WarszawaAn analysis of the exterior Neumann problem for the Poisson equation in connection with a numerical procedure

ROCZNIKI POLSK1EGO TOWARZYSTWA M AT KM ATY CZNEG O Seria III: M ATEMATYKA STOSOVVANA XXXVI (1993)

T o ma s z R o l i n s k i

Warszawa

A n analysis o f th e exterior N eum ann problem for th e P oisson equation in connection

w ith a num erical procedure

(Received April 22, 1992)

This paper is concerned with transformation of the following exterior problem:

j -lu = / on

i du I l ^rlr = /y.

where ft C R w is a bounded region, Q c — int.(R“\f?), into a. problem represented by two erpiations. The first one is posed on a bounded domain and the second one is posed on the outer part of the boundary of the domain. This new problem is suitable for a numerical method based on the coupling of the finite and boundary element.

1. Introduction. In physics we often face the problem of finding a potential of a certain physical quantity. For example, a velocity potential in aerodynamics or an electric field potential in electrostatics. This leads to the exterior Dirichlet or Neumann problem for the Laplace or Poisson equation. The main difficulty in finding a numerical solution in this case is the unboundedness of the domain on which the problem is posed.

In the case of the Laplace equation we can transform the basic problem into an integral equation on the boundary of the domain by using the single or double layer representation of the solution to obtain a Fredholm equation of the second kind.

Another approach to the exterior Dirichlet problem for the Laplace equa-

tion in R1 * 3 was presented by J.C. Ned elec and J. Planchard in [8] where a

Fredholm equation of the first kind (the solution is represented by a single

layer potential) yielded a certain elliptic variational problem, with unknown

being the jump of the normal derivative of the solution across the boundary.

4 T. Rolinski

A similar method was applied by M. N. LeRoux in [6] for the plane Dirich- let problem together with a detailed analysis of convergence of numerical solutions to the exact solution.

A method of solving the exterior Neumann problem for the Laplace equation was presented by J. Giroire and J. C. Nedelec in [3]. The authors transformed the above problem to an elliptic variational problem on the boundary, the unknown of which was the jump of the solution across the boundary.

A method for numerical solution of the exterior boundary problem for the Poisson equation was presented by C. Johnson and J. C. Nedelec in [4] for the Dirichlet condition. The basic problem in this case was transformed to a problem represented by two equations. The first, was a Poisson equation on a bounded domain, the second was an integral equation on the outer part, of the boundary of the domain. The int egral equation was obtained by the Green formula for harmonic functions under the assumption that the solution of the basic problem was harmonic in the exterior of the domain.

In this paper we deal with the t ransformation of the exterior nonhomo- geneous Neumann problem for the Poisson equation into a problem posed on a bounded domain and on the outer part of the boundary of t he domain using the methods described in [4].

The existence and uniqueness of the solution to the basic problem is proved in the space of potentials introduced by M. N. LeRoux in [5]. It is proved that the associated Dirichlet form is elliptic in the quotient space of potentials modulo the constant functions. The proof is based on Lemma (I- 1-1) of [5]. Moreover, a general form of the Green formula from [1] is recalled to prove the equivalence between the basic problem and its variational form (see Th. 2.1). This formula is applied extensively throughout the paper. A similar form of the Green formula in R3 for potential spaces can be found in the paper by J. Giroire and J. C. Nedelec [3] (see Proposition 1-2). In the present paper attention is paid to the exact formulation of the conditions (the choice of proper spaces for the data and the solution, the assumptions concerning the boundary) for the equivalence of the basic problem and the transformed one. The complete exterior problem, i.e. under the assumption that the right-hand side of the Poisson equation can have an unbounded support (see (2.1)), is considered first. Then a problem with the restricted right-hand side is considered (see (3.2)) and the difference between the solution of the basic and the restricted problem is estimated (see Th. 3.1).

Finally, the equivalence between the transformed problem (3.10) and the restricted problem (3.2) is established.

According to the ideas from [3] the solution is sought in the quotient

spaces modulo the constant functions because the solution of the exterior

Neumann problem in the plane is defined up to a constant.

Exterior Neumann problem 5

The existence and uniqueness of the solution to the transformed problem is proved by means of the above mentioned equivalence (see Corollary 3.1), in contrast with [4], where the transformed problem was reduced to a Fredholm equation of the second kind (see 2.18 of [4]) and the uniqueness was proved for the latter equation.

The proof of regularity of the solution (see Th. 4.1) is based on the above mentioned equivalence, which leads to a simpler bilinear form (com- pare (6.4), (6.5) of [4] and the beginning of the proof of Th. 4.1 from the present paper). However, it should be noted that in [4] more general data for the transformed problem wa,s considered, requiring the use of a. more complicated bilinear form.

2. The basic problem. We are looking for a solution of the following problem:

J - V 2Av. = fi on Qc, i du\ _ „ l dv ~ 0 i

where Q C R 2 is an open bounded simply connected set with boundary iH, Qc = R 2 \ f? and : R2 h - R+ is the weight function, defined by V(x ) = y/\ + r2(l T log v/T+72), r2 = .r2 + xr,, x = (.iq, .r2) (the role of this function will be explained later on).

Assume that F is regular, i.e. it can be described by a finite number of C1 functions. In the classical approach we are looking for a solution u £ C2(QC) D C^(1?L), where (Ti' ) is defined in [2] (Tome 1, Chap. II, Par. 1, 3b). Roughly speaking consists of functions in C1 (T?c) n C°(f?L) for which it is possible to define an outer normal derivative in C°(F) on

The formulation of sufficient conditions on the functions / — fi/*P2 : Qc y— M.. g : /'i— R. on the boundary f and on the solution u £ C2(f?c) fl C\(Q ) to obtain existence and uniqueness modulo constants can be found in [2] (see Tome 1, Chap. 11. Par. 2, Sec. 3, Par. 3, Sec. 2, 3, 4, Par. 4. Sec. 4, 5). For example, assume that / £ C°(Q ) satisfies a local Holder condition and |/(.r)| < Cf\x\2As for |x | > It, It,£ > 0 (| • | denotes the Euclidean norm in R n, n £ N), g £ C0(F). We also assume the following solvability condition:

J / dx -1- f g do - 0 ,

S2C r

where F is described locally by a finite number of functions with first deriva-

tives satisfying the Holder condition. Then there exists a classical solution

u £ C2(QC) fl Cl(7t) satisfying | grad u\ »— 0 as |;r| oo, which is unique

up to an additive constant.

6 T. Kolinski

111 this paper we discuss a. more general situation where /j G IF°( i7c), g G / / “1//2( T), the boundary / is Lipschitz continuous, and we look for a solution u belonging to the space IF1(J?C, A). The above mentioned spaces can be considered as subspaces of spaces of distributions, and they are de- fined below. The derivatives in the formulation of the problem are to be understood in the distributional sense.

Let

W °(fic) = {v e V '( n c) : ip-'v G L2{Qc)} . The norm in VF° is denoted by || • ||o,i/',i?c, and defined by

IMIo,v,nc — 11^ Vu G IF°(j?c),

where |j • |jo,/2c denotes the norm in L2(i?c) and V'{Qc) is the space of distributions defined on Qc.

Let

W l(Qc) = {v G IF°(J?C) : axi G /,2(J?C), i = 1,2} . The norm in IF1 is denoted by || • and defined by

v\\i,^,nc — IMIo,0\f?c + dv

Ox I +

o ,r?c

Ov

Oxo Vo G \Vl (Qc).

0.1?<

Let

IL1^ , A) = {v G H ' ffT’ ) : V2Av G lF°(f?c)} . The norm in IF1 (i2c, _A) is denoted by || • || and defined by

IMIi,<£,i?c,^ = ^WvW\,v,s?c + £ IF1 ( ^A ).

The space / / -1/2(. ^j T) is the dual of I I ^ 2( /'). and the latter is the space of traces of functions from the space A definition of this last space can be found in [7],

Notice that for any v G IF1(i?c) the restriction of v to where J? C i? and Q is a bounded domain, belongs to f l 1 (f?|) and ||v||i,/?j <

C\\v\\itf}Qc, where || • ||i,j?i denotes the standard norm of IIA{Q\). For v G H l {Q\) we have tr v\r G / / 1 ^2( / 1). which means || tr c||| /2,r < C||e|| i ^ , where || • \\\/ 2 ,r denotes the standard norm of / / 1 /2( / ’). This property to- gether with a definition of J[l^2( f ) can be found in [7], The above implies that any v G VF1 (T2C) has t r r |r £ f f l^2{ f) : || tr *’|r’||i/2,r <

A generalized solution of (2.1) is defined as a solution of the following variational problem: find u G IF1 (T2C) such that

a(u,v) = (fu v)t + (<h tr»>, Vi; € H'‘(CC),

(

2

2

Exterior Neumann problem

where

a{v, -w) = 1=1 nc J (/ it i CJi dx, Ve, w £ H7l(f?c) , and (-,-)& denotes the scalar product; in IF°( l?c):

(v,u>)tf/= j —p -r/x , Vt\ w £ W°(QC) . nc

From now on we make the convention that round brackets ( , ) denote an inner product, and angle brackets {, ) denote a dual pair. In the latter case the functional is always in the first place. If there is any doubt as to what, spaces we have in mind they will be written explicitly as subscripts.

T h e o r e m 2.1 Problems (2.1) and (2.2) are equivalent, i.e. if u £ IF1 (ftc, _\) is a solution to (2.1), then it is a solution to (2.2). Conversely, if u £ II7l(i?c) is a solution to (2.2), then it belongs to H7l(f?c, A) and it is a solution to (2.1).

P roof. First, we check that the kernel of the trace operator tr : W 1(ftc) , / / 1/2(T) is dense in W°{QC). Obviously, the V{ftc) functions belong to the kernel. Take any 5 > 0 and v £ 1F°(/?C). We can always find a domain ft C ftc such that the function vr : ftc t— R defined by

7,y(;r)

v[x), x £ ft, _ 0, x £ i7c\i7

satisfies ||?;r — u||o)^,r;c < c/2. The restriction of vr to ft is in If1 {ft). Since T>{ft) is dense in Is {ft), we can find tp £ V{ftc) such that ||cr — v?||o.</v<?c <

e/2. Hence ||<^ — < £, and the required density follows.

Due to the above density argument we can apply the generalized Green formula to be found in [1]:

(2.3) ^{a{ v, iv} ) = ( ^{—W2A v , w )$} + ^(Xv, tr ^tv) , Ve £ I F1 ( ^ftc, Ji),

Vie £ IF1 (J?°), where the operator X : IF1 (Qc, _A) i— / / _1/2(T) is an extension of the operator (the interior normal derivative on F).

If u £ l'F1(/?c, W) is a solution to (2.1), then by (2.3) it is also a solution to (2.2). Conversely, if u £ IF1 (J?c) is a solution to (2.2), then by the definition of distributional derivatives it also belongs to IF1 (ftc, _\). Formula (2.3) can be applied once again to show that u £ VI7l(f?c, zi) is a solution to (2.1). ■

Let; us now consider the existence and uniqueness of a solution to problem

(

2

2

8 T. Rolinski

T h e o r e m 2.2. Assume the following solvability condition:

(2.4) ( f i , 1 -f {</. l)//-]/2Xjp/i/2 = 0 .

Then the solution to problem, (2.2) exists and is unique up to a constant.

P ro o f. First, we prove that the form a, from (2.2) is elliptic in W l(ftc)/Po* where / o is the space of polynomial Is of degree zero. Take two smooth functions y?, xf : E 2 y~ R such that

1 ^<p(x) 0 < <p(x) < 1, ^{+ <Hz ) = I ,}

V?(*) = 0, |*| > Ru

<p{x) = 1, |x | < R R < R u

where R is so large that i? C B{0, ll). B{0, R) is the ball with centre 0 and radius R.

Take v G W l (Qc). We have ifv G H'o (l?c), where Vl^(«c) = {» € ll ' u n : trt-lr = 0} . By Proposition I—1—1 of [5],

< CT<Hi,ac*

where

li,nc - dv

dxi

dv

dx) Vt> G Hrl(i?c) Together with the definition of V’ this yields

lkH li.*,tf' < C'{|Mlo,r + h’li,r?^} ,

where T = T{ 0, 7?,, R i ) denotes the annulus with radii R and Ri and centre 0. Moreover (see [7]),

IMI ^o . ^t < c {1 ’|i/r + | J ^{vdx }}

T Hence

Similarly,

+ | f vdx T

Therefore

lb*’lli.*./?c < + | ^{J vdx}

T

|h;||l.</',J?c < 6'{| <’lu?- + | j vdx }•

By choosing an appropriate const ant c we get

||r + f||i,#,/7« < C’|r|i,«<, V v e w ' ( n c).

Exterior Neumann problem 9

From the above we infer

Plk*,tf</Po < c'M i.rt'. v,,-e H''(«'■).

where || • ||i,^.j?c/p0 denotes the standard norm in W l{fic)/P{), v G ? G W l {Qc)f P q . This implies

a(v, v ) = |e|ii/?c > T'2p ||2 ^ c/Po, Vy G IF1 (f?,?).

Hence tlie ellipticity of the form a in \VX{QC)/P q . Condition (2.4) im- plies that the right-hand side of (2.2) can be viewed as a functional in (IT1! i?c )/Po)', and the theorem is a consequence of the Lax-Milgram theo- rem. ■

3. T he transform ed problem . Constructing a finite approximation of problem (2.1) is troublesome due to the unboundedness of Qc. Thus our aim is to construct a problem posed on a bounded domain and approximating problem (2.1). To this end, let Q be an open bounded simply connected set such that Q C 12. We define i?j = Q\Q. 1?> = K2\f?. Moreover, define h r : Cc ^ R 2 by

f fir(x) = f \( ) +

for x G C ],

\/ir(a-,) = 0 for .T G

where the constant c is chosen in such a way that the solvability condition is satisfied:

(3.1) (/lri I)*/' + (.(/' fpi~ ~ 0-

Then the problem to be solved reads: find ur G IT1 (i?r, ) such that j —lP2A ur = h r on J?C

(’>•2) S | _

l 1hT\r ~ fJ-

The above problem approximates problem (2.1) as shown by the following theorem:

T h e o r e m 3.1. If u is a solution to (2.1) and ur is a solution to (3.2), then

||w— 'b’||i,i^j?c/p0 < c{^ J

P ro o f. First notice that problem (3.2) has the following variational formulation: find ur G ITI(l?r ) such that

(3.3) «(ur, v) = ( / lr, <•)* + {,I. trv). Vt> e u ' . i n .

This problem can be derived in exactly the same way as (2.2). The ellipticity

of a in lT1(i?c)//Jo, the formulation of problems (2.2) and (3.2), and the

10 T. Roliriski

solvability conditions (2.4) and (3.1) imply

C || W— ur\\i^Mc/Po < ||/l ~ /l r ||o, Qc || — «r||o,!P ,Qe / PQ » where || • ||o.^,r?^/p0 is the standard norm in IT°( Q c )/P q . Ilence (3-4) II« - Ur\\lL,V,nc/P0 < C\\fi - flr\\o,^J2c ■ Moreover,

(3.5) < ( / c2/'? 2</;t)1/2+ ( j = h + I2 ,

sh n 2

where the constant c is from the definition of f \ r. In view of (3.1) and (2.4) this constant satisfies

This leads to (3.6)

f c/V2dx = f fj/& 2fl.V.

S21 Qo

n <

L : 1 <>' j lx.

f? 2

Now (3.4)—(3.6) yield the assertion. ■

The problem announced at the beginning of this section can be derived from problem (3.2) by decomposition: find mj £ IIA{Q\% _A), no G IT1 (J?2, ) such that

a) — Aui = f Oil Qx

b) — Alio = 0 on i?2

c) tr U\ = tr u -2 on A

d) 1-ILL | r _ 8 ^U-2 1

dn ’ t i Oil

e) ^du-i |

l On O = y on 1\

where is the boundary of f?o, / G Lr{Q\), f(x) = f i A x ) / ^ 2, V:r G ,

«,• = i = 1 ,2 and ^ denotes the interior normal derivative on 7) or T. The space /T1(J?i,zA) is defined as follows:

//'(V i, -1) = {t•€ //'(/?, ) : J r e )} , with the norm

= IMlU + IM»'llo,n,. e J).

The space ITJ(1?2,Z\) is defined analogously to IT1! i?c, _A).

Problem (3.7) can be transformed to a variational problem posed on l?i and Ti since «■> = \n2 1S harmonic in 1?2.

Once again we recall the generalized Green formula from [1] for the spaces

L2(Q i ), H l/2(r) and the trace operator tr : /7l(i? i) 7/1//2(clJ?i).

Exterior Neumann problem 11

This formula and equations (3.7a, e) imply (3.10a), where Ai =

denotes the interior normal derivative. Since uo is harmonic in Q-i (equation (3.7b)) we can apply the Green formula for harmonic functions (see [2]) to obtain a. relation between tru jl/j and Ai:

(3.8) = - ^ J ~^-log|.r - y\u\{y)d(Tll

* ri Uy

- J log \x - 2/|A ⁱ Aay + a0 ,

where ciq is a constant. The Green formula (2.3) for the spaces IT1 (i?2, Zi), 1T°(1?2). 771//2(A ) and equation (3.7b) yield

(i 2 (U‘2, v ) = (Ai, tr v)px, Vu G W 1 (i?2), where

a2{w, e) = ]T f

i—1

dw dv

T h td T i'* ^Vw.\« € w 1{ni).

Hence (Ai,l)/^ = 0. If iio G z\), then Ai G H 1/2(JTi), where H _ 1 /2(A, ) is defined in (3.11) below. In case Ai G Z/2(A )i we have

(3.9) J Ai (ler = 0 .

A

Multiplying (3.8) by a function // G />2(A ) satisfying (3.9) and integrating along A we arrive at (3.10b). The form b\ from this equation is defined for regular functions, i.e. those in L2{J\ ). However (see [5], [6] ), it can be extended continuously onto 7/- l /2(A ) X / / -1/2( A )• Thus the problem to be solved reads: find u = («i,Ai) G H 1(Qi) X / / _1/2(A ) such that

(3.10) where

a) f «i ( ^mi , v) + (Ai, tr v)Fl = (/, v)L* + (g, tr v)r , Vv G H l(Qi), b) I 26i(A!,//) — (//., trMi)r ,+ 2(/t,G '„trtti)ra = 0 , V/iG H 1/2(A ),

(3.11) / f - ’/2(A ) = {/< e i r ' /2(A ) •• ( / « . = 0},

and / / -1/2( A ) is defined analogously to / / _1/2( / ’). The forms and operators in (3.10) are defined as follows:

2

« , ( » , to) = ] T / V t . , <« € # » « ? , ) ,

j=i r?i

12 T. Rolinski

6l((9,/0 = - 1 2 IT f j log I ^{X - y\0(y} )// ( ^{X) day derx,}

i\ r i

V0,// ^g l 2(T i ) n i / -1/3(.r1), (On V)(.1*) = ~ / 27r tMv log |.r - y\v(y) dt ^t (/, V/; € //1/2( / i ).

J i

T h e o r e m 3.2. //?/,. /$• a solution to (3.2), then the pair (v/1, A]), where

u

| , Ai = 1 r t , is a solution to (3.10). Conversely

if (u |.A|) /\$ a

solution to

3

10

then there exists a harmonic extension u £ U 1 ( Q

of Ml to the whole f?c and this extension is a solution to (3.2).

P ro o f. The first part of the theorem comes from the derivation of prob- lem (3.10).

Now assume that we have a solution (u[, A[) of (3.10). Then there exists a solution to : i?2 ^ K to the following Dirichlet problem: find ^m 2 £ VV'1 (/?•>) such that.

j —A u -2 = 0 on f?2 *

\ lr R•_>!/•, = tTMi| r, on 1\.

For this function we can derive an equation similar to (3.10b):

(3.12) 26 ^j (A2,//) - (//, tr io) + 2(//,Gn trio ) = 0. V//. £ / / _1/2( A ), where A2 = - £ Jl~]d2{ /',). If we subtract (3.12) from (3.10b), then, bearing in mind trM]|r, = triol/y. we get

4|(A2 - A 1,//) = 0,V/16 / / - 1/- ( r 1).

Tlie form hi is elliptic in t) (see [5], [(>]). Hence A_> — A.. Let »,■ : Qc h -* R be such that ur |^. = Mj, / = 1,2. We know that

f — A u r = f in 1? i,

| — _ A = 0 in 1?2.

We want to check that

(3.13) {—A u r.<f)x»xv — ( ^ £ Th!?0).

The definition of distributional derivatives implies

{ - A u r ,(p)wxv = « i ( « i ? ¥>) + « 2 ( « 2 ^ ) » Vy? € '!>( J 7 ^{e ) .}

The Green formula, applied to the forms a\ and «2 yields

«i(?ti,^) = ( / ,9 ) l - j - (Ai. tr^>A ,

« 2 ( * 9 ) = ( A •>, t.r 9 ) r , , Vy? € I>( l ? c ) .

Exterior Neumann problem 13

Adding the above equalities and taking into account Aj = A•> and f(x) = f\rix )/W2* V. t G i?i we get (3.13). Hence — iP2Aur = / lr in i?c. Obviously

|r = <h by (3.'10a). This shows the second part, of the theorem. ■ R e m a r k 3.1. A transformation of this type was first, applied in [4] to the Dirichlet problem for the Poisson equation.

COROLLARY 3.1. Assume that the solvability condition (3.1) is satisfied.

Then there exists a solution (?/,i, A j ) of (3.10) and it is unique up to a con- stant in the first component of the solution.

P r oof. Assume that we have two solutions (u], A{), ( uj, Aj) of (3.10). By Theorem 3.2 the harmonic extensions uj., u2r of ?/,j and ?/> satisfy (3.2). The solvability condition (3.1) and Theorem 2.2 imply uj, = u2 + c, where c is a constant. The first, equation of (3.10) yields A] = ^ - |r i- Af = ^ - | r j . Hence A} = Aj. The existence of a solution comes from the solvability condition (3.1) and Theorem 2.2. ■

4. A regularity result. In this section we consider the regularity of a solution u = ( mi , Aj) to (3.10). This is important for the numerical analysis of this problem.

T h e o r e m 4.1. Assume that f G lfm(Qi), fj G //m+' /2(/) , m G N, the boundary isCm+l'1. i.e. it can be described locally by functions with Lip- schilz continuous m+1 derivatives, and there exists a solution u = (?/.j, Aj) G IIx{(h) X H - XI2{1\) to (3.10). Then u G / / m+2(f?i) x ]fm+l/2( r\)-

P roof. The proof is by induction. First, we establish regularity of near the boundary T\ in the direction tangent to the boundary. Then we prove full regularity near the boundary by making use of the relation between partial derivatives given by equation (3.10a) of the problem. Finally, we establish global regularity of u\ and, consequently, the regularity of Aj.

From Theorem 3.2 we obtain

«(Mr,<F) = iflri'p)*, VT €

where ur is the harmonic extension of ?/]. In general. ur cannot be regular near /"| in the normal direction due to the definition of / lr. Thus we check the regularity in the tangent direction. Assume that; the theorem holds for in — 1, i.e.

/ e i r ~ l((io ,ge //" ‘- ' / - ( D => m € A, € //">-'/-’( / j ).

We prove that it holds for m, i.e.

/ € H m(Q\ ),.(/ £ //”*+1/2(7’) => u, € Ilm + !(Oi), A, 6 n m+,ll(r,)-

For rn — 0 the proof is the same as for the induct ion step.

14 T. Rolinski

Let then / £ I t m(l?i ), fj € / / m+1/2( £). Let xq be a certain point on the boundary I\ and C a neighbourliood of ;i‘o in R2.Take a diffeomorphism 0, with 0~l being Cm+I J , transforming C onto a disc V in R2. We assume that the images of Co = C D F] , C_ = C fl i?] and C+ = C D i?> under 0 are P 0 = V D {(ij\, ]j 2 ) : 1/2 = 0}, FI- = V f) {( //1, y -2 ) : 1/2 < 0} and V + = V ft {(t/i, y2) : y -2 >0}, respectively.

Let <fQ be a function from the space C°°(R2) with <^0 = f in the neigh- bourhood of .i’o £ -Ti and supp^o C C. In view of

«.(«r, <y?ov) = (/, ^on)L2(f?1), Vc £ IF1 (Qc) , we arrive at

(4.1) «(**«„ ») = » W „ + £ / <**.

1=1 /?c x 1 1 /

Vr £ H'l (/?c).

Define the form d : {II q (V))2 y— R by

c/(w?, r>) = «(//> o 0, c o 0), Vw, /; £ II q (V) ,

where the bar denotes the extension by zero of a function from H q {C) onto the whole plane R 2 and o denotes the composition of two functions. Then by the Friedrichs inequality,

(4.2) |M|i,z> < C sup I, ' ? \ Vw £ II q (V) . veH^(V) \M\i,v

Let u = (y?owr) o 0 1. Then u £ //<]{V). By (4.1) and the definition of w, (4.3) d(u, iv) = a(ipQUr , w o 0) = (<p0/« w o 61)

^ f 7^90 dw 0 0 dipo dllr---^

+ £ J F i = 1 r?£ 1 u - ^ 7 o r r 0 "J ^{1 1} / v" ' 6 J1° {P) ■ The next step is to prove that ^ * 4 ^{“ € T/}

^{CP). T}

this end let us introduce the finite difference:

A hw(y) =

+ h) - w(y))/h1, y £ V, w £ H i(V ) ,

where h = (h\,0) and w denol.es the extension of w by zero onto R 2. No- tice that if supp w C T>, then Aiiw(y) £ //(}( P) for h sufficiently small.

Furthermore,

(4.4) y " J JL-(Ahw ) o 0 J L v o 0 J ( 0 1) dy

i ^{- 1} V Xt ^{d x i}

Exterior Neumann problem 15

= - J J ^ r w 0 0 l )du+ h(uEV) ,

«=1 V 1

where supp w C V, w, v £ II1 (V), and

< C Z J

i - 1 V

( 7 o O-^—v o f)J( 6 1) dy c)xi\()yi J ()xi

= ~ ^ 2 J ~T~W o()T ~ ( TT“ r ) o0J{9 1) dy + / 2( w, y ), _{i= 1 V} Oxi \ dy { where w £ //(f (T>), v € If 2{V), and

A' < C’IMI ⁱ / ^p IMI ⁱ . ^p * * = .

Formulae (4.4), (4.5) can be rewritten in the following way:

(4.6) r/( v) = -d(w, A kv) + I\ ( «■’, c ),

(4.7) d (*Z - X 7 9P

d { ^ ' V = - d [ w' o i ; ) + Ii(w’v h We use formula (4.7) in times and formula. (4.6) once to obtain

/ <)m v \ , f i)m v \

(4.8, — , , ) = < - ! ) - > , ) + / , ( * , - ) . Expression G can be estimated as follows:

|C (?M;)| < C'j|//||m4-i,p||7;||i,t>.

Now we deal with the first term of the right-hand side of (4.8). Rewrite formula (4.3) as

( d m v \ ( d m v

(‘UJ) d \ % A h d W )

+ £ f ( ^ n J L ( 1 „ ^ 1 ) 0 0 - * I ) A*

dxi dxi

i— 1 (2

— + h •

Change of variables yields l)mv

0y\ dxj d x i \ dy\

( / i 110 71 \ / f l i

^ ' w ) olldx= s v o o O - ' f o 0 - l A„— u r l )d9 . By the Gauss formula

f c 0 U ^ /3-t f ^ /J-1 7/tf"1

16 T. Rolinski

Hence

|c4 | < C '| j/ !| , „ , 4 r _ ||H |i.T > -

It is easy to see that the form 75 can be estimated as follows:

1-^5 | < ^',H wr||m + l,c|| ,;l|l,r • Estimate (4.2) and equalities (4.8), (4.9) yield

, dmur

-T/i .),tlm cah

ci m +1 —

The last estimate implies G // ^q (X>), which in turn yields (4.10) 0 m + 2 u o _ 0 m + 2 u r , , _

---T t i ' I P ) -

We cannot expect that u G II™*2(V) for the reasons explained above, but we can prove that 7?|z>_ G JJm+2(V-) using the relation between partial derivatives given by the equation itself. By virtue of the formula for the derivative of the product of two functions we arrive at

E ()^P0 Oily

~d 7 ^{^~d} 7 ^‘

i - 1 ‘

1 Since by assumption f \ n } G H m{Q\) and u G 7/m+1(i?1), we get

02 (<po^r) . d 2 ^(<p 0 ^ur)

where // G IIm{V_ ). In view of ipour — uoO, by writing out the expressions

~{ax°~ ? * = we arrive at

d2u d 2 a ,

° 7TT + ft o y z a t

0 2 u c

+ 7 Tpv + ^

ih Gift n

where cv, ft, 7 , <*> are sufficiently smooth functions and a > 0. Let us differen- tiate both sides of (4.11) in times: m — l times with respect to y\ and / times with respect to y>, I — 0,1,..., m. As a consequence, by (4.10), we see that all the derivatives of u of order m-f 2 are in L2[V-). Hence Ti G 7/m + 2(X,_ ), which implies <poUr G Ilm+2(C- ). The fact that u\ G 7/m+2(i?i) is a simple consequence of the last remark and general results concerning solutions of elliptic problems (see [7]). These results give us the local regularity and the regularity near the boundary F. Since A] = 1 / \ , it is then easy to see that A! G I /m+1/'2(Fi). ■

In the next paper we will show how to find an approximate solution to

problem (2.1) using the theorems of this paper.

Exterior Neumann problem 17

A cknow ledgem ents. I wish to express my gratitude to Dr. A. Wakulicz for his advice concerning this paper.

References

[1] J. P. Au bin, 1972, Approximation of elliptic boundary value problems, New York:

W iley-Interscience.

[2] R. D a u t r a y , J. L. L ion s, 198-1, Analyse mathematique el calc it I numerique pour les sciences et les techniques, Paris, New York: Masson.

[3] J. G ir o ir e , J. 0 . N e d e le c , 1978, Numerical solution of the exterior Neumann prob- lem using a double layer potential, Mailt. Coin put. 32, 973-990.

[4] C. .foil n son , J. C. N e d e le c , 1980, On the coupling of boundary integral and finite element methods, Math. Com put.. 35, 1063-1079.

[5] M. N. L e R o n x , 1971, Resolution numerique du probleme du potentiel dans le plan par tine methode variationnelle d'etemenls finis, These, L’Universite de Rennes, U.E.R., Matheniatk|ues et Inforinatiqne.

[6] M. N. L e R o n x , 1977, Methode d ’elements finis pour la resolution numerique de problemes exterieurs en dimension 2, RAIRO Ninner. Anal. 11, 27-60.

[7] J. N e c a s , 1967, Les methodes direcles en theorie des equations ellipliques, Prague:

Academia.

[8] J. C. N e d e le c , .1. P la n c h a r d , 1973, (hie methode variationnelle d'elements finis pour la resolution numerique d'urt probleme exterieur dans R'1, RAIRO 7 R-3, 105- 129.

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