84 G. Schmidt (1.3) T : '! u := uj ; where u is the solution of (1.1{2). It is well known (cf. [7], [1]) that the linear operator T : H ( )! e H ( ) i

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1 Numer. Math. 69: 83{11 (1994) Numerische Mathematik c Springer-Verlag 1994 Electronic Edition Boundary element discretization of Poincare{Steklov operators Gunther Schmidt Institut fur Angewandte Analysis und Stochastik, Mohrenstrae 39, D-1117 Berlin, Germany Received October 1992 / Revised version received April 14, 1994 Summary. The construction of a discretization of Poincare{Steklov operators with the boundary element method is given providing the same mapping properties as the nite element discretization of these operators. Mathematics Subject Classication (1991): 65N38, 65N12, 65N55, 65N3, 31A1, 31B1 1. Introduction Poincare{Steklov (PS) operators are natural mathematical tools for the investigation of boundary value problems and their numerical solution with domain decomposition (DD) methods based on the nite element (FE) solution of the subproblems ([1], [9]). The purpose of this paper is to construct discretizations of PS operators for elliptic boundary value problems with the boundary element method (BEM). We will show that the discretizations obtained via a direct Galerkin BEM possess the same mapping properties as the FE discretizations if the boundary elements satisfy some natural conditions. Hence the given construction provides a base for the analysis of dierent DD methods using the BE solution of subproblems, of the coupling of FE and BE methods and related problems. As model problem we consider the mixed boundary value problem for the Laplacian. Let R d (d = 2; 3) be a bounded domain with piecewise smooth boundary such that angles at corners and edges do not degenerate. The boundary is partitioned into 3 nonintersecting domains = D [ N [ and suppose that D 6= ;, \ N = ;. Let ' 2 H ( ) be given and consider the problem: Find u 2 H 1 () such that (1.1) u = ; in ; uj D = ; n uj N = ; (1.2) n uj = ' : Here n u denotes the derivative with respect to the outer normal. The PS operator T is dened as the mapping page 83 of Numer. Math. 69: 83{11 (1994)

2 84 G. Schmidt (1.3) T : '! u := uj ; where u is the solution of (1.1{2). It is well known (cf. [7], [1]) that the linear operator T : H ( )! e H ( ) is bounded and invertible, symmetric with respect to the duality between H ( ) and e H ( ), induced by the scalar product of L 2 ( ), and positive denite (T '; ) = ('; T ) = T '(x) (x)d x ; (T '; ') ck'k 2 H ( ) : Here H s denotes the usual Sobolev spaces: For H s () = fuj : u 2 H s (R d )g ; s 2 R ; H s () = 8 < : fuj : u 2 H s+ (R d )g ; s > ; L 2 () ; s = ; (H s ()) ; s < : H s ( ) = fuj : u 2 H s ()g ; s ; eh s ( ) = (H s ( )) ; s < ; eh s ( ) = fu 2 H s ( ) : ~u 2 H s ()g ; s ; H s ( ) = ( e H s ( )) ; s < ; where ~u denotes the extension of u by zero to. We note that T 1 = S, where (1.4) S : 2 e H ( )! 1 u := n uj and u solves the problem (1.1) together with the Dirichlet boundary condition on (1.5) u = The FE discretization of the PS operator T and its inverse S follows immediately from the variational formulation of (1.1). Let us suppose that we are given a space V h H 1 () of nite element functions on vanishing on D Denote by V h = fv h : v h j D = g ; dim V h < 1 : o V h = fv h 2 V h : v h = g ; ex h = f v h : v h 2 V h g e H ( ) : For given h 2 e Xh we determine u h 2 V h such that u h = h page 84 of Numer. Math. 69: 83{11 (1994)

3 Boundary element discretization of Poincare{Steklov operators 85 and (1.6) a(u h ; v h ) := ru h rv h d! = ; 8 v h 2 o V h : Then a(u h ; v h ), v h 2 V h, denes a linear functional of v h 2 e Xh bounded in the e H ( ){norm. Let P h : e H ( )! e Xh be a bounded projection onto Xh e, by Ph we denote its L2 ( ){adjoint projection and Y h := Im Ph H ( ) can be identied with the dual space of Xh e. Hence there exists h 2 Y h ; k h k H ( ) c such that a(u h ; v h ) = ( h ; v h ) : It can be easily seen (cf. [1]) that the mapping is linear and invertible, S h : e Xh! Y h dened by S h h = h (1.7) ks h h k H ( ) ck hk eh ( ) ; 8 h 2 e Xh ; symmetric (S h h ; h) = ( h ; S h h) = a(u h ; u h) ; where u h solves (1.6) with u h = h, and positive denite Moreover (1.8) (S h h ; h ) = a(u h ; u h ) c 1 ku h k 2 H 1 () ck hk 2 eh ( ) : k(s S h ) h k H ( ) k(p hs S h ) h k H ( ) + k(i P h)s h k H ( ) c c ((S S h ) h ; v h ) sup v h 2V h k v hk eh ( ) + inf ' h 2Y h k 1 u ' h k H ( ) inf ku v h k H v h 2V 1 () + inf k 1 u ' h k H h ' h 2Y ( ) h where u is the exact solution of (1.1) with u = h. We note that the constants (1.7) and (1.8) depend on the norm of P h. Thus, if kp hk is uniformly bounded with respect to h, then estimates (1.7) and eh ( ) (1.8) hold with constants independent of h and h. Now it is clear how to dene the FE discretization of T. Choose Y h and h 2 Y h, nd u h 2 V h such that a(u h ; v h ) = ( h ; v h ) ; 8 v h 2 V h ; and dene T h h := u h. Since T h = S h 1, this operator has analogous properties as S h and T, moreover k(t T h ) hk eh ( ) c inf v h 2V h ku v h k H 1 () ; ;! page 85 of Numer. Math. 69: 83{11 (1994)

4 86 G. Schmidt where u solves (1.1) and 1 u = h. The mentioned mapping properties of PS operators and their FE discretizations were essentially used (sometimes unknowingly) for the formulation and analysis of various DD methods, which are in fact equivalent to the iterative solution of operator equations with PS operators of dierent subdomains. In the following we will prove that a direct Galerkin BE solution of problem (1.1) (considered in Sect. 2) yields discretizations S h and T h of the operators S and T with the same mapping properties mentioned above (proved in Sect. 3). Therefore convergence results for many DD methods remain valid if the FE solution of subproblems is replaced by their BE solution whenever it is possible. 2. A direct Galerkin BEM In this section we discuss some results concerning a boundary integral method for solving mixed boundary value problems for the Laplacian. Let u 2 H 1 (), R d, be a solution of (2.1) u = in : Then we have the representation (2.2) u(x) = 1 2 G(x; y) n u(y)d y 1 2 with the fundamental solution G(x; y) = 8 >< >: 1 lnjx yj ; d = jx yj1 ; d = 3 : ny G(x; y)u(y)d y ; x 2 ; Let us dene the boundary integral operators for x 2 V '(x) := K '(x) := G(x; y)'(y)d y ; K'(x) := nx G(x; y)'(y)d y ; D'(x) := nx The following properties are well known (cf. Costabel (1988)): ny G(x; y)'(y)d y ; ny G(x; y)'(y)d y : V : H + ()! H + () ; D : H + ()! H + () ; K : H + ()! H + () ; K : H + ()! H + () (2.3) are continuous for jj. The operator K is the adjoint of K with respect to the duality between H () and H () induced by the scalar product (; ) L 2 (). Moreover, page 86 of Numer. Math. 69: 83{11 (1994)

5 Boundary element discretization of Poincare{Steklov operators 87 (Dv; v) L 2 () c jvj H () ; c > ; for all v 2 H () ; j j H () denotes the seminorm ; (V ; ) L 2 () c k k H () ; c > ; for all 2 H () if d = 3 ; (2:4) and for all 2 H () with ( ; 1) L 2 () = if d = 2 : To ensure the solvability of boundary integral equations we need that the operator V is invertible. Therefore if d = 2 then we assume in the following that cap () 6= 1, i.e. R 2 has the property (P) If 2 H () solves V = then =. Note that for any R 2 the domains m = fmx : x 2 g, m >, satisfy (P) with the exception of one value m e (cf. Symm (1967)). Now the representation (2.2) and the jump relations for single and double layer potentials lead to the equalities on (2.5) u = 1 2 ((I K)u + V nu) ; n u = 1 2 (Du + (I + K ) n u) : Hence, if we consider the mixed boundary value problem for (2.1) (2.6) uj 1 = g 1 ; n uj 2 = g 2 ; where g 1 2 H ( 1 ), g 2 2 H ( 2 ), 1 [ 2 =, 1 6= ;, and take the limits of u(x) for x 2 2 and of the normal derivative n u for x 2 1, then we get the equation (2.7) D22 K 21 v K 12 V 11 D21 I K 22 g1 = ; I + K 11 V 12 g 2 where the subscripts in D jk, etc., mean integration over k and evaluation on j. Here v := uj 2, := n uj 1 denote the unknown boundary values of the solution u of (2.1), (2.6). If we substitute in (2.7) v = v + lg 1, = + lg 2 with arbitrary extensions lg 1 2 H (), lg 2 2 H () then we obtain a system of boundary integral equations v (2.8) A := D22 K 21 K 12 V 11 v D2 I K lg1 f2 = 2 =: : I + K 1 V 1 lg 2 f 1 Here D 2 for example denotes integration over and evaluation on 2. If g 1 2 H s ( 1 ), g 2 2 H s1 ( 2 ) and s 2 (; 1) then f 2 2 H s1 ( 2 ), f 1 2 H s ( 1 ). Moreover, the operator A : eh s ( 2 ) H s1 ( 2 )! eh s1 ( 1 ) H s ( 1 ) is continuous and satises a Garding inequality in e H ( 2 ) e H ( 1 ), for U = (w; ') 2 e H ( 2 ) e H ( 1 ) we have page 87 of Numer. Math. 69: 83{11 (1994)

6 88 G. Schmidt hau; Ui = (D 22 w + K '; w) 21 L 2 ( 2) + (K 12 w + V 11 '; ') L 2 ( 1) = (D 22 w; w) L 2 ( 2) + (V 11 '; ') L 2 ( 1) = (D ~w; ~w) L 2 () + (V ~'; ~') L 2 () ; where ~w; ~' denote the extension by zero to. Relations (2.4) imply the existence of a compact operator C such that (2.9) h(a + C)U; Ui c kwk 2 eh + ( k'k2 : 2) eh ( 1) Since for d = 3 we can set in (2.9) C = and for d = 2 because of property (P) the system (2.8) has no eigensolutions we derive that (2.1) is invertible. A : eh ( 2 ) H ( 2 )! eh ( 1 ) H ( 1 ) Theorem 2.1. (Costabel and Stephan (1985a), von Petersdorf (1989)). Let g 1 2 H ( 1 ), g 2 2 H ( 2 ) with arbitrary extensions lg 1 2 H (), lg 2 2 H (). Then there exists exactly one solution (v ; ) 2 e H ( 2 ) e H ( 1 ) of (2.8) and v := v + lg 1 j 2 2 H ( 2 ), := + lg 2 j 1 2 H ( 1 ) solve (2.7). Theorem 2.2. Let g 1 2 e H ( 1 ), g 2 2 e H ( 2 ). Then (2.7) is uniquely solvable and c 1 kg 1k eh ( 1) + kg 2k eh ( 2) where c 1, c 2 do not depend on g 1 and g 2. kvk eh ( 2) + k k eh ( 1) c 2 kg 1k eh ( 1) + kg 2k eh ( 2) Proof. The boundary values of the solution of the mixed boundary value problem (2.1), (2.6) satisfy v 2 e H ( 2 ), 2 e H ( 1 ). If we extend g 1 and g 2 by zero, lg 1 j 2 = lg 2 j 1 =, then (v; ) is the unique solution of (2.8) and kvk eh ( 2) + k k eh ( 1) c kf 1 k H ( 1) + kf 2 k H ( 2) c 2 kg 1k eh ( 1) + kg 2k eh ( 2) where we have used (2.3) and (2.1). Since the same considerations can be applied to the boundary value problem for (2.1) with the data uj 2 = v ; n uj 1 = ; ; ; the twosided estimate follows immediately. ut page 88 of Numer. Math. 69: 83{11 (1994)

7 Boundary element discretization of Poincare{Steklov operators 89 The Garding inequality (2.9) yields the convergence of Galerkin methods for the approximate solution of system (2.8). We choose on 2 and 1 nite dimensional sets of approximating functions M h H ( 2 ), N h H e ( 1 ) with lim dim M h = lim dim N h = 1. Furthermore we suppose that g 1 can h! h! be extended by some lg 1 j 2 2 M h and g 2 can be extended by some lg 2 j 1 2 N h which implies that g 2 2 e H ( 2 ). We denote f Mh = M h \ e H ( 2 ) and consider the Galerkin method: Find (v h ; h ) 2 f Mh N h such that (2.11) A v h h ; wh ' h = f2 f 1 ; wh ' h ; 8 (w h ; ' h ) 2 f Mh N h : Under the assumption that [ h fm h and [ h N h are dense in e H ( 2 ) and eh ( 1 ), resp., by standard methods the following results can be obtained. Theorem 2.3. (Costabel and Stephan (1985a)). For any g 1 2 H ( 1 ), g 2 2 eh ( 2 ) and all (suciently small if d = 2) h the Galerkin equations (2.11) are uniquely solvable and kv h k eh ( 2) + k h k eh ( 1) c klg 1 k H () + klg 2 k H () : Moreover, the functions v h := v h + lg 1j 2 and h := h + lg 2j 1 approximate the unknown boundary values quasioptimally kv v h k H ( 2) + k hk eh ( 1) c inf k ' hk ' h 2N eh h ( + inf kv w 1) hk eh w h 2 em ( 2) h where the constant c does not depend on h. Using the regularity of the solution (v ; ) for the case d = 2 (cf. Costabel and Stephan (1985)), the structure of the mapping A and the Aubin{Nitsche Lemma one can estimate the convergence of the approximate solutions in Sobolev norms of lower order. Theorem 2.4. Let d = 2. Then there exist > 1=4 and c > such that kv v h k H (2) + k hk eh (1) c " (h) kv v h k H ( 2) + k hk eh ( 1)! ; where " (h) := sup w2eh + ( 2) '2eH + ( 1) inf w n2 em h ' h 2N h kw w hk eh ( 2) kwk eh + ( 2) + k' ' hk eh ( 1) k'k eh+ (1)! : page 89 of Numer. Math. 69: 83{11 (1994)

8 9 G. Schmidt From the density of [ h fm h and [ h N h we have " (h)! as h!. Note that depends on the inner angles at the corners of and at the points where boundary conditions change (Costabel and Stephan (1985)). We remark that under the condition g 2 2 H e ( 2 ) one can choose lg 2 j 1 =, but for some considerations in Sect. 3 it is useful to admit lg 2 j 1 6=. Now we dene a positive bilinear form for the boundary values of the BE solutions. First of all we note that the Galerkin method (2.11) admits the following equivalent formulation: Find v h 2 M h and h 2 N h such that (2.12) ~v h := g1 v h 2 H (); ~ h := and the boundary values of the function (2.13) u h (x) := 1 2 satisfy the equations (2.14) 1 (u h g 1 )' h d + G(x; y) ~ h(y)d y 1 2 h g 2 2 H () ny G(x; y)~v h (y)d y ; x 2 ; 2 ( n u h g 2 ) w h d = ; 8 (w h ; ' h ) 2 f Mh N h : Under the assumptions of Theorem 2.3 the solution u h exists. From (2.2) and (2.5) we conclude that u h solves (2.1) and on it holds (2.15) u h = 1 2 ((I K)~v h + V ~ h) ; n u h = 1 2 (D~v h + (I + K ) ~ h) : Therefore Theorem 2.3 yields the estimate on the boundary (2.16) ku h uk H () + k n u h n uk H () c inf kv w hk eh w h 2 em ( + inf k ' 2) hk : ' h 2N eh h ( 1) h! Let us now suppose that u h and u h are of the form (2.13) with the densities (~v h, ~ h ) and (~v h, ~ h ), resp., and are solutions of equations (2.14) for corresponding boundary data g 1, g 1 2 e H ( 1 ), g 2, g 2 2 e H ( 2 ). Then we may choose lg 1 j 2 = lg 1j 2 =, whereas lg 2 j 1 ; lg 2j 1 2 N h are arbitrary. We consider the bilinear form (2.17) b(u h ; u h) := g 1 n u h d + g u 2 hd 1 2 In view of (2.14) we get page 9 of Numer. Math. 69: 83{11 (1994)

9 Boundary element discretization of Poincare{Steklov operators 91 b(u h ; u h) = u h n u h d + (g 1 u h ) n u h d + (g 2 nu h) u h d 1 2 = u h n u h d + (g 1 u h ) ( n u h ' h) d + (g 2 nu h) (u h w h )d 1 2 for all (w h ; ' h ) 2 Mh f N h. If we choose w h = v h, ' h = h (cf. (2.12)) as solutions of the corresponding equations (2.11) then b(u h ; u h) = u h n u hd (2.18) = 1 2 (~v h u h ) ~ h n u h (D~v h ; ~v h) L 2 () + (V ~ h; ~ h ) L 2 () here we used (2.15) and the symmetry of D and V. d = b(u h ; u h) ; (2.19) We remark that ~v h u h = 1 (I + K)~v 2 h V ~ h ; ~ h n u h = 1 D~v 2 h + (I K ) ~ h are the boundary values on of the function u a h(x) := 1 2 G(x; y) ~ h(y)d y ny G(x; y)~v h (y)d y ; x 2 R d n : Now we use (2.4), Theorems 2.2 and 2.3 to estimate for the solution u h of (2.13{14) (D~v h ; ~v h ) L 2 () c 1 k~v h k 2 H c () 2 kvh k2 eh + ( kg 2) 1k 2 eh ( 1) c kg 1 k 2 eh + klg ( 1) 2k 2 ; H () (V ~ h; ~ h) L 2 () c 1 k ~ hk 2 H () c kg 1 k 2 eh ( 1) + klg 2k 2 H () (D~v h ; ~v h ) L 2 () c 1 j~v h j 2 H () c jv hj 2 H ( 2) + jg 1j 2 H ( 1) and if d = 3 (V ~ h; ~ h) L 2 () c 1 k ~ hk 2 H () ckg 2k 2 H ( 2) : ; cjg 1 j 2 H ( 1) ; If d = 2 then we represent ~ h = h +!e, where V e = 1 and! = ( ~ h; 1) L 2 ()= (e; 1) L 2 (). Then ( h ; 1) L 2 () = and (V ~ h; ~ h) L 2 () = V h ; h L 2 () +!2 (e; 1) L 2 () :! 2 can be estimated using Theorem 2.4, since page 91 of Numer. Math. 69: 83{11 (1994)

10 92 G. Schmidt j ~ h d j = j k h k eh (1 So for d = 2 we obtain (V ~ h; ~ h) L 2 () c ~h n u d j = j 1 ( h )d j 1 d c " (h) kg 1k eh ( + klg 1) 2k H () kg 2 k 2 H ( "2 2) (h) kg 1 k 2 eh + ( klg 1) 2k 2 H () Now (2.19) and the derived estimates let us conclude Theorem 2.5. Let u h, u h be the BE solutions of (2.1), (2.5) for the boundary data g 1 ; g 1 2 e H ( 1 ) and g 2 ; g 2 2 e H ( 2 ), resp., obtained via (2.13{14). Then the form b(u h ; u h ) dened in (2.17) is symmetric and bilinear. Furthermore, for any extension lg 2 of g 2 with lg 2 j 1 2 N h b(u h ; u h ) c 1 kg 1 k 2 eh ( 1) + klg 2k 2 H () : : and b(u h ; u h ) c 2 jg1 j 2 H ( 1) + kg 2k 2 H ( 2) " 2 (h) kg 1 k 2 eh ( 1) + klg 2k 2 H () ; with constants independent of g 1, g 2 and h. The second term of the right hand side in the last inequality appears only if d = BE discretization of PS operators Now we are in the position to construct discretizations of the operators S and T using the Galerkin BEM. According to the denition of this method we are given spaces of trial functions M h H ( N ) and N h 2 e H ( D ), fm h = M h \ e H ( N ). Furthermore, on we have nite dimensional spaces ex h e H ( ) to approximate the boundary values u and Y h e H ( ) for the approximation of the normal derivatives 1 u. In order to obtain mappings between the BE approximations of u and 1 u providing the same properties as the FE discretizations of S and T we need some relations between the spaces of trial functions. For the existence of an isomorphism S n : e Xh! Y h with (S h h ; h ) 6= ; 6= h 2 e Xh ; or T n : Y h! e Xh with (T h h ; h ) 6= ; 6= h 2 Y h ; it is necessary that e X h \ Y h = ;. Since the spaces of trial functions are nitedimensional this is equivalent to 1. For any h there exists a bounded projection P h : e H ( )! e Xh onto e Xh, such that the L 2 ( ){adjoint projection P h maps onto Y h, Im P h = Y h, and (3.1) kp h fk H ( ) ckfk H ( ) ; 8 f 2 H ( ) : page 92 of Numer. Math. 69: 83{11 (1994)

11 Boundary element discretization of Poincare{Steklov operators 93 In order to prove the required properties of our BE discretization of the operator T we need also the following technical assumption which is fullled in any reasonable setting. 2. Since \ N = ; it is natural to introduce h := f h : h j D 2 N h, h j 2 Y h g. We require that Y h h possess the extension property: For any h there exists a linear extension operator E h : Y h! h ; E h ' h j = ' h, (3.2) ke h ' hk eh ( [ D) ck' hk H ( ) ; 8 ' h 2 Y h : Note that in general we do not assume that the constants in (3.1) and (3.2) are independent of h. But it will be clear that for some estimates concerning the BE discretizations S h and T h as h! we have to require the uniform boundedness of kp hk eh ( ) and ke hk H ( )!eh ( [ D). Due to the boundary conditions uj D =, n uj N = of (1.1) the BE solutions u h are of the form (2.13) with densities (3.3) ~v h 2 H () such that ~v h j D = ; ~v h j N 2 Mh f ; ~v h j 2 Xh e ; ~ h 2 H () such that ~ hj D 2 N h ; ~ h j N = ; ~ h j 2 Y h : Because of \ N = ; the conditions ~v h j N 2 f Mh, ~v h j 2 e Xh are natural. Hence the solutions u h of (1.1) with the direct Galerkin BEM belong to the linear space V h := ( u h of the form (2.13), the densities ~v h ; ~ h satisfy (3.3) and u h ' h d + D N n u h w h d = ; 8 (w h ; ' h ) 2 f Mh N h BE solutions of (1.1) with Dirichlet data h 2 e Xh we denote by R D;h h, with Neumann data h 2 Y h by R N;h h, i.e. h 2 Xh e R! R D;h h 2 V h with ( (R D;h h ) h )' h d = ; 8 ' h 2 Y h ; R h 2 Y h! R N;h h 2 V h with ( 1 (R N;h h ) h )w h d = ; 8 w h 2 Xh e : (3.4) From Theorem 2.3 we deduce that for all (suciently small if d = 2) h the functions R D;h h and R N;h h are uniquely determined. Now we construct the BE discretization of the operator S. Let h 2 Xh e, then due to Theorem 2.4 with b(r D;h h ; R D;h h ) = [ D ~ h n R D;h h d ~ h (x) = h (x) ; x 2 ; ; x 2 D ; ) : page 93 of Numer. Math. 69: 83{11 (1994)

12 94 G. Schmidt and b(r D;h h ; R D;h h ) c 1 k ~ h k 2 eh ( [ D) ; b(r D;h h ; R D;h h ) c 2 j ~ h j 2 eh ( [ D) "2 (h)k ~ h k 2 eh ( [ D) where in the last inequality the second term of the right hand side appears only for d = 2. But ; k ~ hk eh ( [ D) and j ~ hj eh ( [ D) c 3 k hk eh ( ) c 4 k hk eh ( ) with constants independent of h and h. Hence for all (suciently small if d = 2) h we have (3.5) c 1 k h k 2 eh ( ) b(r D;h h ; R D;h h ) c 2 k h k 2 eh ( ) Using (3.1) we dene (3.6) such that (3.7) b(r D;h h ; R D;h ' h ) = S h h := P h 1 (R D;h h ) h S h ' h d = ( h ; S h ' h ) ; h ; ' h 2 e Xh : Theorem 3.1. For all h (< h if d = 2) the BE discretization S h (3.6) of the operator S has the following properties: (i) S h : e Xh! Y h is bounded, (3.8) ks h h k H ( ) ck hk eh ( ) ; 8 h 2 e Xh : If the projections P h are uniformly bounded then the constant c in (3.8) is independent of h. (ii) c 1 k h k 2 eh ( ) (S h h ; h ) c 2 k h k 2 eh ( ), c 1, c 2 do not depend on h 2 Xh e and h. (iii) (S h h ; ' h ) = ( h ; S h ' h ) ; 8 h ; ' h 2 Xh e. (iv) k(s h S) h k H ( ) c inf ku w hk eh w h 2 em ( + inf k N) n u hk, h h 2 eh h ( [ D) where u solves (1.1) with the Dirichlet condition u = h on. If the projections P h are uniformly bounded, then c is independent of h. Proof. (i) From (3.1) and Theorem 2.4 we have ks h h k H ( ) = kp h 1(R D;h h )k H ( ) c 1 k 1 (R D;h h )k H ( ) ck hk eh ( ) : (ii) and (iii) follow immediately from (3.5), (3.7) and Theorem 2.5. page 94 of Numer. Math. 69: 83{11 (1994)

13 Boundary element discretization of Poincare{Steklov operators 95 (iv) k(s h S) h k H ( ) k(s h P h S) hk H ( ) + k(i P h)s h k H ( ) c k 1 (R D;h h u)k H ( ) + inf k 1 u ' h k H ' h 2Y ( ) h The application of Theorem 2.3 and the estimate inf k 1 u ' h k H ' h 2Y ( ) inf k n u hk h h 2 eh h ( [ D) yield the assertion. ut Next we dene the BE discretization of T. Let h 2 Y h, R N;h h 2 V h exists for all (suciently small if d = 2) h and we set (3.9) T h h := P h R N;h h ; such that from Theorem 2.5 and (3.1) (3.1) (T h h ; ' h ) = b(r N;h h ; R N;h ' h ) ; h ; ' h 2 Y h : But in general T h 6= S h 1, as can be seen from the following. Let h 2 ex h. If we denote the densities dening R D;h h via (2.13) by ~v h, ~ h and the corresponding densities of R N;h (S h h ) by ~v h, ~ h then Hence and (3.11) b(r D;h h ; R N;h (S h h )) = = = R D;h h n R N;h (S h h )d (~v h R D;h h ) ~ h n R N;h (S h h ) ~v h n R N;h (S h h )d + h 1 (R N;h (S h h ))d + = (S h h ; h ) = b(r D;h h ; R D;h h ) d ~ h R D;h h d ~v h ~ h d S h h (R D;h h )d b(r D;h h R N;h (S h h ); R D;h h R N;h (S h h )) = (S h h ; h ) 2(S h h ; h ) + (T h S h h ; S h h ) ((T h S h I) h ; S h h ) h S h h d = 1 (D(~v h ~v 2 h); ~v h ~v h) L 2 () + (V ( ~ h ~ h ); ~ h ~ h ) L 2 () c j~v h ~v h j2 H + k ~ () h ~ h k 2 : H () But using the extension property (3.2) we can prove that T h is spectrally equivalent to S 1 h. page 95 of Numer. Math. 69: 83{11 (1994)

14 96 G. Schmidt Theorem 3.2. Suppose that estimates (3.1) and (3.2) are satised uniformly with respect to h. Then for any h(< h if d = 2) the BE discretization T h (3.9) of the PS operator T has the following properties: (i) T h : Y h! e Xh is bounded, kt h hk eh ( ) ck hk H ( ) ; 8 h 2 Y h : (ii) c 1 k h k 2 H ( ) (T h h ; h ) c 2 k h k 2 H ( ), the constants do not depend on h 2 Y h and h. (iii) (T h h ; ' h ) = ( h ; T h ' h ), h ; ' h 2 Y h. (iv) ((T h S h I) h ; S h h ), 8 h 2 e Xh. (v) k(t h T ) hk eh ( ) c inf w h 2eX h ku w hk eh ( ) + inf k n u ' hk ' h 2N eh h ( + inf ku w D) hk eh w h 2 em ( N) h where u solves (1.1) with the Neumann condition 1 u = h. Proof. (i) kt h hk eh ( ) j(t h h ; ')j = sup '6= k'k H ( ) j(t h h ; Ph = sup ')j '6= kph 'kh ( ) Using (3.2), Theorem 2.5 and (3.1) we obtain! kph 'kh ( ) : k'k H ( ) j(t h h ; Ph')j = jb(r N;h h ; R N;h (Ph'))j c 1 ke h hk eh ( [ ke D) h(ph')k eh ( [ D) From (3.1) we get the assertion. (ii) Theorem 2.5 and (3.2) imply ck h k H ( )kp h 'k H ( ) : (T h h ; h ) = b(r N;h h ; R N;h h ) c 1 ke h h k 2 eh ( [ D) ck hk 2 H ( ) :, If d = 2 then (T h h ; ) c 1 k h k 2 H ( ) "2 (h)ke h h k 2 eh ( [ D) c 1 (1 " 2 (h))k h k 2 H ( ) : Note that the boundedness of P h is not used. (iii) and (iv) follow from Theorem 2.5, (3.1) and (3.11). (v) k(t h T ) hk eh ( ) k(t h P h T ) hk eh ( ) + k(i P h)t hk eh ( ) sup '6= j((t h P h T ) h ; ')) k'k H ( ) + c inf w h 2eX h kw h T hk eh ( ) : page 96 of Numer. Math. 69: 83{11 (1994)

15 Boundary element discretization of Poincare{Steklov operators 97 We have j((t h P h T ) h ; ')j k'k H ( ) c j((t h T ) h ; P h ')j kp h 'k H ( ) and since ((T h T ) h ; P h') = [ D (R N;h h u) E h (P h')d ' h R N;h h d =, 8 ' h 2 N h, and u denotes the solution of (1.1) with D 1 u = h. Therefore, using (3.2) k(t h T ) hk eh ( ) c sup '6= [ D (R N;h h u) E h (P h') d ke h (P h ')k eh ( [ D) c kr N;h h uk H ( [ D) and Theorem 2.3 yields the assertion. ut 4. Remarks In these concluding remarks we give an example of BE functions satisfying (3.1) and (3.2) and consider the BE discretization of PS operators in the case =. Let d = 2 and suppose that u and 1 u shall be approximated by linear boundary elements, i.e. e Xh and Y h consist of piecewise linear functions. It is natural to use ex h = o S1 () ; the set of continuous piecewise linear functions subordinate to a mesh = fx k g n k= on and vanishing at the endpoints x and x n of. There are many dierent possibilities to choose a projection P h onto Xh e bounded in H e ( ) and so to determine Y h = Im Ph. For example, denote by w k, k = 1; : : : ; n 1, the hat functions, i.e. w k 2 o S1 (), w k (x j ) = kj, k; j = 1; : : : ; n 1 ; and set X n1 w (x) = 1 k=1 w k (x) ; x 2 : Then S 1 () = span(w k ; k = ; : : : ; n 1) is the set of periodic linear splines, let P : L 2 ( )! S 1 () page 97 of Numer. Math. 69: 83{11 (1994)

16 98 G. Schmidt be the orthoprojection. By f' k g n1 S k= 1() we denote the biorthogonal base of fw k g n1 k=. We introduce P u := n1 X k=1 (u; ' k ) w k : Clearly P is bounded in e H ( ), Im P = o S1 () and Im P = span(' k ; k = 1; : : : ; n 1) = Y h S 1 (). The projections P are uniformly bounded in e H ( ) for any sequence of quasiuniform meshes, i.e. := max jx k x k1 j c min jx k x k1 j 1kn 1kn for any mesh = fx k g n k= with a constant c independent of. Indeed, for quasiuniform meshes we have and k' k L 2 ( ) kw k L 2 ( ) c kp uk L 2 ( ) kp uk L 2 ( ) + k(u; ' )w k L 2 ( ) ckuk L 2 ( ) : For u 2 e H 1 ( ) the interpolating spline Q u = which implies n1 X k=1 ku Q uk L 2 ( ) c kuk eh 1 ( ) ; u(x k ) w k j(u; ' )j = j(u Q u; ' )j c kuk eh 1 ( ) k' k L 2 ( ) : satises So the uniform boundedness of P and the inverse property of splines yield kp uk eh 1 ( ) kp uk eh 1 ( ) + k(u; ' ) w k H 1 ( ) Hence by interpolation kuk eh 1 ( ) + c kw k H 1 ( ) kuk eh 1 ( ) k' k L 2 ( ) c kuk eh 1 ( ) : kp uk eh ( ) c kuk eh ( ) ; 8 u 2 e H ( ). The extension property (3.2) holds for a wide class of piecewise polynomial functions, as shown in [11] for the case of Sobolev spaces H m ( ), m nonnegative integer, and mentioned in [6] for arbitrary H s ( ). Especially, (3.2) holds for the piecewise linear functions in H ( ). Now we apply the results of Sects. 2 and 3 in order to construct S h and T h if =, i.e. we consider the BE discretizations of the operators which map the boundary value of a harmonic function onto its normal derivative and vice versa. page 98 of Numer. Math. 69: 83{11 (1994)

17 Boundary element discretization of Poincare{Steklov operators 99 Since the Neumann problem for the Laplace equation in a bounded domain is not uniquely solvable these operators are not bijective and some modications are necessary. Let us denote by R() the set of constant functions on. It is well known that ker (I+K) = R() ; Im (I+K) = fv 2 H () : (v; V 1 1) L 2 () = g : (4.1) Then Im V 1 (I + K) = f 2 H () : ( ; 1) L 2 () = g =: H () ; and from (2.5) we obtain Su := n u = V 1 (I + K)u is a bijective mapping from the factor space H ()=R() onto H (). Remark that the two spaces are in duality with respect to the L 2 (){scalar product, (H ()) = H ()=R(). Therefore (2.4) implies that D : H ()=R()! H () is bijective, by D (1) we denote its inverse. From (4.1) we see that I K : H ()! H () ; hence the second equality of (2.5) leads to T n u := u = D (1) (I K ) n u : The relations DK = K D and KV = V K (cf. [5]) show that the above mentioned properties of the PS operators remain valid if T and S are considered as mappings acting between H ()=R() and H (). Under the natural assumption that R() X h where X h is the space of traces of nite elements v h 2 V h H 1 () we can construct as in Sect. 1 the FE discretizations of T and S which are symmetric isomorphisms between X h =R() and its dual. We show that the BE discretizations S h and T h constructed analogously to Sect. 3 preserve these properties. Suppose we are given spaces of trial functions X h H () for the approximation of uj and Y h H () for the approximation of n uj. As mentioned in Sect. 3 we have to assume that dim X h = dim Y h ; R() X h and X h \ Y h = ; : Then there exists a bounded projection onto X h, P h : H ()! X h such that Im P h = Y h, i.e. Y h can be identied with the dual of X h. For given h 2 X h the density ~ h of (2.13) is determined from page 99 of Numer. Math. 69: 83{11 (1994)

18 1 G. Schmidt (V ~ h; ' h ) L 2 () = ((I + K) h ; ' h ) L 2 () ; 8 ' h 2 Y h ; or, equivalently, from the projection equation (4.2) P h V ~ h = P h (I + K) h : Since R() = ker (I + K) X h it suces to restrict to h 2 e Xh := X h =R() = P h H ()=R() : On the other hand it follows that ker Ph (I + K )j Yh of (4.2) belongs to the set 6= ; and the solution ~ h Y h := f h 2 Y h : V h ker P h(i + K )j Yh g : But in general P h V e h 62 R() for some e h 2 ker P h (I + K )j Yh, such that ~ h 62 e Xh = fh 2 Y h : ( h ; 1) L 2 () = g = P h (H ()) =: e Yh : Therefore the mapping h 2 Xh e! ~ 2 Y h does not provide the properties of PS operators, it can happen that this mapping is not bijective, in general. But dening as in Sect. 3 we see that S h h = Ph(D h + (I + K ) ~ h) = (PhD + Ph(I + K )Ph(P h V Ph) 1 P h (I + K)) h S h e Xh = e Yh = e Xh : Applying the ideas of the proof of Theorem 3.1 we conclude that for S h : e Xh! ey h the assertions of this theorem are valid. For h 2 e Yh the construction analogous to Sect. 3 leads to T h h = (P h V + P h (I K)P h (P hdp h ) (1) P h(i K )) h ; which can be considered as mapping from e Yh to e Xh and the assertions of Theorem 3.2 remain true. Here (P h DP h) (1) denotes the inverse of P h DP h : e Xh! ey h. Finally we mention the matrix representation for S h and T h. Let fw k g n1 X k= h, f' k g n1 Y k= h be biorthogonal bases, (w j ; ' k ) L 2 () = jk, then where S h = D h + (I h + K h)v 1 h (I h + K h ) ; T h = V h + (I h K h )D (1) h (I h K h) ; (V h ) j;k = (V ' j ; ' k ) L 2 () ; (K h ) j;k = (Kw j ; ' k ) L 2 () = (Kh ) k;j ; (D h ) j;k = (Dw j ; w k ) L 2 () ; (I h ) j;k = (w j ; ' k ) L 2 () = jk : page 1 of Numer. Math. 69: 83{11 (1994)

19 Boundary element discretization of Poincare{Steklov operators 11 Especially we note the case X h = Y h which is often used. For a given basis fw k g n1 k= X h the biorthogonal basis is given by f' k g n1 k= = M 1 h fw kg n1 k= ; where M h denotes the mass matrix, (M h ) j;k = (w j ; w k ) L 2 (). Hence V h = M h 1 V hm h 1 with (V h ) j;k = (V w j ; w k ) L 2 () ; K h = K h M h 1 with (K h ) j;k = (Kw j ; w k ) L 2 () ; which implies the formulas S h = D h + (M h + M h 1 K h M h)v h 1 (M h + M h K h M h 1 ) ; T h = M h 1 Vh + (M h M h K h M h 1 )D(1) h (M h M h 1 K h M h) M 1 Acknowledgement. Thanks are due to Dr. H. Strese whose numerical experiments with a number of DD algorithms based on the BEM showed that naive BE discretizations of PS operators lead to nonstable iterative procedures and so stimulated the theoretical investigations. The author thanks also Prof. W.L. Wendland and Dr. B.N. Khoromskij for stimulating discussions and remarks. This work has been partially supported by the German Research Foundation under Grant Ko 182/2{1. h : References 1. Agoshkow, V.I. (1988): Poincare{Steklov's operators and domain decomposition methods in nite dimensional spaces. In: Glowinski, R. et. al., eds., Domain Decomposition Methods for Partial Dierential Equations I, 73{112, SIAM, Philadelphia 2. Costabel, M. (1988): Boundary integral operators on Lipschitz domains: elementary results. SIAM J. Math. Anal. 19, 613{ Costabel, M., Stephan, E.P. (1985): Boundary integral equations for mixed boundary value problems in polygonal domains and Galerkin approximations. In: Fiszdon, W. et. al., eds., Mathematical Models and Methods in Mechanics (1981), PWN, Warsaw, 175{ Costabel, M., Stephan, E.P. (1985a): The method of Mellin transformation for boundary integral equations on curves with corners. In: Gerasoulis, A. et. al., eds., Numerical Solutions of Singular Integral Equations, IMACS, 95{12 5. Costabel, M., Stephan, E.P. (1985b): A direct boundary integral equation method for transmission problems. J. Math. Anal. Appl. 16, 367{ Hebeker, F.K. (1991): On the parallel Schwarz algorithm for symmetric strongly elliptic integral equations. In: Glowinski, R. et. al., eds., Domain Decomposition Methods for Partial Dierential Equations IV, SIAM, Philadelphia, 382{ Lebedev, V.I., Agoshkow, V.I. (1983): Operatory Poincare{Steklova i ich primenenija v analize. OVM, Moskva 8. von Petersdor, T. (1989): Randwertprobleme der Elastizitatstheorie fur Polyeder{Singularitaten und Approximation mit Randelementmethoden. Dissertation, TH Darmstadt 9. Quarteroni, A., Valli, A. (1991): Theory and application of Steklov{Poincare operators for boundary value problems. In: Spigler R., ed., Applied and Industrial Mathematics, pp. 179{23, Kluwer, Dordrecht 1. Symm, G.T. (1967): Numerical mapping of exterior domains. Numer. Math. 1, 437{ Widlund, O.B. (1987): An extension theorem for nite element spaces with three applications. In: Hackbusch, W. et. al., eds., Numerical Techniques in Continuum Mechanics, pp. 11{122, Vieweg, Braunschweig This article was processed by the author using the LaT E X style le cljour1 from Springer-Verlag. page 11 of Numer. Math. 69: 83{11 (1994)