ROCZNIKI PO LSKIEGO TOVVARZYSTWA M AT 15 M AT Y C- Z N EG O Seria III: M A TE M A TY K A STOSO W ANA XXXVI (1993)
To m a s z Ro l i n s k i
Warszawa
A m eth od for solving the N eum ann problem for th e P oisson equation
on th e exterior o f a polygon
( Received April 22, 1993)
This paper is concerned with a numerical method of solving the following problem:
J An = / in Q c,
l
where Q C ./?“ is a polygon, T is the boundary of i?. The method is based on the coupling of the finite and boundary element techniques. To compensate the loss of smoothness of the solution u near the corners of the polygon /? we refine the triangulation without changing the number of triangles. We apply the affine triangular Lagrangean element of degree A: £ N and the Lagrangean boundary element of degree A — 1 to obtain the optimal order of convergence via the Galerkin projection.
1. In tro d u c tio n . The problem announced in the title appears when looking for a field potential, e.g. the velocity potential in aerodynamics or elect ric field potential in electrost atics.
The main difficulties in finding an approximate solution to this problem are the unboundedness of the domain in which a.solution is to be found and t he irregularity of the boundary. Bot h difficult ies appear naturally if we look for the velocity potential of a perfect incompressible fluid around an aerofoil profile or the electric field potential induced by a conductor represented by the domain Q.
To overcome the difficulty related to the unboundedness of the domain we have chosen the technique known as the coupling of FEM and BEM.
AMS(MOS) subject, classifications 65N30. (>5N35, 65N50, 65NT15
20 T. Roliriski
A mathematical justification for this procedure was first given in the paper by C. Johnson and J. C. Nedelec [9]. This idea was given treatment in many papers, to mention but three: W. Wendland [19], G. 0. Ilsiao [S] and M. Costabel [3]. Generally, we apply FEM to domains where inhomogeneities might appear (the function /) , and BEM to exterior domains. Coupling of both methods seems to be particularly efficient when both problems are present, which is the case for our problem.
The basic problem is formulated bv making use of some spaces of poten- tials (see (2.2)). A similar application of these spaces was presented in the papers by M. N. LeKoux [12]. [11]. According to the procedure of coupling of FEM and BEM problem (2.3) is transformed to a problem posed on a bounded domain Q\ and the coupling boundary 1\ (see Fig. 1 and (3.1)), under the assumption supp/j C / U i?j. 'Then the problem of existence, uniqueness and regularity of a solution to the transformed problem (3.1) is considered. The results of the above problems, coming partly from [15], are combined in Sec. 2 and 3.
In contrast to [15] we introduce irregular boundaries / ’ and / j , which leads to different regularity results (see Th. 3.2, Th. 3.3; compare TJi. 3.3 of the present paper and Th. 1.1 of [15]). The proof of Th. 4.1 of [15] was given under the assumption supp f\ C Q\, the assumption in this paper being supp fi C / U Q \ . The former, being more general, leads to a more complicated proof because we cannot expect local regularity of the har- monic extension of the solution onto the set Qc = int(R2\f?) (see proof of Th. 3.2).
In Section 4 we prove the Garding inequality, which implies strong ellip- ticit.y, for problem (3.1) under the assumption that; the angles at the corners of the coupling boundary 7j (see Fig. 2 ) are close enough to the straight
The Neumann problem 21
angle. This can be viewed as an approximation of the case of the smooth boundary . It is known from [9] that if I\ is smooth strong ellipticity is a consequence of compactness of a. double layer operator (see (3.1)). In our case compactness is violated as pointed out in the paper by M. Costabel and E. Stephan [4]. By redefining this operator it is easy to represent it as the sum of an operator with small norm (see Lemma 1.1) and a compact operator, which gives the desired (larding inequality (see Th. 4.1). Due to this approach we can define exact triangulations of
We propose the Galerkin method with finite-dimensional spaces based on the Lagrangean finite element of degree k £ N (see Sec. 6) and the La- grangean boundary element of degree k — 1 (see Sec. 7). In the case of strong ellipticity of the operator the approximation error can be estimated by the interpolation error (see Th. 4.2, Cor. 4.1), which in turn depends on the smoothness of the exact solution. The smoothness is violated if the boundary f is not regular, but the behaviour of the solution for the two-dimensional case is well-known (see V.A. Kondrat’ev [10], P. Grisvard [7], M. Costabel and E. Stephan [4]). This can be used for construction of finite-dimensional spaces in such a way as to have the same approximation error as for a smooth solution. The construction in this paper is based on definition of triangulations so that the triangles are condensed in the neighbourhood of the corners (see Lemma 5.1). A similar technique of refined triangulations for solving elliptic problems in irregular domains had already been studied for some cases (see e.g. V. V. Shaldurov [17] for Lagrangean elements of order 1, and A. IT. Shatz and L. B. Wahlbin [16] for general approximating spaces). In [16] the authors obtained estimates in the maximum norm. In the present paper we consider Lagrangean finite element s of degree k £ N and obtain estimates in the / / 1 Sobolev norm, the definition of 1 riangulations being simple and general (see Sec. 5).
In Section 6 we define finite element spaces based on the refined trian- gulatioiis. The main result here is Theorem 6.2 concerning interpolation of power type functions from the representation of the solution. It turns out that using the Lagrangean finite element of degree k £ N and refined trian- gulations satisfying a certain condition (see (6.3b)), we get the k — th order of convergence.
In Section 7 we complete the numerical analysis. We define finite element spaces on fi\, and boundary element spaces on G . In the latter case we use the interpolation result of [12] (see (7.8)). By choosing the Lagrangean finite element of degree k £ N, the Lagrangean boundary element of degree k - 1, the triangulations defined in Section 5 satisfying condition (7.1), and partitions of the boundary G conforming with the triangulations, we obtain the k — th order of convergence under the assumption / £ l l k~x(Q\ ), g £ JIk~l / 2( J\ ) (see Cor. 7.1).
22 T. Rolinski
2. Basic problem . In this section we present the problem to be solved numerically:
(2.1) ■V2Au = /i
tlvtin r = 27 in Qc, on r ,
where i? C M2 is an open simply connected domain with Lipschitz boundary V, and i?c = int(]R2 \ 12); ^ : M2 i—- R + is a weight function defined by:
&(x) = a/ 1 + r 2 ( l + log v71 + r2), r2 = .r f + a-.2, .r = ( x {, .r2).
Problem (2.1) is to be understood in the distributional sense; in partic- ular, fi G V { Qc) and the outward normal derivative S i p is an element of
■D'iD.
Let us define the following spaces:
IT°( i?c) = {v G V'{QC) : \P~'v G L2(f?c)} ,
(2.2) = {(- € : Y v 6 [ i 3{/?e)]2}-,
w ' ( n \ A ) = { » e i i 'V c ) } , with .norms defined as follows:
I ??I VV ° ( L ‘ (S7C)-
ll^llu'Mt?0) “ Mh/0(i?c) + I^Hf/y-(rc)p ?■
— IMIiV'tr?*) + \ ^ “ A ^’1 Iv'°(q c) ’
where V = The above spaces were used in a similar context in the literature (see [14], [6], [12], [11]).
We will assume g to belong to // ~ l//2( / ’). the dual of 7/1//2( / ’), the latter being the space of traces on F of functions from the Sobolev space
(see e.g. [13]).
Now it is possible to state problem (2.1) in a rigorous way: find u G W X{QC, A) such that
(2.3) — iJ/2A u = f [ in f2°,
S i r = (J on 1\
where f x G H/0(/?c), g G H ~ ' / 2( F).
Since v G IP 1 (12'') has a trace tr v on F (see [15]), we can look for a gen- eralized solution of (2.3 ) as a solution of the following variational problem:
find u G H7l (J?c) such that (2.4)
where
a{«, v ) = (/]. v )<j, 4- (g, tr v). Vc G IT1 (Qc)
'=i nc Vv. w G IT1 (Qc),
The Neumann problem 23
and (•, -)q, denotes the inner product in 1F°( Qc):
(V. «■’V = f ^ dx, Vo, tv € H/0( 12c) . Qc
We jnake the convention that; the round brackets ( , ) denote an inner prod- uct;, and the 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.
Let us cite two theorems of [15] concerning problems (2.3) and (2.4) (see proofs of Ths. 2.1 and 2.2 of [1.5]):
'Theo rem 2.1. Let f\, g. u, f be a.s above. 'Then problems (2.3) and (2.4) are equivalent., i.e. if u £ IT1( P C,_A) is a solution to (2.3), then it is a solution to (2.4). Conversely, if u £ I f 1 (Qc) is a solution to (2.4). then it belongs to IT1 (Qc, A) and is a solution to problem (2.3).
Th eo rem 2.2. Let /j, g, ?/,. F be as in Theorem 2.1, and assume the following solvability condition:
(2.5) (/i1 1 )$- + (<L 0 / / - 1/2<r)x//1/-(r) — 0-
Then the solution to problem (2.4) exists and is unique up to a constant.
3. C oupled problem . The unboundedness of J?c is the main obstacle in constructing finite-dimensional approximation of problem (2.3). Let i? be an open bounded simply connected domain satisfying Q C 12, with piecewise smooth boundary L\. Let i?y = i?\i? (see Fig. 2). If we assume that /]
in the right-hand side of (2.3) satisfies supp /j C Th U then problem (2.3) can be transformed to a problem posed on $1\ and I \ , namely: find (mi. A,) £ / / ‘(P i) x i i - l/ 2(J\) such that
(3 l) a) { ">("1' '•) + (Ai,tr-/.’)r, = if: v )iT + (if.tf ?>), Vc £ l C { Qy ) , b) l (iL VA)r, - (//. t i + {//., AJ(trui ))r.t = 0. V// £ /V-1 /2( l\), where
W -1/20 '1) = {/<€ / / - 1/2( A ):</«, )} = 0}.
A x ) = h (O / 'P H x ) , V.r 6 1?,, and
« , ( » , « > ) = T , J
i— 1 S21
<9v.’ <9<r Vt\ tc £ ),
(W)(.T) = - 1
7T
(ALn)(.r) = n_i_
7T / T ” log |.r - /y| c(?/) c/cxy, dny V v e n l/2( r i ),
24 T. Roliiiski
where ay is the arc measure on 7\. We assume the normal derivative ^ to be directed to the interior of Q \.
The operator V is a single layer potential operator, and K is a dou- ble layer potential operator. To define these operators we assume that the boundary /A is piecewise smooth.
The relation between problems (2.3) and (3.1) is explained by the fol- lowing theorem:
THEOREM 3.3. Let / , u,\, A], l\ be as above, and let ry, F be as in Theorem 2.1. If u is a solution to (2.3), then the pair (?/|,A|), where ui = u\nt , Ai > ls a solution to (3.1). Conversely, if is a solution to (3.1), then there exists a harmonic extension u G VF1(f?c) of Ui onto fic which is a solution to (2.3).
R e m a r k . A proof of Theorem 3.1 can be found in [15]. Equation (3.1a) comes from a generalized version of the Green formula (see [6], [5], [1]).
Using this formula, Aj = can be defined as an element of II~ xl 2(T\).
The second equation (3.1b) is based on the assumption supp/i C !2| U / ’.
Thus the solution of (2.3) is harmonic in i?2 = R 2\J? (see Fig. 1) and we can use the Green formula for harmonic functions (see [11], [5], [ 1]). A coupling of this type was first applied in [9].
Using Theorem 3.1 it is easy to prove existence and uniqueness of (3.1) in view of Theorems 2.1 and 2.2.
Co r o l l a r y 3.1. Let / , u\. X\ , /A, <y, F be as in Theorem 3.1. and assume that the solvability condition (2.5) is satisfied. Then there exists a solution ( u,i, Ai) of (3.1) and it is unique up to a constant in the first component.
Now we establish some results concerning regularity of solution to (3.1).
First, assume I.\ to consist of line segments 7’j, j — 1 ,..., .J\. The boundary F is assumed to be of class C fr+1,1, k G N, i.e. it can be locally described by functions with Lipschitz continuous k -f 1-st derivative. Following [-1] we define
//* + ,/5( r , ) = {i>|r, : r € //*+1(R2)},
( 3 '2 > J l k+ ' r - { r i ) = { t ,|r , : „ € / / * + ' / i ( A ) } ) j = 1 ...
Th e o r e m 3.2. Assume that J € H k{Qc), <j € / / fc+|/ 2( f ) , k € N (/', l\
defined above), and there exists a solution (?/j, At ) G f/ 1 (Q\ ) X // ~1 / 2( f \ ) to (3.1). Then («,, A,) € H 1+2( « , ) X I ^ i
R e m a r k . Notice that only the boundary F is assumed to be smooth, the boundary I\ being a polygonal line.
P r o o f o f T h e o r e m 3.2. Let (//1, A () be a solution to (3.1). Then there exists u G \Vl (f?c) such that. = u\ and ?/|^2 is harmonic. By the
The Neumann problem 25
Green formula for harmonic functions (see [11], [5], [4]) we can relate tr to X2 = ^ | r i , where ~ is directed towards Q\\
(3.3) (/c VX2)r\ - (fiAru)pl + (/<,N’(tr u))ri - 0, V// € //~ 1/2(iY).
By subtracting (3. Lb) from (3.3), in view of tr u\rx = t r |/^, we arrive at (//, V(Aj - A,)) = 0. V //G //" 1/2( / ,1).
The form [ / / _1/ 2( T, )]2 9 (fi.O) i - (//., V’tf)A € R is / / " 1 /2( A ) elliptic (see [12], [11]), which implies At = X>. On the other hand,
f - A n = / in i?j ,
\ = 0 in Q-2 •
The above together with the fact that A| = A> in ) implies - Au = h / V 2 in 17C (for details see the proof of Theorem 3.2 of [15]). By referring to the well-known results concerning regularity of solutions of elliptic boundary value problems (see e.g. [13]), we get the desired local regularity of u in Qc and the regularity at the boundary f . This yields the regularity of ui = w| 17, : u i G f f k+2{Q\). Now we apply Lemma 2.11 of [4], which describes the properties of the operator U\ jA , to obtain Aj G f l / i i /G'+1'/2(T/). ■ Let r consist of line segments l ' \ j = 1 , . . F = jj^_, ^ ' (T2 is an open segment). Let uY be the exterior angle between f J and T-,+1 at the corner point tJ (see Fig. 2).
In this case Theorem 3.2 is no longer true. However, the behaviour of t/,}
from (3.1) is well-known. In the next theorem we cite some results of [10], [7] (see also [4]).
26 T. Rolinski
Th e o r e m 3.3. Assume that f G JJk(i?i), g G H kJrl (T), /,■ G N ( l\ as in I'h. 3.2, r defined above), and there exists a sedation (//1, Aj) G 7/l(J?i)x 7/ - '/ - ( / ’j ) /0 (3.1). Then the first eompement of the sedation admits the following el eeenn posit iem foroj'/ir ef Z (Z the set of integers):
.]
(3.4) »i = Rio + Cj l V1;jl i
7 = 1 l</<(A:+l)^/?r ic/icrr w10 G FIk+2(f21), cji are constants, cm <7
f 'lii7'1/ ^ cos(hr/wJ)<pj, if hr/u)J £ Z ,
| rl.ii(rf luJ \ogrjCOs(helu3)vj - m\(l7r/^J)ipj),
{ if hr /uj:i G Z;
here gg G Cq^ R 2) cmc .swr7/ f/m/ t l £ supp //,/,/ / j . / , j G { 1 , . . . / } , rmr/
there exists a neighbourlwod O ' o f t ' such that i)ji\o i = h fid d ly , ( r , , 0 j ) are poleir coordinates with P as pole (see Fig. 2).
R e m a r k . By [10], [7] the singular funrt.ions (3.5) are eigenfunctions of the Neumann problem on an infinite angle with measure u/J. Decomposition
can
4. G rd in g ’s inequality and th e ap proxim ate problem . In this section we prove the G rding inequality for problem (3.1).
Let us redefine the space 7I l l2( T\) with the help of a partition of unity } on the boundary 7 1: y. > 0. \. G C'(f'(R 2). /[ ^ suppy., j / *, j, > G 7|). i \ (•>•) = b V.?; G F\, where tj is the point where F( and F f+1 meet (see Fig. 2). Let; v : F\ t— R. The function \ v. can be identified via parametrization with a pair of functions (y . c ) _, (y.t>)+
such that; ( \ u)± : R+ R. Thus v can be identified with {(\ .c)_, (\y r)+ , j = 1,.. .,.7j). The space J J ^ 2( T \) can be redefined this way (see [1]):
/ / , / 2( / i ) = {» : (\;.i:)± € R+) <uul - (\ .<’)+ € /~/l/2( R +)} . where H l/2( R +) = : v G 77 l/'2(R) and c|f_ = ()}, with the norm
■h
IHlfp/^yy) = ^ ( l l ( \ I l 7 / 1/2(if_|.) + IK '/r )-llyyJ/2(]&+)) • 7=i
Let K-jk = y X \ . . j . k = 1, ... J\. The operator Kjk can be viewed as
j i ■ w.»
acting on the infinite angle F, 1 with vertex if and sides f + 1 = R +. T_ 1 = (cosuij, sin oy-j )R+ (see Fig. 3).
The Neumann problem {
operator Ay/,- can be identified with a 2 x 2 matrix (see [ 1]):
o a:*
a:* o where
K ^ { x ) = - I Tin (—
7T J '»*'
0 X(-
)<?(y) fly, V9 g C(f [o. oc). x e R + ,
lo stands for urj\ j — 1____ J i , i = \f-V.
Le m m a 4 .1 . The norm of the operator : l f ] l 2( E + ) i— 7 / 1/ 2( R + )
satisfies
ii*:. l / / 1/2 ( I + )►
P r o o f ’. Let 9 G C’^ ’fO. oo) and M<p(X) = x tX ]<p(x)dx, A G C (4/
is the well known Mellin transform). By Lemma 2.3 of [4], M//I/2(I+) = - J (ffA cothttA - 1 )\M9(A)I2(IX ,
Im A = 0
where
I‘t?I//1/2(]£+) - J J By Lemma 2.13 of [4],
o o
■ M *) - ^(y)1: k - ij\2
M ( K u<p)(\) = - S"'l'.(7! ^ AM si nil 7r A a )■ Im A € (-1.1).
28 T. Roliriski
Hence
|A.'.^sp|//i/2(l+ )
< sup
iii) ,\=o sinh"7T/\
On the other hand (see [4], Tli. 2.19),
1 /* (7TA coth 7rA - 1) sinh2 ( it - ui)A
- / ---—t—t---p'V/y( A)|" dX
'7r si n il “ 7r A
I Sr'I /■/ 1/2(1+ ) ’ Vv? G Cq (O .o c)
1 in A = 0
sinh2(7r — vj)A o
7r —
Sill 2 ^ d + )
sinh (it — a/)A ,7T — ui. . .. .. 0
The estimate supImA=0 — — < I—^— U the identity ||^||;/ i'/2(1+} = Ml2(i+ ) + lv?l//.i/2(i+ ) together with the above estimates yield the assertion.
■
Let us reformulate problem (3.1). Let A : V x l ’ >— R, V = H l(Q\) x / / _1/ 2( r i ), be defined as follows:
A{v,w) = a,(v,w) + (//., tr w)Fl + (O.Vp)rx - {0.trr)r, + (0 ,£ (trn ))r, , Vr = (n,/./.), Vic = (u\0).?\ w G l r.
Let / : F h- R be defined as follows:
(/'*>) = (.A + {.(/Ou,c)/Y-i/2(r)></fi/2(r ) . Now problem (3.1) reads: find n = (ui, A0 G 1 such that
(4.3) A(«, c) = (/, c), V ? G l '.
Th eo r em 4.1. Let. the form .1 b< as above and assume that the quantity max \uj{ — it I
is small enough. Then there exists a compact bilinear form C : V x V'' h- R such that
(4.4) A(v, v) > C\\v\\y/P{i - C(v. v), Vc G V' ,
where ||^||yyp0 — lf71 //1 ) 3- ll/<ll//-i/2(/-’1)? v — (*’> t1) € ^ ? I l//'(/?i) ^ lf:
first seminorm from the Sobolev space II1 (l?i).
P ro o f. With the helj) of the partition of unity {\.j---- ,.Yy } the form A can be decomposed as follows:
A(v, w) = B(v, w) + C(v, w), Vc, w G V', where
./i
B(v, w) = a(v. w) + (//, trw*)r, -f (#* Vy)r\ - (0, trv )ri + ^ ( 0 , Af/j(tr w))r, , j = i
The Neumann problem 29
C(v,w)= ■h (0, K-Jk( tr V))n • j,k= 1 ,j^k
With the help of (4.1) and (4.2) we can estimate the norm of K'jj:
llv /'V /’H/P/^r,) = ll\i ( ^ \ / ’)-||?/1/2(1+) + ll\i (^\V l’)+H?/*/2(i+)
= l | \ ;(^ w( \ J-?,)+)-||/./l/2(|:+) + ||\'y-(A''w(-Vj l,)~ )+ ll//1/2(l+ )
< C “\u - 7r|'{||(\y<0+||}/i/-’(i+) + l l( \'/) - ll^ /2 ( i+)} ’ V?>£ / / 1/2(A ), u; = o;/.
By t he above estimate
■h
X K' u r
j = i
< C max la;/ - t t( II i’ll )■ ^
/7»/2( r ,) ~
Moreover, the form (0, Vfi)n-*/2{ri)xH1/*(r1) ,s H *^2( A ) elliptic (see [12], [11]). Thus
B(vS>) > |i’|//>(/?,) +^i||/i||yy-./2(ri) + ^ ( / / , ^ tre)■h
./=!
^ My/qr?,) + 11/^1//-i/2(rj)
- C2 ntax |u;/ - t t| || tr e||/P/2(A)||//||//- 1/2(ri) . j£v>....hi
Since /!(?, ?) = .1(1?, ??), V? = (?;,//) £ V\ where ?? = (v + e.//), c is a constant, we can always assume f Q vdx = 0. By the inequality ab <
h(u2 -f I r ), a, b 6 R, the trace property of v and the Poincare inequality we obtain
B{Tk v) > [1 - Tj-C2C3(121) . max |w/ - Trllh’lz/UJ?*) .?€■{ i ,-»•?*; i }
+ [Cl - 5C2 max K - ?r|]||/(||}/-,/5(r ),
where the constant ) comes from the trace property and the Poincare inequality. Generally, this constant may depend on the boundary of fi\.
However, if we assume that a family of polygonal lines {l \ c- : 5 > 0} with vertices lying on a smooth curve I \ H satisfies dist( I \ e% F\a) < £, then the constants C^£ for the corresponding domains j?le are uniformly bounded by a -constant C3. This property is sufficient, for our case since we have some freedom of choosing the coupling boundary 1\ .
30 T. Rolinski
Hence the form D is V J^-elliptic, where VjPo — I I 1 (i?]) /IJo x I I ~1 (A ) ( H 1({?i)/Pq is a quotient space, Pq being the set of constant functions), provided that the angles ujf approach tc.
Now we show that the form C is compact. To this end it is enough to show that the operators AC;*, j ^ k, are compact. In case |j — k\ > 2 the kernel of K-jk = \ . K - x . is in C™; hence AC,** is compact. In case |j — Aj = 1, supp( \ 0 I \ ) fl supp(\. n A ) c / ’i, wliere i = j or i = k. The definition ofk .7 K implies that if ^ G I f ), then = 0; hence AT?/.- ,s compact, and the compactness of C follows. ■
To solve problem (4.3) numerically let us introduce a family Vh of finite- dimensional spaces such that. V), C V\ V), = IF/,, x II/,, \Vh c u ' w i), IIh C / / ~ / "(A b where h G (0,//o). Assume that the approximation is convergent:
(4.5) lim „ inf |v - } + ||// - = 0, vh = (vh, p h) , V7 = (l\ /./) G V - The approximate problem reads: find w/i € F/, such that
(4.6) A(uh,v h) = { f, v h), Vc/( € FT .
There arises the question of well-posed ness and approximation properties of problem (4.6). If we compare (4.3) and (4.6), then we obtain
A{uh,A ) = A(u, vh), VA G Vh ,
and we can pose the question of existence of the Calerkin projections V/P0 3 ii i—- Uh = G V)JI\). wliere V},,/Pq = IF*/P0 x 7//,, and the tilde ~ denotes a class in the quotient space.
Th e o r e m 4.2. Assume that the form A satisfies (4.4), and the family of Vh satisfies (4.5). Then the mapping <3h exists for h < ho and the norms II • ||yyp0_.v7Po of the family {C]h : Ii < hQ} are uniformly bounded.
R e m a r k . By Theorem 4.2 the solution u/, is unique up to a constant in the first component Uh- A proof of Theorem 4.2 can be found in [18].
Co r o l l a r y 4.1. For h < ho the following estimate hotels:
(4.7) \ \ Z - M v/p0 < C _ inf ||w- A | |k/Po- ii/, € V), / Po
R em a rk . Corollary 4.1 is a consequence of Theorem 4.2 and projection properties of {(//, : h < ho}.
The Neumann problem 31
5. T riangulations. In this and the next sections we define a family of spaces {F/, : h < h0} satisfying (4.5) and ensuring appropriate conver- gence by making use of the estimate (4.7). First we must define a family of triangulations.
R e m a r k . The triangulations differ from those satisfying the standard regularity condition (see [2]) in that the diameters of the triangles behave in the neighbourhood of /■>, j = 1 (see Fig. 2) in a strictly controlled way to compensate for the loss of smoothness of uj of the solution u = («i, Ai) (see (3.4)). Notice that, by Theorem 3.2. nx is smooth at /j, j = 1 ,..., , i.e. at the corners of f \ so it is not, necessary to modify the triangulations there. ■
Let S be a sector of a disk. 0 = (0,0) the vertex of .S', u> the vertex angle, 0 < u; < 2ir, and / the radius of the disk (see Fig. 4).
Let {'7/, : 0 < // < ho} be a family of affine triangulations of S (the triangulations are not exact) 7 G T/t. D{7) = diani(e), <l{7) = sup{diam B : B C 7, B a disk }. Assume
(5.1a) (5.1b)
max{D{7) \7 - h . 3ir > 0 V/? G(0,/?o) max{ D{7) : 7 £ Th}
< a .
min{('/(c) : 7 G Tk}
R em a r k. The condition (5.1b) is different from the standard regularity condition (see [2]). It ensures that the triangles 7 G 7), are evenly distributed all over t he sector S. m
Now we transform these triangulations by transforming the vertices of the triangles 7 G 7), via. the Injective mapping <j : S •— S defined by
(5.2) !Ji = I " V i + ,
V-2 = + X l f /2X'2 ,
32 T. Rolinski
where x = ( x l ,X2 ),y = (2/i,2/2) € 5. I11 polar coordinates mapping (5.2) reads g(r) = l~dr l+ifi 0 < r < /, r2 = .r2 + .r:>. Let {a{ ~ : /’ = 1,2,3} denote the vertices of e G 7/t. Then aue = <7(<7;<(§). /■ = 1.2,3. are vertices of a new triangle <. tdius defining a family of triangnlations {7),(,V/() : 0 < h < ho}, where ,9/, = (J{c : e G 7),}, the bar denoting the closure of a set.
Le mma 5.1. The family of Iviangulations {Tt fS'iA : 0 < h < h0} satisfies
(5.3a) 3rr > 0 V/?. G(0,/io) Ve G Th — —^<<7,d{e) (5.3b) Vfi G Th{0 i e => D(e) ~ [dist.(e,0)] A fc} ,
where h is defined in (5.1a), and fl is the parameter of g. The symbol ~ denotes that the first quantity can be estimated from above and from below by the second one times a constant independe nt of e G 7), and h G (0. ho).
R e m a r k . Condition (5.3a) is the standard regularity condition as can be found in [2]. Condition (5.3b) shows the wav the triangles are distributed in the sector 5. It can be seen that they are condensed in the neighbourhood of 0, the overall number of triangles remaining the same as for 7/t.
P r o o f of Lem m a 5.1. Proof of (5.3a).
1. The case 0 ^ e.
Any triangle e G 7/, can be inscribed in a sector S{e) limited by circle arcs A \ B 1, C\D\ and radii A\D\, B\C\ (see Fig. 5).
Let 7*1, 7*2, r 7„ 7 be t he quantities as in Fig. 5. Let A2 B2 C2 D2 I'h Le the images of .4] B\ C\ Di E\ under g (see (5.2)). These points yield a new sector with circle arcs A.0B2, C_>7L> and radii A i /L>, /LCT We have the
The Neumann problem 33
following estimates:
(5.4a)
(5.4b)
1 4 - / 3 B i C \
(1 + <t)P DiCi c \ F r x < c n ±
Bx “ B2E D2C2
1 < (1 + a )
IhCi
B XE1
The above estimates are consequences of 1 +4 1+0
<
<
.1 + 0 < (1 + 0)r 2 - i'[
7‘o /••7
n /'I < l'n < Vj
and the estimate ^ < 1 -f a (see (5.1b)). Estimates (5.4a.b) imply (5.3a) for e such that 0 $ e.
2. The case 0 £ r.
In this case the triangle can be inscribed in a sector 0 A 1 B t limited by radii OA\. OBi and a circle arc . l ( //] (see Fig. (j).
Let /■[. r-2 be the radii as in Fig. 6, and let. A2 B2 E 2 be the images of Ai B[ Ei under g. We have the following estimates:
(5.4c) O E i 0 E 2
0.41 " 0.42 which is a consequence of
< (1 + ! ) ) S'*—
O /l]
( 1 4 - / ? ) !^ 1 < < ( ! 4 - ^ ) ( p -
r 1 r i +fi \ n
r -2 - r 1
n 7*1 < Vo
together wit h the estimate ^ < <7. Estimate (5.4c) implies (5.3a) for e such that 0 € e. Thus all triangles e € % satisfy the regularity condition (5.3a).
34 T. Kolinski
To prove (5.3b) take a triangle e such that 0 g e. Let : i = 1,2.3}
be its vertices, and </(«-~) = a*,e, i = 1,2.3. It is easy to see that (5.5a) dist(e.O) = [dist(c, 0)]1 + a .
Let ri. /’o be the radii as in Fig. 7.
The following estimate holds:
(5.5b) D(e) ~ P(e)[dist(c, 0)]/3.
This is a consequence of (see Fig. 7)
(1 + - r , ) < r \ * e - r|+# < (1 + /))(! + S)'’rf(r2 - r , ).
Bv combining (5.5a) and (5.5b) with D{7) ~ h (see (5.1a,b)) we arrive at
6. F in ite elem en t spaces. Now we define finite element spaces on the sector .S'. Then we will obtain an interpolation result for power type functions to be used for interpolation of the singular functions from the representation (3.5).
Let us take a Lagrangean finite element (c, P, 27), where f C l ' is the basic triangle, P = Pf.\p, being the space of polynomials of degree less or equal k £ N. The set 27 = {(pi £ P ' ., N ] is a set of functionals (for terminology see par. 2.3 of [2]) for which there exists a set of functions {T>i € P : i = 1...iV) such that, pi form a basis of P and (pi(pj) = Sij, 1 < /, j < Ar, &ij being the Kronecker symbol. Let us define the interpolation operator 77 : C{e) h* P by (llv){x) = Yl'iLi £*(**)?*» and let us take the family of triaiigulations 7}(,9/,). h < /?o- defined in the previous section. For every e £ 7/t we can define an affine operator e B x t— x = Ff (x) £ e. The invertibility of Fe allows us to define a family of finit e element s (e, P, 27) by
The Neumann problem, 35
a) ( e = Fe(e) ,
(6.1) b) < P = {p : e h— R : p = j?o Ff-1, p G P) ,
c) [ 17 = {*« : P R. / = ,V : vMp) - &(/>)} •
Let 77e_:C(e) ^ I \ (.//ey f= 77 n, Vt> G C(e), 9 = r o Tfc, ( .f = (•) o Fe. Let 77 : C{Sh) • * £ 0 0 77 »|* = //e(^ic)* Vv € C’( .S '/T h e operator //e is a local interpolation operator, and II is a global interpolation operator. We assume that the set £ is chosen in such a, way that II : C{Sh) h“* C (S /,,). A definition of such sets can be found in [2].
In what follows we shall use the Sobolev spaces IF p(e) (1 < p < oc):
W$(e) = {v € : 7TS; G Lv(e) W : |6| < A-, 6 = (6iJ 2)}
with the norm || • j|iv*(e):
\\VHe) ~ 5 5 VV^(e)’ MvV'J(t) ~ 5 5 ^ V^LP(f) ^01' ^ - P < ° ° ’
t=0
IMIw£(e) - (_max^ i r I e) - jllax too(e)
In case \V^(e) <■— C {(). where <— denotes continuous embedding, we can interpolate functions from W^(e). Following [2] we have
Th e o r e m 6.1. Let p.q G (hoc), W£+](e) ^ W'J(e), W£+l(e) ^ C(e) for some k G N. Then
(6.2) \v - IIf e\n.-i(r) < C(e. I \ L')a[nu^s{e)]l/q~l/pD(e)k'\v\yvk+,[eV
V r G H p H 1 ( e ) .
where meets denotes Lehesgue measure, m Now we are ready to prove
T h e o r e m 6.2. Let £ : S e— R he a function satisfying (6.3a) |/A(-<-)l < ni/iD C -l^llogi-l.
where 6 = ( ^ l,^ ) G N X N, r 2 = xj + .rj, cv G (1/2, oc). Let LI : C(.$h) C( S’h) be the interpolation operator defined above, and let
h — a
(6.3b) i3> --- ,
a
where 1 + (3 is the exponent of the function g (see (5.2)). Then (6.4) |{ - ] H \h <(S>) < C a - + l)C(/),a.k)C(c. P , S ) a h k ,
where C(k -f 1) the constant from (6 .3 a ) , C(e, P,H) is the constant from
( 6 .2 ) , A- G N , C (/j, n , A-) - oo a s |(3 - - 0.
P ro o f. 1. The case 0 (f c.
36 T. Kolinski
Using (6.2) for q = 2 and p — oc yields
K - < C’(?, ? , £)a[nio88(Ol,/s/? (0 ‘'l < l < « („ • Assumption (6.3a) leads to the following estimate:
(6.5) |{|< + .(e) < C(k + I )[disl(t,0)]“" A'- 1|log(dist(f. 0 ))|.
In view of (6.5) and (5.3b),
|« - U S hh o < CU‘+ i)C(e,P,
x [<list(c,0)]“- ‘'_ l|log(c1ist(<?,0))|.
Now take the sum of squared seminorms | • \h '(c) over c E 7/> : 0 ^ e:
(6.6) Y . I< “ < e J«r + 1 )C2(f, P, S ^ h 311
p€Th: Oge
x ^ meas(e )[dist(e, 0)]T+JT [dist(c, 0)]2(fv“ k~1 * | log( dist (e, 0))|2
e€T,,:Oge
= C *(k + 1 P, S ) a 1l r k 1(h) ,
where /(/i) — / J ^ re log2 rtht'i r/r, a.s /, — 0 (5 = + 2(o — /.' — 1)).
The quantity I(h) can be estimated uniformly by a constant C(,S.a,k) if the integral is finite, which is equivalent to s + 2 > 0, which in turn yields (6.3b).
2. The case 0 E e.
We have £ E /71 (c) and we make use of (6.2) for p = 2, q = 2. k — 0 to obtain
(6.7) |{ - 77,f|w.(e) < C’(f, P, r)<r|{|„i(e).
Let us estimate |^|//i(e). In view of (6.3a) we get 2 ,t r - V +/j
J |6;, I2 <lx < J J C'2(l )r2a~21 log r|2 r dr djj
e 0 0
< C 2{ 1 )/)(1+/b2o| log/(|2 ,
where C( 1) is the appropriate constant from (6.3a). Since the integral of
|6.2|2 can be estimat ed in the same way, we obtain If !«■<,> < C '(l)A "+i3,“ |log/1|.
In view of estimate (6.7),
(6.8) |f - HSuHr) < C( 1 )C'(7, P, log A|.
The Neumann problem 37
To preserve order k we must, assume a + (3 a - k > 0, which is (6.3b). Since 3rn G N V/>. G (0, /?o) card{e G T/, : 0 G e} < m, t he estimates (6.6) and (6.8) yield (6.4). ■
7. In terp o latio n . In this sect ion we consider global interpolation of the solution u - (»i,Ai) of (3.1). Since a G V = / / 1 (T21) X / / _1 / 2 ( ) we define finite element spaces on the whole domain i?| and the boundary F\ .
We start, with triangulat ions. Let Sj denote a sector: / ?- the vertex of S'ji ujj the vertex angle, 0 < < 2jt; and /,- the radius of the circle. The length lj is so small that Si fl Sj — 0, j ^ /, /, j = 1.../. Since the boundary of Q\ consist s ofjine segments it is always possible to construct, a family of triangulations {Th : 0 < h < />0} of Q\ satisfying conditions (5.1a,b) and
Ve G Th .Tj, x 2 € deD Sj => xixo C Oe ,
where stands foi- the circle arc of S j . The above condition was added to make use of the results of the previous section.
We use the same symbol Tp from Section 5 for a triangulation on the whole domain l?i for the sake of brevity. We also use some other symbols from Section 5 if it does not lead to confusion.
Let {(jj : Sj h— Sj : j = be a family of mappings defined analogously to the mapping of (5.2). Let g : D\ i— f?j be defined by
.'/(*) = gj(x) for x G Sj, x for .r G f h \ { j S j .
For every gj we impose the condition (comp. Th. 6.2 (6.3b)):
(7.1) 3 j > I; - a j
j = 1 ,..., J, a j = — .
Now take 7 G 7/,; let a; ~. i — 1,2,3, be the vert ices of 7. Let al f — g( ai ~) be the vertices of a. new triangle e G 7/,. Thus we have defined a family of triangulations {7), : 0 < h < I)o). Let [7.P.U) be an element as in Section 6; let {{(. l\ . ) : e G T),} be a family of finit e element s defined as in (6.1): let If : C(J?i) C(Q\) be the global interpolation operator defined as in Section 6. We define the finite-dimensional space M’/,. to be the image of (7(1?i ) under //.
Let k G {1,2,3 . ..}. If / G I f k~ l ( l? i), g G I f k~T'2( f ) and the solvability condition (2.5) is satisfied, then by Theorems 2.2 and 3.3 the solution u = («i, Ai) to (3.1) exists and v\ can be expressed by the following formula:
j
(7.2) tii = Mio + Cjiuiji*
j = 1 1 < / < A- / o,
38 T. Hoi in ski
where Uiq € U k^ ]( Q\ ), c.ji is a constant , a _/ = 7r/uy, and
7.3) « t j / +,/ K e ~j/ay for la j (f Z ,
i)ji Re( log ) for /a ; E Z
(comp. (3.4), (3.5)), where zj = i^e"'’’ (i — >/— l ) and (rj.<pj) are the polar coordinates for .S'/.
THEOREM 7.1. Let TI : C ( Q \ ) Wh be the global interpolation operator and let uj be given by ( 7.2). Then the following estimate holds:
(7.4) |«i — 77w-x|//i( ) < C(e, P, E)ahk { | )
./
+ ^ 5 3 cjiCj{k + 1 )C{aj,/3j, Ar) j , 7 = 1 i<l<k/nj
where C{e, P, E) is the constant from (6.2), and the constants Cj{k + I), C'( O' j , j3 j, A’) are from (6.4).
P roof. By the linearity of 77 and the triangle inequality,
( 7 .5 ) |« i - 7 7 « i | / / i ( ^ 1) < | a io ~ H
+ 5 3 5 3 - A/'//.i//)|y/1(o J) .
j=l 1 </<*?/«>
In view of (7.2), (7.3) it is easy to see that the functions u\ ji satisfy the assumptions of Theorem 6.2 for a — oj = 7r/l oj. Thus
(7.6) \uiji - 11 u\ < Cj{k + l)C(/3/,Oj,/v)6'(e, P , E)ahk . On the other hand, the well-known result (see [2]) yields
(7.7) |«io - 77 «io|/yi(/2j) < C(e, 7J, £,)frhA’|M1o|//*+i(oJ) . Thus the estimates (7.6) and (7.7) together with (7.5) yield (7.4). ■
Now we define finite-dimensional spaces approximating 77- l / 2( If ) (Ai E //-!/-( Pi)). Let {'/)' : 0 < h < 1/q} be the family of partitions of If induced by the family of exact triangulations {7), : 0 < h < ho) defined above.
Let (e/, P \ E 1) be a. finite element such that e* is a segment, P1 =
E' — : 1 < i < TV'} is a set of functionals for which there exists a set of functions {/5- E P' : 1 < i < Ar/} such that p* form a basis of P and (p'fp'j) = bij, 1 < i < j < A7'. Let {(e', P€' , E (j ) : e' E '7);} be a family of finite elements defined analogously to the family {(c, Pe> E f ). e E 7/,} in
The Neumann problem 39
(6.1). Let
tth = j/D1 € C(J\) : \fe' 6 ‘T,[ //•/, o Fc» G P' and j ;//( <7<r = o | , A
where 7v : e' i-+ e' is defined as /re *ui Section 6. In case k = 1, //* is a space of piecewise constant- functions. In case k > 2 we assume VC G 7)'
<*i,eG«2,e' € XV*, where « i , «2,e» are the ends of C, which yields continuity of functions from 77/,.
Let us refer to [12] (Lemma 3.5) for the following result:
(7.8) ||A, - S A 1||„ - ,/=<;-,)
where S is the orthogonal projection from 12{P\ ) to 7Z/j, the spaces / / A'+1/ 2, k G N, are defined in (3.2) (see also [4]).
COROLLARY 7.1. Let k — 1,2... let IT/,, Up be the spares defined in this section, let u be a solution of (3.1) and uh a solution of (4.6). Them
IIS - S /Jl.va < + l|A,||;,,.-1/>(ri)}1/2, where
G » —j y—^
J - 111 io l / / A+J ( CJ1 ^ ( a :i? P j 7 k ) j = i l < l< k /a j
P ro o f. The corollary is a simple consequence of (7.8), (7.1) and Corol- lary 1.1. ■
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