scaled arguments and with coecients exlicitly given by the oint values of u, we get semi-analytic cubature formulae for volume otentials, which rove to be high order aroximations of the integrals. Our aroach is based on an aroximation method roosed by the second author in [2] which use generating functions forming only an aroximate artition of unity. Given a function u, dened and somewhat regular on R n, the aroximate aroximation oerator M h;d is dened as the quasi-interolant M x? hm h;d u(x) = D?n=2 u(hm) ; (1) m2 h D n where h is the ste size, D is a ositive arameter and satises some decay and moment conditions. In [7] it is shown that for any integer N it is easy to nd a generating function such that at any oint x, ju(x)? M h;d u(x)j c u; ((h D) N + " 0 (; D)): (2) A roer choice of the arameter D allows to make the saturation error " 0 (; D) as small as necessary, e.g., less than the machine recision. Formula (1) is the basis of the semi-analytic cubature formulae for the aroximation of various integral and seudo-dierential oerators. It suces to nd the action of the corresonding oerator P on the generating function of the quasi-interolant M h;d :? P u(x) PM h;d u(x) = u(hm) P hm (x): m2 h D n Some imortant examles are analyzed in [3] and [9], including in articular, the harmonic, elastic, hydrodynamic, diraction and other otentials. Such cubature formulae erform well and satisfy estimates similar to (2) only if the aroximated function u is dened and somewhat regular on the whole sace or can be continued outside the domain of denition with reserved regularity. For functions dened only in bounded domains, we develo multi-resolution schemes for accurate aroximation u to the boundary by alying iteratively aroximate aroximations on ner grids. The mesh renement is achieved using the analytical factorization of the oerator M h;d M h;d = M h;d f Mh;D ; 0 < < 1; where Mh;D f is another quasi-interolant of the form (1). These iteration schemes not only retain, but increase the accuracy of aroximation at oints lying nearer to the boundary. The 2

4 rocedure results in the aroximation formula: B M u(x) = M m2q k c k;m x? hk m ; h k = k h; 0 < < 1; (3) h k D which is accurate on the whole of excet on a boundary layer of width decreasing exonentially with M, the number of stes made in the iteration scheme from which B M u originates. The sets Q k n are such that the mesh oints h k m h k Q k lie in boundary layers of width exonentially decreasing with k and the coecients c k;m are given by u(hm) ; k = 0, c k;m = u(h k m)? Mhk?1 f ;Du(h k m); k 1. Of course, reresentation (3) can be used not only near the boundary, but also locally at other regions where higher accuracy is needed. Clearly, the multi-resolution oerator B M retains also the quasi-interolation character of the M h;d which grants an easy comutation of the coecients c k;m. Moreover, in similarity to wavelet bases and other techniques built uon orthogonal basis functions, the introduction of new higher-frequency terms in (3) does not require re-comutation of the coecients c k;m. The good accuracy rovided by (3) for functions on domains can be used to successfully aroximate a large class of integral oerators. Given an integral oerator P with density u dened on a domain, one obtains a cubature formulae for its calculation by setting P u(x) P h u(x = PB M u(x) = M m2q k c k;m P? hk m h k D (x): (4) In the cases of many otentials from mathematical hysics, including the harmonic, elastic, hydrodynamic and diraction otentials, integration can be erformed analytically (cf. [2],[3] and [9]). Since the density is reroduced accurately near the boundary if M is large enough, the cubature formula (4) admits error estimates similar to (2). More recisely, in section 7 we rove the following theorem: Let u 2 W N () with N > n= and suose that P mas L (R n ) into the Bessel otential sace H m (Rn )). For any " > 0 there exists D > 0 such that kp u? P h uk H m (R n ) c 1 (Dh) N kr N uk L() + c 2 h 1= M kuk L 1() + "kuk W () : If additionally P 2 L(H?m (R n ); L (R n )) then kp u? P h uk L(R n ) (c 1 (Dh) N + c 2 h 1=+r M )kuk W N () + " hm kuk W () ; where 0 < r < m=n; r (? 1)=. We note that a signicant reduction of the comutational cost can be achieved through anisotroic mesh renement in direction normal to the boundary which will be studied in a forthcoming aer. The outline of the aer is as follows. In section 2 we briey review some results of quasiinterolation on uniform meshes with smooth and raidly decaying basis functions. Section 3 is devoted to aroximate renement equations for those functions resulting in the factorization and multiresolution decomosition of the corresonding quasi-interolation oerators. In section 5 we dene the boundary layer aroximants (3), the aroximation errors in integral and weak norms will be studied in section 6. In the nal section obtain error estimates for cubature formulae and give examles of semi-analytic cubature for otentials. 3

5 2 Aroximate aroximations on domains In this section we derive some estimates for the aroximation roerties of the quasi-interolant (1) for the case when u is dened on a domain with comact closure and Lischitz boundary and is continued by zero outside. 2.1 Notation We will suose that the generating function belongs to the Schwartz class S(R n ) and that for some N > 0, the following moment conditions are satised: R n (x) dx = 1; For a given multi-index, we introduce the numbers R n x (x) dx = 0; 0 < jj < N: (5) " = " (; D) := D? m? m?n=2? x m2 D D R (x) dx n n = (; D) := D? m? m?n=2 L : m2 D D 1(R n n ) From Poisson's summation formula one obtains immediately ; L 1(R n ) " jf x7 (x (x))( D)j; 0 jj < N; (7) m6=0 (6) where F is the Fourier transform Fu() = We dene also the monotone function g ;D (t) = D?n=2 su x2r n jx?mj>t R n u(x) e?2ihx;i dx; x? m x? m ; D D and note that since 2 S(R n ), g ;D (t) decays far out faster than any negative ower of t. Of course, if is continuous, then evidently (; D) = g ;D (0). For r > 0, let B(x; r) be the closed ball centered at x of radius r. Finally, if is a bounded domain in R n, we dene the subdomain r and the equidistant r-neighbourhood + r of by r = fx : B(x; r) g; + r = fx : dist(x; ) < rg: (8) 2.2 Accuracy of aroximate aroximation in domains In [7] it is shown that if u if N-times dierentiable and the generating function satises the moment conditions (5), the quasi-interolant M h;d u aroximates u at a rate O(" 0 + (h D) N ). The quantity " 0, dened by (6), is referred to as the saturation error. Since 2 S(R n ), by (7) the values of ", 0 jj < N, can be made as small as needed if D is chosen large enough. Note also that the bound (7) for the saturation error is indeendent of the ste size h. Clearly, the boundedness of = su u does not imly boundedness of the suort of M h;d u. Nevertheless, as is in the Schwartz class, M h;d u(x) decays fast with the dist(x; su u): 4

6 Lemma 1 Suose that u is a bounded function and = su u. Then jm h;d u(x)j g 0;D (h?1 dist(x; )) kuk 1 : Since g 0;D 2 S, one can nd a number N s > 0, such that g ;D (N s ) " (; D); 0 jj < N: (9) In other words, Lemma 1 assures that if N s is a ositive number such that (9) holds, the essential suort of M h;d u is the N s h-neighbourhood + N sh of, in the sense that jm h;d u(x)j " 0 kuk 1 whenever x 2 R n n + N sh : (10) Note also that since jm h;d u(x)j decays far out more raidly than any ower of dist(x; ), the quasi-interolant on R n n + N sh is of the order of the saturation error " 0 even in integral norms. Remark 1 Another consequence of (10) is that the comutation of M h;d u requires to take only the (2N s + 1) n summands in (1) for which jx=h? mj N s, since the error introduced by neglecting the other terms is smaller than the saturation. In order to show the aroximation roerties of M h;d for functions dened on domains and continued by zero outside, we begin by investigation of the behaviour of the quasi-interolant under truncation of the summation. Theorem 1 Suose that 2 S(R n ) satises the moment conditions (5) and let N s > 0 be such that (9) holds. If u is N-times continuously dierentiable in the ball B(x; N s h), then where M (B) h;d j(i? M (B) )u(x)j h;d 2 jj=0 (h D) jj " (; D) + (h D) N denotes the truncated quasi-interolant M (B) h;d = D?n=2 hm2b(x;n sh) j@ u(x)j jj=n u(hm) (; D) x? hm h D : k@ u(x)k C(B(x;N sh)); Proof. Set for brevity B = B(x; N s h) and m = x?hm h. The Taylor exansion of u(hm) around D the oint x yields M (B) h;du(x) = D?n=2 jj=0 (? u(x) hm2b +D?n=2 m( m ) jj=n (? Dh) N u(y m ) m( m ); where y m lies on the segment connecting the oints hm and x. If we slit the summation over n and n n B, we obtain for the rst inner sum in the right-hand side whereas D?n=2 hm2b D?n=2 m ( m ) " (; D) + g ;D (N s ); 0 jj < N; hm2b m ( m ) (; D); jj = N: Choosing N s as in the statement of the theorem comletes the roof. 5

7 2.3 Examles As an examle, consider the generating functions based on the radial Gaussian 2M (x) =?n=2 L (n=2) M?1 (jxj2 ) e?jxj2 ; M = 1; 2; : : : ; (11) where L () (t) denote the generalized Laguerre olynomials dened by k L () k (t) = t? e t k Since the corresonding Fourier transforms are (cf. [3]) d k dt k t k+ e?t ; >?1: (12) F 2M () = P M?1 ( 2 jj 2 ) e?2 jj 2 ; P m (t) = m t k k ; (13) these functions satisfy the moment conditions (5) with N = 2M and hence, by Theorem 1, give rise to quasi-interolation formulae (1) of aroximate order of convergence O((h D) 2M ). Furthermore, using (7), the saturation error " 0 can be estimated by " 0 ( 2M ; D) m2 n nf0g P M?1 (jmj 2 r 2 ) e?jmj2 r 2 = O(r 2M +n?4 e?r2 ) ; r = D: Note that since e?2 5:17 10?5, already D = 4 ensures a saturation error in the range 10?15 10?12 for 1 M 3 and sace dimensions n = 2 and L -estimates We recall that our main goal is to use quasi-interolants for aroximation of densities of integral oerators, many of which are known to be continuous maings from L to the Sobolev sace W l, l > 0. Thus, in order to derive estimates for the aroximation of the integral oerators, it will be necessary to have L -estimates for the aroximation of the corresonding densities. By Theorem 1, only the values of the function in a small neighbourhood of the oint x aect the aroximation results, and hence, modulo the doubled saturation error, the truncated oerator M (B) h;d ossess identical aroximation roerties as it's untruncated counterart M h;d. This means also that functions belonging to C N () are aroximated at the rate O(" 0 + (h D) N ) in the subdomain Nsh (cf. (8)), i.e., at all internal oints which lie on a distance larger that N s h from the Generally, if u belongs to the Sobolev sace W N (), the following L -estimate holds (cf. [9]): Theorem 2 Suose that 2 S(R n ) satises the moment conditions (5) and that N s is as in (9). Further, let be a domain in R n with comact closure and Lischitz boundary and u 2 W N () with N > n=, 1 1. Then, k(i? M (B) )uk h;d L (Nsh ) 2 jj=0 where Nsh is the sub-domain dened in (8). (h D) jj " (; D) k@ uk L(Nsh ) + (h D) N jj=n (; D) k@ uk L() ; 6

8 We note that under the requirements in Theorem 2 u is continuous on and thus the quasi- (); then the result of Theorem 2 can be interolant M h;d u is well-dened. Clearly, if u 2 W N extended to the whole sace R n instead of Nsh. In order to estimate the accuracy of aroximation of integral oerators, besides the bounds inside the domain given by Theorem 2, one needs estimates for the discreancy (I? M h;d )u on the whole sace. Theorem 3 Suose that the conditions of Theorem 2 hold. Then for any t > 0, k(i? M h;d )uk L( + th n Nsh) c h 1= (1 + 0 (; D))(N s + t) 1= kuk L 1() km h;d uk L(R n n + th ) hn= kg 0;D (j j + t)k L(R n ) kuk L 1(); where c is a constant deending only on the domain. The roof is based on the following lemma: Lemma 2 Suose that is a domain in R n with comact closure and Lischitz boundary. For h > 0, denote by Sh the characteristic function of the boundary layer fx 2 : < hg. Then, the following estimates hold: k Sh uk L() ch (t?1)=t kuk Lt() ; 1 ; t < 1 ; (14) k Sh uk L() ch r kuk W s (); 1 < 1; 0 < r < s=n; r 1= ; (15) k Sh uk (W s ()) 0 chr kuk L=(?1) (); 1 < 1; with constants deending only on. 0 < r < s=n; r 1= ; (16) Here (W s ())0 denotes the dual sace of W s () with resect to the L 2 scalar roduct. Proof. The rst inequality follows from j Sh uj dx n o 1=t n juj t dx S h o (t?1)=t dx = (meas Sh ) (t?1)=t kuk : Lt() To rove (15), we note rst that since u 2 W s (), s > n=, then u 2 C(). Hence j Sh uj dx max ju(x)j meas S h c meas S h kuk x2s W ; s h () so that k Sh uk L() ch 1= kuk W s (): Since evidently k Sh uk L() kuk L(), we obtain by interolation k Sh uk L() ch = kuk W s () ; 0 1; s > n=: Setting r = =n yields (15). Finally, since the oerator Sh is symmetric, there holds which roves (16) and the lemma. k Sh k L=(?1) ()7(W s ())0 = k S h k W s ()7L () ; 7

9 Proof of Theorem 3. Let for brevity S denote the boundary stri S = + th n N sh. Then by the roof of (14) k(i? M h;d )uk L(S) k(i? M h;d )uk L 1() (meas S)1= (1 + 0 (; D))(measS) 1= kuk L 1() : To obtain the second estimate in the formulation of the theorem, we note that km h;d u(x)k R n n + th R n n + th D?n=2 h n kuk L1() hm2 dist(;h?1 )>t u(hm) ( x=h?m D?n=2 D ) m2h?1 dx (?m D ) d : By the construction of the set + th we have and hence j(x=h? m)j t + inf jh?1 (x? y)j; x 2 R n n + y2 + th ; hm 2 ; th j? mj t + dist(; h?1 + th ); m 2 h?1 ; dist(; h?1 ) > t: Lemma 1 rovides the estimate and therefore The roof is comleted. D?n=2 m2h?1? m g0;d (t + dist(; h?1 + D th )); km h;d u(x)k R n n + th h n kuk L1() jjt fg 0;D (t + jj)g dx: Combined, Theorems 2 and 3 give L -estimates for the aroximation error on the whole of R n. By Theorem 2, the quasi-interolant M h;d u is a good aroximation of u at internal oints, lying at a distance larger than N s h from the boundary. The error is then of order O(" 0 +(h D) N ) and can be controlled eectively by a roer choice of the ste-size h and the arameter D. The second estimate from Theorem 3 assesses the error accumulated outside of the th-neighbourhood of su u. Since g 0;D is in the Schwartz class, kg 0;D (j j + t)k L(R n ) 0 more raidly then any ower of t, so this term can be made of the same order of magnitude as, e.g., the saturation error "(; D), by choosing t larger. Thus, the main contribution to the overall error comes from the boundary stri + th n N sh, where, by the rst estimate in Theorem 3, the error is of order O(h 1= ) if u does not vanish Clearly, it will be numerically very exensive to make this term small by choosing h smaller, esecially in higher sace dimensions. In what follows, we concentrate our eorts to build local mesh renements near oints of where the quasi-interolant M h;d u does not aroximate with satisfactory accuracy, in articular, near the boundary of the domain. 3 Aroximate renement equations In this section we concentrate on the construction and roerties of the cornerstone of aroximate multi-resolution techniques, namely, the renement equations of the tye (x) = ~() (x=? ) + small remainder term: (17) 2 n 8

10 3.1 Construction It was roven in [10], that an aroximate renement equation of tye (17) is true for 2 S(R n ) if the Fourier transform F 6= 0 and that ~ can be determined from F ~() = F() F() : (18) More recisely, the following theorem holds: Theorem 4 Suose that (18) holds for some ositive < 1 and that, ~ satisfy Then 2 S(R n ); ~ 2 S(R n ); F > 0: x m D = D?n=2 ~ D m2 n where the remainder R ;;D 2 S(R n ) is given by R ;;D (x) = m2 n nf0g x? m D + R ;;D (x) ; (19) e 2ihx;mi= R n F ~()F( + Dm)) e 2ih;xi= D d : (20) Moreover, for any " > 0 there exists D = D(; ) > 0 such that jr ;;D (x)j < ". In the sequel, the function ~ dened by (18) will be referred to as the adjoint function corresonding to. For examle, the generating functions (11) based on the Gaussian satisfy the requirements of Theorem 4, since by (13) they ossess ositive Fourier transforms. The analytic exression of adjoint functions ~ 2, ~ 4 and ~ 6 in the case of one sace dimension are: where = 1? 2, ~ 2 (t) = e?t2 = ; ~ 4 (t) = 1 2 h ~ 2 (t)? W( ~ 6 (t) = 1 4 n ~ 2 (t)? 2 <h 1 + i i W ( W(z; t) = e?t2 and w(z) is the scaled comlementary error function 2 (1+i) ; ; t ) f w (i(z + t)) + w (i(z? t))g; w(z) = e?z2 erfc(?iz) = e?z2 1? 2 0 t i ) ; io?iz e?t2 dt : Of course, these formulae allow to obtain analytical reresentations for the adjoint functions in any sace dimension when (x) is a roduct of one dimensional functions: (x) = 2M (x 1 ) : : : 2M (x n ): Note that for comutations we do not need the analytic exression of the functions ~. In the following section we will show that for our uroses it suces to recomute the values of ~ just in several oints, which can be done with some numerical method for inverse Fourier transform. ; (21) 9

11 3.2 Proerties of the adjoint function ~ Suose that in addition to the requirements of Theorem 4, is subject also to the moment conditions (5). Since these conditions can be rewritten by Fourier transformation as F(0) = 1; F x (x (x))(0) = 0; 0 < jj < N; relation (18) guarantees that they are satised by ~ as well. Then, by Theorem 1, ~ gives rise to a quasi-interolant f Mh;D featuring the same rate of aroximate convergence as M h;d, which is generated by. Hence, in similarity to (9) one can introduce the ositive integer e Ns = e Ns (D), so that ~g ;D ( e Ns ) ~" ; 0 jj < N; where ~g ;D (t) = D?n=2 su x2r n jx?mj>t x? m x? m ~ : D D and ~" = " (~; D) are dened as in (6). The same estimate as (7) holds also in this case, and consequently, the saturation error ~" 0 0 as D 1 more raidly than any ower of D. For examle, for the adjoint functions ~ 2M to 2M (cf. (11)), one obtains by (13) and (18) that ~" 0 m2 n nf0g P M?1 (jmj 2 r 2 ) P M?1 (jmj 2 r 2 ) e?(1?2 )jmj 2 r 2 = O(r 2M +n?4 e?(1?2 )r 2 ) ; r = D: 3.3 Quasi-interolants based on the remainder term In the following we meet quasi-interolants generated by the remainder term R ;;D (x) of the form R h;d u(x) = D?n=2 u(mh) R ;;D (x=h? m) (22) m2 n By Theorem 4 these quasi-interolants are roerly dened, since we have raid decay in x. For instance, when is the Gaussian, the corresonding function ~ 2 is by (21) also a scaled Gaussian: ~ 2 ( x ) = 2 ( D x x D(1? 2 ) ) = 2( ); D e = D(1? 2 ): ed The aroximate renement equation for this case takes the form e?jxj2 =D = ( e D)?n=2 and the remainder term R 2 ;;D(x) is given by m2 n e?jmj2=e D e?jx=?mj2=d + R2 ;;D(x) R 2 ;;D(x) = 2 x D [(I? f M;D )1 (x )] = 2 x D [(I? M ; e D )1 (x )]; where M ; e D 1 is the quasi-interolant M ; e D u for u(x) 1 and x = (1? 2 )x. Thus by Theorem 1 jr 2 ;;D(x)j ~" 0 and the quasi-interolant R h;d u satises the uniform bound jr h;d u(x)j " 0 (~ 2 ; D)kuk L 1 = " 0( 2 ; D(1? 2 ))kuk L 1: In following lemma, which we state without roof, we establish the remainder terms in the renement equations R 2M ;;D for M > 1 exhibit similar behaviour as R 2 ;;D: 10

12 Lemma 3 Suose that 2M is dened by (11) and 0 < < 1 is a xed arameter. Then there exist ositive univariate olynomials Q 1 and Q 2 of degree M? 1 such that for any suciently large D jr 2M ;;D(x)j Q 1 (jxj 2 =D) e?jxj2 =D m2 n nf0g Q 2 (Djmj 2 ) e?2 D(1? 2 )jmj 2 : As a consequence we obtain that the generating function of the quasi-interolant R 2M ;;D has amlitude of the same order as the saturation error, and the rate of decay of 2M : Corollary 1 Suose the conditions of Lemma 3 are met. Then, there exists a constant C R, such that jr 2M ;;D(x)j C R " 0 (~ 2M ; D) j(x)j: and, hence, the quasi-interolant R h;d u dened by (22) admits the uniform estimate jr h;d u(x)j C R ~ 0 ~" 0 : 4 Factorization and multiresolution decomosition of quasi-interolation oerators In this section we use the aroximate renement equation (20) to factorize the quasi-interolation oerator M h;d. Such a factorization allows to obtain an aroximate multi-resolution decomosition of the oerator on the highest resolution M M h;d from which one obtains the desired boundary layer aroximate aroximation (3) after an aroriate truncation of the summation. In what follows, we suose that and ~ satisfy the requirements of Theorem 4 and the aroximate renement equation (19), and that M h;d, ~M h;d are the corresonding quasi-interolants. Given a sequence of ste sizes fh k g M, where we will use the notation h k = k h; 0 < h; < 1;?1 2 ; A k = M k h;d; ~A k = f M k h;d; R k = R k h;d; k = 0; 1; 2 : : :; (23) where R k is the quasi-interolant (22) based on the remainder term in (19). Theorem 5 (Aroximate oerator factorization) Suose that and ~ are generating functions satisfying the requirements of Theorem 4 and let A k, ~A k and R k be dened by (23). Then A k = A k+1 ~A k + R k ; k = 0; 1; 2 : : :: (24) Proof. Set for brevity D (x) := D?n=2 (x= D) and let ~ D be the corresonding adjoint function, dened by (18). Then, using the aroximate renement equation (19) one obtains A k u(x) = u(mh k ) D (x=h k? m) m2 n = ;m2 n u(mh k ) ~ D (m) D [x=(h k )? m=? ] + D?n=2 2 n u(mh k ) R ;;D (x=h k? x): 11

13 Since?1 is an integer, k = +?1 m 2 n. Thus, after re-indexing and taking into account that h k+1 = h k one arrives at the reresentation A k u(x) = Finally, as k = h k+1k h k, we recognize k;m2 n u(mh k ) ~ D (k? m) D [x=h k+1? k] + R k u(x): A k u(x) = which is recisely the claimed identity. k2 n ~A k u(h k+1 k) (x=h k+1? k) Theorem 6 (Aroximate multiresolution decomosition) Suose that the aroximate oerator factorization identity (24) holds, and let f k g M be a set of linear oerators. Then A M M = A M A k ( k? A ~ k?1 k?1 )? M?1 Proof. By the aroximate factorization identity (24) one has and the theorem follows by induction. A k k = A k?1 k?1 + A k k? A k?1 k?1 = A k?1 k?1 + A k k? A k ~A k?1 k?1? R k?1 k?1 = A k?1 k?1 + A k ( k? ~ A k?1 k?1 )? R k?1 k?1 ; R k k : (25) Corollary 2 Under the conditions of Theorem 6, suose that k = I, k = 1; : : :; M. Then A M = A 0 + M A k (I? A ~ k?1 )? M?1 Adding identity I to both sides in the above corollary and moving A M to the right yields Corollary 3 (Multi-resolution decomosition of identity oerator) I = A 0 + M R k : A k (I? A ~ k?1 ) + (I? A M )? M?1 5 Boundary layer aroximate aroximations In this section we use the multi-resolution decomosition (25) to construct a boundary layer aroximate aroximation oerator B M. If is a bounded domain and u a suciently regular function with su u =, then B M u is an accurate aroximation of u on the whole of excet on a thin boundary layer of width decreasing with M. Moreover, the oerator B M can be dened in such a way that the essential suort of B M u does not extend outside. Throughout this section we suose that ; ~ satisfy the requirements of Theorems 1 and 4, and that M h;d and f Mh;D are the quasi-interolants generated by and ~ resectively. Finally, we suose that there exists a constant C R, indeendent of the ste size h such that g 0;R;h;D (t) = su x2r n jx?mj>t R k : jr ;h;d (x? m)j C R ~" 0 g 0;D (t); k = 0; 1; 2 : : :: 12

14 For instance, if is one of the functions dened in (11), such a condition follows from Corollary 1. We begin by sketching a straightforward way to construct a boundary layer aroximate aroximation oerator B M of tye (3). Corollary 2 shows that modulo the saturation terms P M?1 R k, the multi-resolution oerator A 0 + MP A k (I? A k?1 ) erforms as the quasi-interolant A M on the nest resolution. Thus, if u is smooth in, the multi-resolution aroximation M A k ~u k = A M u + M?1 R k u; ~u k = u; k = 0, (I? A ~ k?1 )u; k 1, achieves high accuracy inside and leaves only a thin boundary layer of width N s h M = M N s h 0 where the error is large. Of course, the use of such a scheme is meaningless since one could have alied A M at once. Also, its numerical cost of order O(h?n M ) becomes unaccetable if we wish to make the boundary layer very small by making a large number of iterations M. On the other hand, if u 2 C N (), Theorem 1 guarantees that j~u k (x)j = O(~" 0 + (h k D) N )); x 2 n e Nsh k?1 ; whereas for oints outside the domain, one has j~u k (x)j = j A ~ k?1 u(x)j ~g 0;D (h?1 dist(x; )); k?1 so j~u k (x)j ~" 0 if dist(x; ) > Ns e h k?1. Hence, if we can truncate those terms in A k ~u k which contain ~u k (h k m) with argument h k m such that (h k m) > N ~ s h k?1 and neglect the saturation terms, then (26) reduces to the boundary layer aroximate aroximation (3) with u(h0 m); k = 0, c k;m = ~u k (h k m); k 1, and Q k = fm 2 n : mh0 2 g; k = 0, fm 2 n : (x) e Ns h k?1 g; k 1. Such a truncation retains the ability of the initial scheme to diminish the remainder boundary layer exonentially with M, while the comutational cost is reduced to O(h n?1 M ). The rice aid is the introduction of an error of order O((h 0 D) N )). (26) 5.1 Boundary layer aroximate aroximations with suort inside In this section we use Theorem 6 to introduce boundary layer aroximate aroximations of the tye (3) with suort essentially contained in the domain of denition of u. Here we use the term essentially to describe the fact that jb M u(x)j is of order O(" 0 kuk L ) for x 1() and decays to zero faster than any negative ower of dist(x; ) if x 2 R n n. Otherwise, if u is smooth enough in, then B M u(x) is a high order of (aroximate) aroximation for x in n S M +1, where S M +1 is a boundary stri of width decreasing exonentially with M. For k = 1; 2; : : :, we introduce the boundary layers (see Figure 1) in S k = where N o is a free arameter such that ( ; k = 0, n (No+Ns)hk?1 ; k 1, N o > N s 1? : 13

15 x n..... S1 = n (Ns+N o)h0... S2 = n (Ns+N o)h1... S3 = n (Ns+N o)h2... S4 = n (Ns+N o)h x1. = Figure 1: Illustration for the nested subdomains (Ns+No)hk comlements Sk in resect to. We dene also the oerators of multilication by characteristic functions and the multi-resolution oerator? k u(x) = ( u(x); x 2 Noh k 0 ; otherwise. M B M? := A 0? 0 + A k ( k?? A ~ k?1? ); k?1 and their where fa k g M, f ~ 0 A k g M 0 are the quasi-interolants from (23). In analogy with the notation in the beginning of 5, one can introduce also discreancy functions ~u k, and write B? Mu = M A k ~u k ; ~u k = (? 0 u; k = 0 ( k?? A ~ k?1? k?1 )u; k 1. Now we will show that B M? u is essentially suorted in. characteristic function of some set, then by Lemma 1, we have By Theorem 6, and hence jm h;d u(x)j g 0;D (h?1 dist(x; su )): B? M u = A M? M u + M?1 R k? k u We notice rst that if is a ja M? M u(x)j g 0;D (N o + h?1 M dist(x; ))kuk C() ; x 2 Rn n ; as dist(@; su k ) = N o h k by denition. In other words, B? M is of the same order as the saturation error if N o > N s, and jb M? u(x)j decreases faster than any ower of dist(x; ) for large x as we declared in the beginning. In the resent form, however, the summation is erformed uon the whole of R n, due to the unbounded suort of ~u k, so it remains to truncate using the idea in the same sirit as we did in the beginning of section 5. In virtue of Theorem 1, ~u k is of order O(~" + Dh N k?1 ) for x 2 (No+ Ns)hk?1 e, so the contribution to A k ~u k from oints in (No+ Ns)hk?1 e can be neglected. In the following denition, we introduce the oerator B M in which the summation is erformed layer by layer with only minimal overlaing: 14

16 Denition 1 Let f k g M 0 be the oerator sequence 0 =? 0 ; k u(x) = u(x); x 2 : No h k (x) (N o + e Ns )h k?1 0 ; otherwise. Then the multi-resolution oerator B M := A 0? 0 + M A k k ( k?? A ~ k?1? k?1 ): (27) is called the boundary layer aroximate aroximation oerator subordinate to f k g M 0. Alternatively, as we indicated in the beginning of this section, we can rewrite (27) in the form with coecients B M u(x) = c k;m = u k (h k m) = M m2q k c k;m x? hk m ; (28) h k D (? 0 u(h 0 m); k = 0 (? k? ~ A k?1? k?1 )u(h km); k 1. (29) and Q k = fm 2 n : mh0 2 g; k = 0, fm 2 n : N o h k (x) (N o + Ns e )h k?1 g; k 1. Remark 2 The ractical imlementation of Theorem 6 does not require an exlicit formula for ~. Indeed, in order to calculate B M u(x) by (28) one has to comute the coecients c k;m, i.e., to tabulate (? k? ~ A k?1? k?1 )u at the oints h km (cf. (29)). By Remark 1, the comutation of A? k?1? k?1 u(h km) requires only summation for indices, for which jh k m=h k?1? j = jm? j e Ns ; where e Ns is such that (9) holds for ~. These (?1 (2N s + 1)) n (or just?1 (2N s + 1), if ~ is a radial function) values can be re-comuted using numerical Fourier inversion of (18). 6 Accuracy In this section we estimate the error if functions belonging to certain function saces over are aroximated with the oerator B M. Since the cubature formula for the integral oerator P is obtained by P u(x) PB M u(x) = M m2q k c k;m P? hk m h k D for the study of the cubature error it is therefore sucient to estimate (I? B M )u in integral norms, for examle in L or weak Sobolev norms, but on the whole of R n. 6.1 L -estimates Theorem 7 Suose that is a domain in R n with comact closure and Lischitz boundary and let u 2 W N () with N > n=. For any " > 0 there exists D > 0 and a boundary layer aroximation B M such that ku? B M uk L(R n ) c 1 (Dh) N kr N uk L() + c 2 ( M h) 1= kuk L 1() + "kuk W () : (x): 15

17 su? k su k?1;? k?1 su k+1 su(i? k+1 ) su(i? k su k S L n S k+1 N o h k (No+ e Ns)hk (N o +N s )h k N o h k?1 (N o +N s )h k?1 (N o + e N s )h k?1 Figure 2: Sketch of the mutual disosition between the layer Sk n Sk+1 and the suort of the cut-o oerators k,? k, I? k, etc. The bottom numbers denote distance to the Proof. We will estimate the L norm of (I? B M )u on each of the layers S L n S L+1, S M +1 (cf. Fig. 2) and on the exterior domain R n n. To estimate kb M uk L(R n n) we decomose B M by Theorem 6: M B M = A 0? 0 + A k k ( k?? A ~ k?1? k?1 ) Ak? k? A k?1? k?1 + R k?1? k?1? A k (I? k )(? k? ~ A k?1? k?1 ) M = A 0? 0 + = A M M?? M Ak (I? k )(I? A ~ k?1 )? R k?1? k?1 ; where we used in the last equation that (I? k ) k? = (I? k)? k?1. Thus, by Theorem 3 we get immediately kb M uk L(R n n) h n= M kg 0;D(j j + N o )k L(R )kuk n L 1() M + h n= k kg 0;D (j j + (N o + N ~ s )?1 )k L(R n ) + h n= kg k?1 0;R ;;D (j j + N o )k L(R ) n kukl 1() c h n= kg 0;D (j j + N o )k L(R n )kuk L 1() : Setting for brevity we obtain analogously ku? B M uk L(SM +1 ) ku? A M? M uk L (SM +1 ) + M d N = N o + N s ; and ~ d N = N o + ~ N s ; h n= k kg 0;D (j j + ~ d N?1? d N M?k )k L(R n ) + h n= kg k?1 0;R ;;D (j j + N o? d N M +1?k )k L(R ) n kukl 1() M ku? A M M? uk L(SM +1 ) + hn= kuk L kn= kg 1() 0;D (j j + d ~ N?1? d N M?k k L(R n ) + kg 0;R;;D (j j + N o? d N M +1?k )k L(R n )) : (30) (31) 16

18 To estimate ku? B M uk L(SL ns L+1 ) we use the reresentation B M = M L A k k ( k?? A ~ k?1? k?1 ) + A L L? + Ak (I? k )(I? A ~ k?1 )? R k?1? k?1 : k=l+1 By Theorem 3 we obtain M k=l+1 A k k (? k? ~ A k?1? k?1 ) u L(SL ns L+1 ) h n= kuk L 1() M k=l+1 kn= kg 0;D (j j + d N L?k?1? ~ d N?1 )k L(R n ) (32) as well as Ak (I? k )(I? A ~ k?1 )? R k?1? k?1 uk L(SL ns L+1) k L?1 L?1 h n= kuk L 1() kn= 2~ 0 kg 0;D (j j + N o? d N k+1?l )k L(R n ) + (k?1)n= kg 0;R;;D (j j + N o? d N k?l )k L(R n ) ; (33) showing that these terms are small if N o and N s are chosen large enough, and additionally tend to zero together with h. Consequently, besides the estimate ku? A L? L uk L (SL ns L+1 ) (h L D) N + 2 jj=0 jj=n (h L D) jj " (; D) which follows immediately from Theorem 2, it remains to study In view of (; D) k@ uk L() k@ uk L(SL ns L+1 ) ; k(a L (I? L )(I? ~ A L?1 )? R L?1 )? L?1uk L(SL ns L+1 ) : and kr L?1? L?1uk L(SL ns L+1 ) (meas (S L n S L+1 )) 1= kr L?1? L?1uk L 1(S L ns L+1 ) (meas (S L n S L+1 )) 1= C R ~" 0 0 kuk L 1() ; k(a L (I? L )(I? ~ A L?1 )? R L?1 )? L?1uk L(R n n (No+ ~Ns?Ns)h L?1 ) we are left with the estimation of 2h n= L ~ 0kg 0;D (j j + N s )k L(R n )kuk L 1() ; (34) (35) ka L (I? L )(I? ~ A L?1 )? L?1 uk L (GL ) ka L (I? L )(I? ~ A L?1 )~uk L(GL ) + (meas G L ) 1= 0 k~g 0;D (j j + ~ N s )k L(R n )kuk W N () ; where G L = S L \ (No+ ~N s?n s)hl?1, and ~u 2 W N (Rn ) is the extension of u 2 W N k~uk W N (R n ) = kuk W N. () () with 17

19 The function (I? L )(I? ~ A L?1 )~u(x) is discontinuous on G L. In order to aly Theorem 2 we introduce the smooth counterart ' L of the characteristic function k. That means, we require that ' L 2 C N 0 (Rn ) is constant with the excetion of small neighbourhoods of the jums of L not containing grid oints and that ' L (h L m) = L (h L m), m 2 n. Obviously such a function with k@ ' L k L 1 c N h jj L ; 0 jj N ; exists. Furthermore, we introduce the continuous analogue of the quasi-interolant ~ A L?1 ~K L?1 u(x) := ( DhL?1 )?n R n ~ x? y DhL?1 u(y)dy: and the function ~ U L = (I? ' L )(I? ~ K L?1 )~u. With this notation we have A L (I? L )(I? ~ A L?1 )~u = A L ~U L + A L (I? ' L )( ~ K L?1? ~ A L?1 )~u: (36) and from Theorem 2 we obtain ka ~ L U L k L(GL ) k U ~ L k L(GL ) + k(i? A L ) U ~ L k L(GL ) k U ~ L k L(GL ) + (h L D) N (; D) jj=n Now the rough estimate k@ ~U L k L(R n ) C N =0 k@ ~ U L k L(R n ) + 2 jj=0 (h L D) jj " (; D) k@ U ~ L k L(GL ) : (? ) hjj?jj L k@ (I? ~ K L?1 )~uk L(R n ) together with the moment condition of ~ imlies k@ ~U L k L(R n ) C N kr N uk L() =0 (? ) hjj?jj L (h L?1 D) N?jj R n j~(x)jjxj jj dx; resulting in k@ ~ U L k L(R n ) c ;D h N L?1kr N uk L() (37) with a constant c ;D deending only on, D and. The second term in (36) can be written as the dierence between an integral oerator and its semi-discretization A L (I? ' L )( ~ K L?1? ~ A L?1 )~u = h?n L?1 with the smooth kernel function L (x; y) := D?n R n L x h L ; m=2h?1 L su L y ~u(y)dy? h L?1 x? m m? y ~ D D j2 n L x h L ; j : ~u(jh L?1 ) This dierence can be estimated by using the Taylor exansion of ~u 2 W N (Rn ) in the following form (cf. [7],[1]): ka L (I? ' L )( K ~ L?1? A ~ L?1 )~uk L(GL ) c(dh L?1 ) N kr N uk L() + (Dh L?1 ) jj k@ uk L(GL ) jj=0 =0 (; D) "? (~; D) (? : ) 18

20 with some constant c not deending on ~u and h. Summing u the last estimate together with (30)(35) and (37) we see that for u 2 W N () ku? B M uk L(R n ) ku? A M M? uk L(SM +1 ) + c(dh) N kr N uk L() + (Dh) jj k@ uk L() + h n= (1) kuk L 1() ; jj=0 where the numbers, which deend on " and ~", can be made arbitrarily small for D large enough, and (1) is determined by the functions g 0 and is suciently small if the arameters N 0 and N s are aroriately chosen. Thus we have only to aly Lemma 2 (see also Theorem 3) and the roof of Theorem 7 is comlete. 6.2 Pointwise estimates In a similar way one can show the following ointwise result Theorem 8 Suose that u 2 C N () and the boundary layer aroximate aroximation oerator B L is dened by (27). Then for any " > 0 and x 2 n S M +1, there exist D > 0 and ositive integers N s and N o, such that the accuracy of aroximation satises the estimate j(i? B M )u(x)j c( Dhk ) N kr N uk L 1() + "kuk C () ; where 0 k M denotes the index for which x 2 S k n S k+1. Thus the behaviour of B M u(x) is actually very close to that of A k u(x) for some ositive k M, where k increases as the distance from x to the boundary decreases. This leads to the eect that the aroximation becomes better in oints x 2 n S M +1 which lie nearer the 6.3 Estimates in weak norms Quasi-interolation on uniform meshes of the form (1) has the remarkable roerty that it converges in weak norms, since the saturation error, which is caused by fast oscillating functions, converges weakly to zero. The same roerty holds for the case of nonuniform meshes considered here. In the roof of Theorem 7, the aroximation error (I? B M )u was decomosed into M (I? B M )u = (I? A M M? )u + Ak (I? k )(I? A ~ k?1 )? R k?1? k?1 u : The second term consists of functions with L -norms which do not exceed c(dh) N kr N uk L() and h n= (1) kuk L 1(), resectively, lus small oscillating functions. similarly to [7] that for s > 0 M Ak (I? k )(I? A ~ k?1 )? R k?1? k?1 u + h n= (1) kuk L 1() + c s h s jj=0 H?s Therefore one can show c (Dh) N kr N uk L() (h D) jj " (; D) k@ uk L() ; where H s = Hs (Rn ) denotes the Bessel otential sace equied with the norm kuk H s = kf?1 ( j j) s=2 Fuk L = k(i? ) s=2 uk L : 19

21 Thus it remains to estimate k(i? A M? M )uk H?s k(i? A M? M )uk H?s. For integer s > 0 we have c(ka M? M )uk L (R n n) + k(i? A M? M )uk (W s q ()0) with q = =(? 1), and from Lemma 2 one gets for 0 < r < s=n; r 1=q k SM +1 (I? A M? M )uk (W s q ()0 chr Mk SM +1 (I? A M? M )uk L () ch r+1= M kuk W N () : Furthermore, k(i? SM +1 )(I? A M? M )uk (W s q ()0 = c (Dh) N kr N uk L() + c s h s M jj=0 so that the following aroximation result is valid. su (I? A M M? )u ' dx k'k W s q () =1 ns M +1 (h M D) jj " (; D) k@ uk L() ; Theorem 9 Suose that is a domain in R n with comact closure and Lischitz boundary and let u 2 W N () with N > n=. Then for any " > 0 there exists D > 0 and a boundary layer aroximation B M such that ku? B M uk H?s (R n ) (c 1(Dh) N + c 2 ( M h) 1=+r )kuk W N () + " hs kuk W () ; where 0 < r < s=n and r (? 1)=. 6.4 Numerical examles Here we give some numerical examles to illustrate the overall aroximation roerties of the oerator B M dened by (27), and esecially the behaviour of the error near the boundary. We shall use the boundary layer aroximate aroximation (28) generated by the functions 2, 4, 6 based on the Gaussian (see (11)), roviding second, fourth, and sixth order of aroximate convergence. The corresonding adjoint functions ~ 2, ~ 4, ~ 6 are given by (21). In all cases we use D = 3, which assures saturation levels of magnitude 1 10?12, 1 10?11 and 1 10?10 for quasi-interolants M h;d based on 2, 4, 6, resectively. The ste renement ratio in all examles is?1 = 3. We recall that by Theorem 8, B M erforms aroximately as A k on the k-th boundary stri S k n S k+1, i.e., the nearer the boundary, the better aroximation. The aroximation results are lotted over the boundary layer S M +1 n S 0 = fx 2 : (N o + N s )h M +1 (N o + N s )h 0 g in order to illustrate the interlay between the dierent quasi-interolants building the oerator B M. Since the ste-size used by B M is roortional to the distance from the boundary, one can determine the order of the formula used by the sloe of the error lot j(i? B M )uj against the distance to the boundary in logarithmic scales. Consider the lot in Fig. 3a showing the error from the aroximation of cos(1000t) near the boundary using the second-order formula based on the Gaussian. One can clearly see the ste-wise increase of the accuracy towards the boundary until a saturation is reached. The error remains unchanged within S k n S k+1 for xed k, since the ste does not change there. Observe also the sloe of the staircase it is aroximately two. In Fig. 3b the same function is aroximated using the sixth-order formula based on 6. Here the sloe is aroximately 6 : 1, but the saturation error is higher. 20

22 u(t) = cos(1000*t); D = 3; µ -1 = 3; u(t) = cos(1000*t); D = 3; µ -1 = 3; 1 O(h 2 )-formula 1 O(h 6 )-formula e-06 1e-06 1e-08 1e-08 1e-10 1e-10 1e-12 1e-12 1e-14 1e-14 1e-10 1e-08 1e e-10 1e-08 1e Figure 3: Boundary layer error lots for (I? BM )cos(1000t) using a) O(h 2 )-order formula, and b) O(h 6 )-order formula. The last examle reresent boundary error lots for aroximation of the function u(x 1 ; x 2 ) = cos(100 jxj 2 ); x 1 > 0, x 2 > 0, 0 ; otherwise, as an illustration for the action of a two-dimensional oerator built as the roduct B M = 1B M1 2B M2 of one-dimensional oerators i B Mi acting on the i-th argument of x = (x 1 ; x 2 ). These one-dimensional oerators are based on the generating functions 2 and 6, which rovide aroximate order of convergence of O(h 2 ) and O(h 6 ), resectively. In similarity with the revious examles, we use D = 3 and ste renement ratio in all examles is?1 = 3 in both the x 1 and x 2 -directions. Again, the aroximation results are lotted in logarithmic scales only in the interesting area near the vertex of the angle. 1e-05 1e-05 1e-10 1e-15 1e e-06 1e e-06 1e Figure 4: Boundary layer error lots for the function cos jxj with suort on the rst quadrant of R 2 using roduct of one-dimensional multi-resolution oerators roviding a) order O(h 2 ) of aroximate convergence; b) order O(h 6 ) of aroximate convergence. Precisely as in the one-dimensional examles, one can see clearly the gradual increase (Fig. 4a) of accuracy in the direction towards the boundary when the second-order formula is used. The lot in Fig. 4b shows the aroximation results when a sixth-order formula is used. In this case the saturation level is reached already after two iterations.? 21

23 7 Cubature of otentials in domains In this section we derive some estimates for the cubature of integral oerators, that often aear in roblems of mathematical hysics. As mentioned in the beginning, the cubature formula P h u for the integral oerator P u(x) = k(x? y)u(y)dy : is easily obtained from the boundary layer aroximate aroximations of the density u and dened as M y? hk m P h u(x) := PB M u(x) = c k;m k(x? y) dy ; (38) h k m2q R n k h k D if is chosen such that the integrals can be obtained analytically or by simle one-dimensional quadrature. For instance, the aroximation by (38) of the harmonic otential H using the generating functions 2M from (11) is obtained after calculating H 2M (x) =?( n 2? 1) 4 n=2 = R n 2M (y) jx? yj dy 1 jxj2 M?2 4jxj n=2?2 e? d +?n=2?jxj2 e n?2 n=2 0 j=0 L (n=2?1) j (jxj 2 ) : 4(j + 1) Here L () k denote the generalized Laguerre olynomials (12). Some further examles for the action of dierent otentials of mathematical hysics on the generating functions 2M in any sace dimension, including the elastic, hydrodynamic and diraction otentials, can be found in [2], [3] and [9]. It is well known that many interesting oerators are bounded maings P : L () W m ( 1) ; (39) with ; 1 R n ; we write P 2 L(L (); W m ( 1)). Note that the case m = 0 corresonds to singular integral oerators, whereas the volume otentials associated with artial dierential equations satisfy relation (39) with m > 0. In any case the kernel function k(x?y) is singular at the diagonal x = y, so that the aroximation of such multivariate integrals is quite comlicated. If the oerator P is such that (39) holds with = 1 = R n, Theorems 7 and 9 imly immediately: Theorem 10 Let u 2 W N () with N > n= and P 2 L(L (R n ); H m (Rn )). For any " > 0 there exists D > 0 such that kp u? P h uk H m (R n ) c 1 (Dh) N kr N uk L() + c 2 ( M h) 1= kuk L 1() + "kuk W () : If additionally P 2 L(H?m (R n ); L (R n )) then kp u? P h uk L(R n ) (c 1 (Dh) N + c 2 ( M h) 1=+r )kuk W N () + " hm kuk W () ; where 0 < r < m=n; r (? 1)=. However, very often the integral oerator P fullls (39) only for bounded domains ; 1 R n. Imortant examles are the harmonic or elastic otentials. In this case we are interested in the estimation of P u? P h u on some bounded domain 1. Since in general su B M u = R n we have to consider also integrals of the form k(x? y)b M u(y)dy ; x 2 1 : R n n 22

24 To this end we choose a ball B R with radius R around the origin such that ; 1 B R and suose that the kernel satises the estimate j@ k(x? y)j r (jyj) ; for x 2 1 ; y 2 R n n B R ; for some function r (x) of at most olynomial growth and the multi-indexes 0 jj m. Lemma 4 For any N > 0 there exists constants c N;;R such k(? y) B M u(y)dy c j;;r h N (meas 1 ) 1= kuk L : R n 1() nb L(1 ) R If R 1 then c N;;R 0. Proof. We estimate R n nb k(y? y) h k m2q k c k;m ckuk L1() ckuk L1() y? hk m h k D dy h k m2q k y? hk m h k D dy r (jyj) R n nb R y r (jyj) g 0;D (dist( ; Q k ) dy : h k h k R n nb R Let r (y) c j jyj j for jyj 1. From the raid decay of g 0;D one obtains g 0;D (dist( y ; Q k )) = g 0;D (N o + h?1 k dist(y; )) c N h N k dist(y; )?N h k h k for any N. Now it is clear that for N > n + j the inequality roves the assertion. R n nb R r (jyj) g 0;D (dist( y h k ; Q k h k )) dy ch N k Now we are in a osition to rove RnnBR jyj j dist(y; ) N dy Theorem 11 Let u 2 W N () with N > n= and P 2 L(L (); W m ( 1). Under the assumtions made above for any " > 0 there exists D > 0 such that kp u? P h uk W m ( 1 ) c 1 (Dh) N kr N uk L() + c 2 ( M h) 1= kuk L 1() + "kuk W () : If additionally P 2 L((W m =(?1) ())0 ; L ( 1 )) then kp u? P h uk L(1 ) (c 1 (Dh) N + c 2 ( M h) 1=+r )kuk W N () + " hm kuk W () ; where 0 < r < m=n; r (? 1)=. Proof. Fix the ball B R and slit P h u(x) = PB M u(x) = P BR B M u(x) + P (1? BR )B M u(x) : The W m ( 1)-norm of the second term is bounded by ch N kuk L 1() due to Lemma 4, whereas the dierence kp u? P BR B M uk W m ( 1 ) c R ku? B M uk L(BR ) 23

25 can be estimated using by Theorem 7. The same arguments aly also for the assertion concerning the L ( 1 )-norm of P u? P h u, if we use the inequality and Theorem 9. kp u? P BR B M uk L(1 ) c R ku? B M uk (W m =(?1) (B R)) 0 cku? B Muk H?m (R n ) Summarizing, for a large class of domain integral oerators with singular kernels one can dene cubature formulae retaining the order O(h N ) lus some small saturation error, if the boundary layer aroximate aroximations of the density is used with aroriate arameters and M. Let us consider two simle examles: Examle 1. Consider the logarithmic otential Note that the maing H 2 u(x) = 1 2 u(y) 1 log jx? yj dy H 2 : L () 7 W 2 (); 1 < < 1; is bounded, if is a bounded domain. Thus, in this case Theorem 11 yields the estimate kh 2 u? H 2;h uk W 2 () c 1 (Dh) N kr N uk L() + c 2 ( M h) 1= kuk L 1() + "kuk W () : Consequently, if the boundary layer aroximate aroximations are such that M is of the same order of magnitude as h, we get the aroximation order O(h N ) modulo saturation error. If we measure the error in a weaker norm than W 2, the small saturation error tends to zero together with h. For examle, if u 2 W N 2 () with N > 1 we obtain kl 2 u? L 2;h uk L2 () (c 1 (Dh) N + c 2 M h)kuk W N 2 () + " kuk h2 W ; 2 () such that already the choice M h leads to O(h N ) order lus a very small error term converging to zero with the rate O(h 2 ). Note that Sobolev's imbedding theorem can be used to rove the convergence of the cubature L 2;h with resect to the uniform norm. Examle 2. The Poisson integral P n '(x; t) = 1 (2 t) n R n '(y) ex jx? yj? dy ; x 2 R n ; t > 0 ; 4t gives a artial solution of the homogeneous heat equation with the initial condition u(x; 0) = '(x). If the basis function is the Gaussian or some related function then obviously the integrals 1 (2 t) n R n y? hk m ex h k D? jx? yj 4t have simle analytic exressions. Since for xed t > 0 the kernel function is smooth the Poisson integral generates a bounded maing from Sobolev or Bessel otential saces of arbitrary negative order into usual function saces. Therefore from Theorem 11 it follows that dy kp'(; t)? P h '(; t)k L2 (R n ) (c 1 (Dh) N + c 2 M h)k'k W N 2 () ; with constants deending on t > 0 but not on ' and h. Hence, P h reresents a semi-analytic cubature of order O(h N ) without saturation errors. 24