BEM COUPLING METHOD

MATHEMATICS OF COMPUTATION Volume 66, Number 220, October 1997, Pages 1407{1440 S 0025-5718(97)00878-8

ANALYSIS OF A FEM/BEM COUPLING METHOD FOR TRANSONIC FLOW COMPUTATIONS H. BERGER, G. WARNECKE, AND W. L. WENDLAND This work is dedicated to Professor Dr. Klaus Kirchgassner on the occasion of his 60th birthday Abstract. A sensitive issue in numerical calculations for exterior ow problems, e.g. around airfoils, is the treatment of the far eld boundary conditions on a computational domain which is bounded. In this paper we investigate this problem for two{dimensional transonic potential ows with subsonic far eld ow around airfoil pro les. We take the arti cial far eld boundary in the subsonic ow region. In the far eld we approximate the subsonic potential ow by the Prandtl{Glauert linearization. The latter leads via the Green representation theorem to a boundary integral equation on the far eld boundary. This de nes a nonlocal boundary condition for the interior ring domain. Our approach leads naturally to a coupled nite element/boundary element method for numerical calculations. It is compared with local boundary conditions. The error analysis for the method is given and we prove convergence provided the solution to the analytic transonic ow problem around the pro le exists.

1. Formulation of the problem 1.1. The boundary value problem. Let R2 be an open bounded domain surrounding a given simply connected wing section P R2: The boundary of

consists of three parts (1.1) @ := ?1 [ ?P [ ; whose interiors are mutually disjoint and where ?1 and ?P are disjoint closed Jordan curves connected by . The curve ?1 2 C 1 is the arti cial exterior boundary of which is taken in order to obtain a bounded computational domain. The curve ?P is the common boundary between and the pro le P , which has a corner, the trailing edge (TE), and is C 1 otherwise. We denote by a slit in ; joining the trailing edge with ?1 : The unbounded far eld domain exterior to ?1 will be denoted by (1.2)

c = R2n [ P : The prolongation of the slit in c to in nity will be denoted by c : Without loss of generality, we assume that the travelling velocity is given by a constant vector eld ~v1 which is parallel to the x1 {axis, see Figure 1. Received by the editor August 13, 1993 and, in revised form, September 18, 1995. 1991 Mathematics Subject Classi cation. Primary 65N30, 68N38, 76H05, 49M10, 35L67. Key words and phrases. Transonic full potential equation, arti cial boundary conditions, nite elements, boundary elements, asymptotic error analysis. The research reported in this paper was supported by the \Stiftung Volkswagenwerk". 1407

c 1997 American Mathematical Society

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H. BERGER, G. WARNECKE, AND W. L. WENDLAND

Figure 1. Computational domains for the coupling

In we consider a stationary, compressible potential ow induced by a subsonic travelling velocity ~v1 at in nity. A simple model is the full potential equation, see Landau and Lifschitz [42], (1.3) div (jruj2)ru = 0 in : This equation models inviscid, steady, isoenergetic, homentropic, planar ows of an ideal gas. Its generalized weak formulation admits transonic solutions with shock discontinuities in the velocity eld. The equation (1.3) can be derived from conservation of mass, momentum and energy, see Berger et al. [9]. The density function (s) is obtained from Bernoulli's law and the assumption of homentropic

ow. It is given as ? 1 ?1 1 2 (jruj ) = 0 1 ? 2a2 jruj2 (1.4) : 0 Here > 1 is the adiabatic gas constant, e.g. = 1:4 for dry air. The constants 0 ; a0 are the density and the local speed of sound, respectively, for the motionless gas. The local speed of sound a(jruj2) is given by a(jruj2)2 = a21? (jruj2)?1 = a2 ? ? 1 jruj2 0 0 0 2 and the local Mach number is M := jrauj . Note that the density (s) is only 2 de ned for jruj2 k2?a01 . This bound for the velocity will be assumed to hold in all further considerations. The dierential equation (1.3) with1(s) given by (1.4) changes type at the sonic a20 2 , i.e. M = 1. For jruj < a , i.e. M < 1,

ow speed jruj = a = a := 2+1 the equation is elliptic and the velocity is subsonic; for jruj > a , i.e. M > 1, it is hyperbolic and the velocity is supersonic. For sonic speed, M = 1, the equation is degenerate, see Courant and Hilbert [23], Courant and Friedrichs [24].

FEM/BEM COUPLING METHOD FOR TRANSONIC FLOW COMPUTATIONS

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In order to have a potential ow with circulation and lift we introduce the slit across which we assume that the velocity eld ru is continuous, whereas the potential u has a nite constant jump. This implies (1.5) u+ ? u? = and @n u+ ? @n u? = 0 on : Here, ~n is a unit normal vector eld on and @n u = ru ~n. By u+ ; u? we denote the one{sided boundary values on . The constant is an additional unknown and gives the circulation of the ow. To determine the jump ; we need the additional

Kutta{Joukowski condition

F ( ) := ru+ 2T E ? ru? 2T E = 0 (1.6) at the trailing edge TE which follows from the requirement of continuous pressure there. Here we shall assume that the ow near TE is subsonic which implies (1.6), too. (See Theorem 1.3.) On the pro le we impose the homogeneous Neumann boundary condition of non-penetration, which is equivalent to a vanishing mass ux, i. e. (1.7) @n u = 0 on ?P : For completeness, we still need a boundary condition at ?1 : It is generally assumed that a physically correct boundary condition for exterior ows is the requirement that the velocity ~v tends to the constant travelling velocity ~v1 at in nity. A simple and widely used method for approximating this condition is the following mass ux condition at ?1 , i. e. (1.8) (jruj2)@n u = (j~v1 j2)~v1 ~n on ?1 : This is the condition used in Berger et al. [9]. A slight improvement of (1.8) can be obtained by replacing ~v1 on the right{hand side by ~v0j?1 , where ~v0 is the velocity of the incompressible potential ow in [ c [ ?1 , see Berger et al. [8]. In this paper, however, we will consider the coupling with Prandtl{Glauert ow exterior to ?1 : As is well known, the linear Prandtl{Glauert equation is (1.9) (1 ? M12 )'x1 x1 + 'x2 x2 = 0 for x = (x1 ; x2) 2 c : This is a linear approximation of (1.3). Here, the perturbation potential ' is de ned by (1.10) p 2 ' := u ? with (x; ) := ~v1 (x1; x2)+ 2 arctan 1 ? xM1 x2 ; 1 where M1 is the Mach number at in nity, see Zierep [57] for details. For the arctan the branch must be chosen in such a way that the jump occurs at the slits and c which meet at ?1 . Of course, we assume that the perturbation velocity eld r' is continuous across the slit c and the pertubation potential '; by de nition (1.10), is continuous across c, too. This gives (1.11) '+ ? '? = 0 and @n'+ ? @n'? = 0 on c : For the perturbation velocity r' we prescribe the radiation condition at in nity, (1.12) r'(x) ! 0 for jxj ! 1 : In addition, we need two more transmission conditions for the coupling of u with ' at ?1 : Denote by ~n = (n1 ; n2) the outer unit normal eld on ?1 , and set @nu = ru ~n. The rst transmission condition is simply obtained by de nition

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(1.10) and continuity across ?1 : For the second transmission condition we require equality of the mass ux (jruj2)@n u to the corresponding expression de ned by linearization about the Prandtl{Glauert solution. The latter leads for the density to 1 := (j~v1 j2) in c . The coupling conditions become (1.13) u = ' + on ?1 and (jruj2)ru ~n (1.14) = 1 r ~n + (1 ? M12 )'x1 n1 + 'x2 n2 on ?1 : Collecting the equations (1.3), (1.5), (1.6), (1.7), (1.9), (1.11), (1.12) and the coupling conditions (1.13), (1.14), we get the following system of equations, boundary and transmission conditions:

Coupled boundary value problem

Find the functions satisfying the

u; ' in appropriate function spaces and the constant 2 R

Interior full potential problem, div (jruj2)ru = 0 in ; @nu = 0 on ?p ; (1.15) u+ ? u? = on ; @n u+ ? @nu? = 0 on ; + 2 F ( ) = jru jT E ? jru?j2T E = 0 ; Exterior Prandtl{Glauert problem, in c ; (1 ? M12 )'x1 x1 + 'x2 x2 = 0 r' = o(1) for jxj ! 1 ; (1.16) + ' ? '? = 0 on c ; + ? @n ' ? @ n ' = 0 on c ; Coupling conditions, (1.17) and

u = ' + on ?1

(jruj2)@n u ? = (j~v1 j2) r ~n + (1 ? M12 )'x1 ; 'x2 ~n on ?1 : Note that, for a solution (u; '; ) of this coupled problem (1.15){(1.18), one also has the solution (u + c; ' + c; ) with an arbitrary constant c. We therefore x this constant by the requirement Z ' ds = 0 :

(1.18)

?1

With the Euler equations in instead of the full potential model (1.15), Sofronov and Tscincov present a similar coupling formulation in [55].


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1.2. The weak formulation. Let W s;p ( ); s 2 R; p 2 [1; 1] and W s;p (?1 ) be the usual Sobolev spaces equipped with norms k kW s;p ( ) and k kW s;p (?1 ) ; respectively. We de ne h; i to be the duality pairing between H s(?1 ) := W s;2(?1 ) and the dual space H ?s (?1 ) with respect to the L2 (?1 ) inner product, (1.19)

h; i :=

Z

?1

(s) (s)ds for all (; ) 2 H s(?1 ) H ?s(?1 ) :

We further introduce the spaces

(1.20)

V := v 2 W 1;2( ) j v+ ? v? = on ; 2 R ; V 0 := v 2 W 1;2( ) j v+ ? v? = 0 on ;

9 8 Z = < 1 ds = 0; H := : 2 H ? 2 (?1 ) ?1

and the set of admissible functions (1.21)

2 Ks0 := v 2 V jrvj2 s0 < 2?a01 a.e. in :

Ks0 is a closed, convex subset of V in W 1;2( ). In order to simplify the notations we de ne the following nonlinear form (1.22) Z a(u j v; w) := (jruj2)rv rw dx for all triplets (u; v; w) 2 Ks0 V V :

For solving the exterior problem (1.16) we shall use a boundary potential formulation based on the Green representation theorem. To this end, one transforms the Prandtl{Glauert equation in (1.16) with constant coecients into Laplace's equation and then uses classical potential theory. Any suciently smooth solution ' satisfying (1.16) has the behavior

e 21 2 + '1 + o(1) for jxj ! 1 ; + x '(x) = 2 log 1 ?xM 2 2 1

with constant '1 . One obtains with the fundamental solution

G(x; y) := 21 log p 1 2 [x1 ? y1 ]; [x2 ? y2 ] (1.23) 1 ? M1 and the kernel of the double layer potential (1.24) K (x; y) := 21 ([x1 ? y1 ]; [x2 ? y2 ]) ~n(y) 2 p1?1M 2 [x1 ? y1]; [x2 ? y2] 1

!

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the Green representation formula in the form (1.25) Z '(x) = p " 2 f(1 ? M12 )'y1 n1 (y) + 'y2 n2(y)gG(x; y) dsy 1 ? M1 ? 1

Z " '(y)K (x; y) dsy + "'1 +p 1 ? M12 ? 1

with " = 1 for x = (x1; x2) 2 c and " = 2 for x 2 ?1 . In the latter case, (1.25) de nes a boundary integral equation relating the Cauchy data (1 ? M12 )'x1 n1(x)+ 'x2 n2 (x) and '(x) to each other. It can easily be shown that (1.25) corresponds to the choice of zero for the additional constant mentioned at the end of Section 1.1 and used in the de nition of H in (1.20). We introduce the co{normal derivative of ' on ?1 by (1.26) (x) := (j~v1 j2)f(1 ? M12 )'x1 n1(x) + 'x2 n2 (x)g and de ne the boundary integral operators of single and double layer potentials Z 2G(px; y) V (x) := ? 2 (y) dsy ; ?1 (j~v1 j2) 1 ? M1 (1.27) Z 2K (x; y) p K'(x) := 2 '(y) dsy : ?1 1 ? M 1 Thus, for x 2 ?1 ; we may write (1.25) in short as (1.28) '(x) + V (x) ? K'(x) = 2'1 : For smooth ?1 we have the following well known result, see Hsiao and Wendland [38]. Lemma 1.1. For a C 1 {boundary ?1; and any 2 R, the boundary integral operators

(1.29) are continuous.

V : H ? 21 (?1 ) ! H + 12 (?1 ) and K : H + 12 (?1 ) ! H + 23 (?1 )

Further, we de ne the bilinear forms (1.30) b(; ) := hV ; i and d(; ) := hK; i for ; 2 H ? 21 (?1 ) and, nally with the function = (x; ) given by (1.10), for any 2 R the functionals `1 (v; ) := 1 hr (; ) ~n; vi and `2 ( ; ) := h(I ? K) (; ); i for all v 2 W ;p ( ) and all 2 H ? 12 + (?1 ). For every xed 2 R, these are linear and bounded functionals.


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Lemma 1.2. The nonlinear form a( j ; ) satis es the following estimates : ja(u j v; w)j 0 k v kW 1;2 ( )k w kW 1;2 ( ) for all (u; v; w) 2 Ks0 V V ; (1.31) ja(u j u; w) ? a(v j v; w)j C k u ? v kW 1;2 ( )k w kW 1;2 ( ) for all (u; v; w) 2 Ks0 Ks0 V :

a0 ; then we obtain uniform monotonicity for these Moreover, if we choose s0 < 2+1 purely subsonic ow elds, i.e.

2

(1.32) a(u j u; u ? v) ? a(v j v; u ? v) 1 k u ? v k2W 1;2 ( ) for all (u; v) 2 Ks0 Ks0 : 1 For a C {boundary ?1 and 2 R there exists a constant C such that the form b(; ) satis es jb(; )j C kkH ? 21 (?1) k kH ?? 12 (?1 ) : (1.33) Moreover, there exists a constant 2 > 0 such that b( ; ) 2 k k2H ? 21 (?1 ) for all 2 H : (1.34) For the linear forms we have with some constant C the estimates j`1 (v; )j j%1 j fj~v1 j + C j jgkvkH ? 21 (?1 ) ; j`1(v; 1 ) ? `1 (v; 2 )j C j 1 ? 2 j kvkH ?1 (?1 ) ; and

j`2 ( ; )j C fj~v1 j + j jgk kH ? 21 (?1 ) ; j`2( ; 1 ) ? `2 ( ; 2 )j C j 1 ? 2 j k kH ? 12 (?1 ) : Proof. The inequalities (1.31) follow from the facts that j(s)j 0 and j(s) + 2s0 (s)j c hold for all s 2 [0; s0]. The coerciveness h a20 inequality (1.32) is a conse. The properties of the form quence of (s) + 2s0 (s) 1 > 0 for all s0 2 0; 2+1 b are shown by Hsiao and Wendland in [38], the continuity estimates for `1 follow from (1.10) and those for `2 from Lemma 1.1. Remark. 1The coerciveness inequality (1.34) and the mapping properties of V and K : H + 2 (?1 ) ! H 1(?1 ) in (1.29) remain valid with jj 21 even for Lipschitz curves ?1 due to Costabel [22]. If we multiply equation (1.28) by a test function 2 H and integrate over ?1 , we obtain (1.35) hV ; i + h'; i ? hK'; i = b(; ) + h'; i ? d('; ) = 0 for all 2 H : This is the weak formulation for the boundary integral equation (1.28). The weak formulation for a solution of problem (1.15) can be obtained by the usual variational approach. Find u 2 Ks0 and 2 R such that (1.36) a(u j u; v) ? h(jruj2)@n u; vi = 0 for all v 2 V 0 :

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The coupling of the weak formulation (1.36) in and (1.35) in c can be obtained with (1.17), (1.18) in variational form via the ux balance (1.14) through ?1 and reads:

Variational problem

Find the three quantities

(u; ; ) 2 Ks H R such that

0 8 > b(; ) + hu; i ? d(u; ) = `2 ( ; ) for all 2 H (?1 ) and :F ( ) = jru+j2 ? jru?j2 = 0 : TE

TE

Note that the solvability of this problem can be proved rigorously, up to now, only in the subsonic case. For transonic ows, however, we shall assume the existence of an appropriate solution. This includes that an appropriate choice of s0 is possible. In the following section we will discuss the Kutta{Joukowski condition in (1.37). 1.3. The Kutta{Joukowski condition. For subsonic ows, i.e. with suciently small ~v1 ; (1.36) is a nonlinear elliptic Neumann problem in variational form. The set of admissible functions (1.21) excludes an arbitrary growth of jruj at the trailing edge TE: The solution of a linear and also a nonlinear elliptic problem would in general develop a singular growth of jruj at the reentrant corner TE if is not chosen appropriately. In case of a singularity, however, there would be a supersonic region at TE which we exclude for physical reasons. Theorem 1.3. Let the trailing edge angle ! satisfy 0 < ! < . Let (u; ; ) 2 Ks0 H R be the variational solution satisfying a(uju; v) ? h; vi = `1 (v; ) for all v 2 V 0 ; b(; ) + hu; i ? d(u; ) = `2 ( ; ) for all 2 H :

a0 and a radius Let us further assume that there exists some bound sT E 2 0; 2+1 r0 > 0 such that around the trailing edge jru(x)j2 sT E for all x 2 U := fx 2 jx ? TE j < r0 g ; (1.38) i.e. the ow is subsonic around TE . Then the velocity eld ru is Holder continuous in U and already satis es the Kutta{Joukowski condition

2

ru+jTE = ru?jT E : (1.39) The proof of this theorem will be given in the paper [21]. At a rst glance, for a variational solution u in W 1;2( ), the point condition (1.39) seems not to be well de ned. Since, on the other hand, it also means that the stress intensity factor associated with TE and the subsonic ow is to be zero, this is a continuous functional on the solution space. For subsonic Prandtl{Glauert linearizations, this requirement determines the circulation uniquely. In the paper [21] this relation will be used. Incorporating (1.39) into the solution space yields a fast deterioration of the condition numbers of the associated discretizations. Here, however, we analyze the method described in [10] where a simple relaxation procedure is used enforcing (1.39) in every iteration step.


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For purely subsonic cascade ow, Feistauer et al. in [29] also use the Kutta{ Joukowski condition for nding the appropriate nite element solution providing circulation. 2. Existence, uniqueness and the entropy condition The coupled boundary value problem (1.15){(1.18) is an approximation to the:

Exterior Boundary Value Problem

jruj2 < 2?a01 satisfying div (jruj2)ru = 0 in [ ?1 [ c ; @n u = 0 on ?p ; u+ ? u? = on [ c ; @nu+ ? @n u? = 0 on [ c ; ~v(x) ?! ~v1 for jxj ! 1; + 2 ? 2 F ( ) = jru jT E ? jru jT E = 0 :

Find an appropriate function u with

(2.1)

2

For the subsonic solution to this exterior problem see Bers [11] and Bojarski [13]. Here, however, we allow the solution to be transonic. Moreover, the dierence to the coupled problem (1.15){(1.18) lies in the fact that there we take the linearization on c and the corresponding coupling boundary conditions on ?1 . Under the restriction (2.2) krukL1 (G) < a in the subdomain G \ ?1 [ c ; the problem (2.1) is elliptic there, i.e. the ow is subsonic on G. A unique solution exists for all ~v1 small enough to imply that (2.2) holds everywhere. This result has a long history, see e.g. Frankl and Keldysh [34], Bers [11] and the references given there, Bojarski [13], Brezis and Stampacchia [14], Ciavaldini, Pogu and Tournemine [20], [19], [51] and Feistauer and Necas [31]; most authors treated the problem in the stream function formulation. The upper bound (2.2) as a global condition depends on ~v1 and the geometry (thickness and form) of the pro le P . Therefore, it has not been possible up to now to explicitly give a priori conditions that imply (2.2). Due to the jump across the slit [ c , the solution is not in H 1 in a neighbourhood of the slit. It is locally only in L2 because ru is a locally bounded measure that has a singular part (see Federer [28]) supported on [ c which is weighted by the jump strength . But, the fact that the coupling conditions imply ru+ = ru? on [ c means that the absolutely continuous part of the measure ru lies in L2 even across [ c. The singular part is unavoidable in a potential formulation with circulation in domains that are not simply connected, however, only the absolutely continuous part of ru de nes the velocity eld. Elliptic regularity theory, which is applicable wherever (2.2) holds, gives interior H 2 regularity of the solution there. Across the coupling boundary ?1 , even if ?1 is smooth, one cannot expect more than H 1 , as is well known for elliptic problems with discontinuous coecients, see Fix and Strang [33] for a simple example. The question of regularity at the trailing edge is complicated by the fact that we put the slit there. This was done entirely for numerical convenience. For the moment we may suppose that touches the pro le somewhere else. Then generically

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the solution will have a corner singularity at TE and be only in H 1 there. Based on results for the linear theory by Djaoua [27], the stream function formulation by Ciavaldini, Pogu and Tournemine [20], [19], and 3{D potential ows by Dauge and Pogu [26] it is shown in [21] that the Kutta{Joukowski condition selects the unique value such that the solution has H 2 regularity up to the boundary also at TE . Now let us take a brief look at the existence of subsonic solutions of the weak, coupled boundary value problem (1.37). For the subsonic case after subtracting the circulation term, the problem (1.37) can be seen as an exterior nonlinear elliptic problem having discontinuous coecients across ?1 . Via Kelvin transformation, this problem becomes an interior corresponding nonlinear elliptic problem for which in the subsonic case existence follows as in [15], yielding existence for (1.37). To treat the full problem with unknown circulation , also in the transonic case, we present a nonlinear iteration in function spaces satisfying the Kutta{Joukowski condition (1.39) by correcting the iterates in every step with appropriate {values to enforce regularity at TE as described in [10]. For subsonic ows, a similar coupling involving the stream function formulation with Dirichlet conditions was treated in [29]. 2.1. The entropy condition. For the case in which the geometry of the problem and the boundary conditions lead to a locally supersonic ow near the pro le, the situation changes quite dramatically. In the supersonic regions the equation (1.3) becomes hyperbolic and the monotonicity property (1.32) is lost. In this case the existence of solutions to (2.1) is still open, even for problems with bounded domains, see Feistauer and Necas [31] and Morawetz [48]. A further complication comes from the fact that mathematical analysis by Morawetz [45], [46], [47], as well as physical and numerical experience with the problem, show that one generally has to expect solutions that have discontinuous derivatives ru, i.e. contain shocks. Hence, one has to consider weak solutions which in turn lead to non{uniqueness and the existence of non{physical solutions to the variational problem (2.1), see Necas [50]. This fact, which is well known from the related theory of systems of conservation laws, see Smoller [54], is the reason why Bristeau et al. in [15], [16], [17] need to supplement the problem with an additional admissibility condition for generalized solutions. We will concentrate here on the speci c numerical entropy condition used for our numerical implementation which we adopted from Glowinski and Pironneau [36]. Further discussion on admissibility of transonic shocks may be found in Necas [50], Key tz and Warnecke [40], and Warnecke [56]. Due to the assumption of isentropy, the shocks in transonic potential ow conserve entropy but not momentum. The practical entropy condition we have used requires that the divergence of the

ow eld must be bounded from above, i.e. (2.3) div ru B with an appropriate constant B 2 R : In Gohner and Warnecke [37] it is shown that this inequality is equivalent to the solution with compressive shocks and is violated by the non{physical expansion shocks. In weak form this means that the inequality (2.4)

Z

Z

? ru r dx B

dx for all 2 C01 ( ) with

0

must hold. The implementation of (2.4) via penalization is shown in Section 4.


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3. The coupled FEM{BEM formulation of the problem In order to discretize the coupled problem (1.37), we approximate the domain

by a family of polygonal domains h where h denotes the parameter of meshwidth. The outer boundary ?h1 of h is supposed to be a polygonal curve with nodes on ?1 which approximates ?1 . In the same way we de ne ?hP as the approximation of the pro le boundary ?P . Without loss of generality we may assume that the slit is already a part of the boundary of h . Together S with h we introduce now a family of regular triangulations fTh gh>0 with Th := i2D Ti where D is a nite subset of the natural numbers N. The nodal points of the triangulation are denoted by pi ; i = 1; : : : ; N . For convenience we assume that the nodes on the slit will be taken to be the rst 2L points, i.e. p+j ; p?j ; j = 1; : : :L, with p+j and p?j having the same coordinates but characterizing the limits from above and below, respectively. By whi ; i = 1; : : : ; N , we denote the usual piecewise linear hat functions satisfying whi (pj ) = ij , i; j = 1; : : : ; N , which form a basis of nonnegative functions for the piecewise linear continuous nite elements. By Ai we denote the areas of the corresponding supports, i.e. Ai =meas (supp whi ); i = 1; : : : ; N . For a triangle Ti 2 Th let i denote the smallest angle. We say that a family of regular triangulations fTh gh>0 satis es the angle property if there is a minimal angle > 0 such that for any h > 0 and any Ti 2 Th one has i . We denote by Si the segments on ?Sh1 given by the edges of the boundary triangles Ti of Th . By fSh gh>0 with Sh := i2B Si we denote the corresponding induced family of polygonal approximations of the boundary ?h1 , where B is a nite subset of N. Without loss of generality we assume that our triangulation h is chosen such that the corresponding family fSh gh>0 guarantees the validity of an inverse assumption, see Ciarlet [18, (3.2.28)]. This implies inverse estimates, see Ciarlet [18, Theorem 3.2.6]. The error analysis is carried out for a Galerkin discretization, also for the boundary element method. However, we implemented the boundary element method using point collocation. Then it can be shown that the asymptotic error estimates used in Section 5 still remain valid due to [2]. The family fThgh>0 itself does not need to be quasiuniform. For all further considerations, the parameter h will stand for the maximum diameter of all triangles Ti 2 Th . Let C 0( h ) denote the set of all continuous functions on h , having one{sided limits on the slit and @ h , respectively. For the discretization of (1.37) we introduce the following nite{dimensional spaces on the polygonal domains h ,

0 e (3.1) Vh := fevh 2 C ( h ) evhjTi is+linear? on every Ti 2 Th and evh ? evh = on with any 2 Rg ;

(3.2)

n

(3.3) n He h := eh 2 L2 (?h1 )

e jS is constant on every S 2 S and h e ; 1i = 0o i h

and the set of admissible nite elements (3.4)

o

Veh0 := veh 2 Veh evh+ ? evh? = 0 on ;

i

Ke s0 := fevh 2 Veh jrevh j2 s0 g :

h

h

h

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Here we use the notation (3.5)

Z

h; ih := (s) (s) ds ?h1

for the L2 {scalar product on ?h1 . For the following, we need an approximate kernel Kh which is de ned as in (1.24) where ~n(~y) is to be replaced by ~nh (~y), the linear interpolant of the normal vectors to ?1 at the vertices of ?h1 . The associated operator will be denoted by Kh . We further de ne the discrete forms Z ah (uejev; we) := (jruej2)rev rwe dx ;

h

eb (e; e) de (ue; e) è1 (ve; ) è2 ( e; )

:= hV e; eih ; (3.6) := hKh ue; eih ; h h := 1 hr (; ) ~nh ; evih ; h := h(I ? Kh ) (; ); eih : The function was de ned in (1.10). Now the discrete analogue of problem (1.37) reads as follows: h

The discrete variational problem Find (ueh ; e h ; h ) 2 Ke s0 He h R He h R such that

(3.7)

ah (ueh jueh; evh ) ? heh ; evh ih = èh1 (evh ; h ) for all evh 2 Veh ; ebh(eh ; eh) + hueh; ehih ? deh(ueh; eh) = èh2 ( eh ; h ) for all eh 2 Heh subject to F ( h ) = 0. Problem (3.7) leads to a system of nonlinear equations, where we have one unknown per node in the triangulation Th of h , one unknown per segment Si of the polygonal boundary ?h1 and the unknown circulation h . For the error analysis below we convert problem (3.7) into a form which allows the use of subspaces to the admissible function spaces used in (1.37). We will de ne subspaces Vh ; Vh0 and Hh satisfying the conformity inclusions Vh V; Vh0 V 0 and Hh H . This reformulation of problem (3.7) will enable us to present an analysis similar to that for conforming nite and boundary elements, see Johnson and Nedelec [39]. We introduce a mapping h : ?h1 ! ?1 , where h (x) is the point on ?1 closest to the point x 2 ?h1 . For h suciently small, the mapping h becomes a bijection, see LeRoux [43]. Hence, the inverse mapping ?h 1 exists. It transforms integrals along ?h1 into integrals along ?1 by (3.8)

Z

e(s) ds =

Z

e ?1(s)J (?1 (s)) ds ; h

h

?1 ?h1 ? 1 where J (?h 1 ) = j @ @sh j is the one{dimensional Jacobian. Here

dierentiation in the tangential direction. We now de ne the conforming boundary space o n (3.9) Hh := = J (?h 1 ) eh ?h 1 eh 2 He h H :

@ @s

denotes the


1419

We also need an appropriate subspace Vh V based on the de nition of Veh . Let evh 2 Veh be an arbitrarily given function. Then veh is well{de ned in \ h by taking the restriction. In the skin h !1 := ( n h ) \ fx j dist(x; ?1) hg ; however, we need an extension of evh which we de ne by setting vh (y) := veh (x) h with x 2 ?1 and any given y lying on the line segment between the two points x 2 ?h1 and h (x) 2 ?1 : In the same way as described above, we de ne the function veh in the skin !Ph = ( n h ) \ fx j dist(x; ?P ) hg : This de nes an extension operator which is only slightly dierent from Zlamal's operator used by Feistauer and Zenisek [32]. Our version is chosen in accordance with the boundary element approximation de ned by LeRoux in [43] and used by Johnson and Nedelec [39]. Note that for the piecewise C 1 {curve ?P we have measf(supp whi ) \ !Ph g ch3i hi Ai : We now denote by Vh the space of all functions vh de ned from evh 2 Veh in the above way. Since on h n the functions vh are well de ned, Vh consists of functions given on [ h while their restrictions to de ne a subspace of V . By Vh0 we denote the set of all functions vh 2 Vh , which are continuous across the slit . By changing the integrations from ?h1 to ?1 and using the de nitions of Vh ; Vh0 and Hh , we can reformulate problem (3.7) as follows. We note that ah (uh vh ; wh) := ah (ueh evh ; weh) ; and de ne (3.10) ZZ ??1(x); ?1(y) ds ds ; 2 p bh (; ) := ( y ) ( x ) G x y h h (j~v1 j2) 1 ? M12 ? ? 1 1 ZZ ? 2 u(y) (x) K ?h 1(x); ?h 1(y) [J (h?1 (y)]dsx dsy ; dh (u; ) := p 2 1 ? M1 ? ? 11

`h1 (v; ) := 1 h(r (; ) ~nh ) ?h 1; vi ; `h2 ( ; ) := h (; ) ?h 1; i ? dh ( (; ) ?h 1; ) : Note that the transformations h do not always map all functions of Ke s0 into Ks0 . But taking s00 < s0 , the functions on Ke s00 are mapped into Ks0 provided h > 0 is small enough. This distinction of s00 and s0 , however, is not signi cant for our error analysis. Therefore, in the following we carry out the analysis for the admissible approximations in Vh \ Ks0 . Using these de nitions we obtain: The conforming discrete variational problem. Find (uh ; h; h) 2 (Vh \ Ks0 ) Hh R such that (3.11) ah (uh juh ; vh ) ? hh ; vh i = `h1 (vh ; h ) for all vh 2 Vh0 and bh (h ; h ) + huh ; h i ? dh (uh ; h ) = `h2 ( h ; h ) for all h 2 Hh subject to F ( h ) = 0.

1420


The following consistency estimates essentially follow from the corresponding results by Feistauer and Z enisek [32] and by Johnson and Nedelec [39], respectively. Lemma 3.1. Provided the polygonal approximation Sh of ?1 satis es an inverse assumption, we have the following estimates for the bilinear and linear forms (3.10): ja(uh juh ; vh ) ? ah (uh juh; vh )j chkuh kW 1;2 ( )kvh kW 1;2 ( ) (3.12) for all (uh ; vh ) 2 (Vh \ Ks0 ) Vh ; (3.13) (3.14) (3.15)

jb(h ; h ) ? bh (h ; h )j chkh kH ? 12 (?1 ) k h kH 12 (?1) for all (h ; h ) 2 Hh Hh ; jd(v; h ) ? dh (v; h )j ch3=2kvkW 1;2 ( ) k h kH ? 12 (?1) for all (v; ) 2 V Hh ; j`h1 (v; ) ? `1 (v; )j (c1 + c2 j j)hkvkW 1;2 ( ) for all v 2 V; j`h2 ( ; ) ? `2 ( ; )j (c1 + c2 j j)hk kH ? 21 (?1) for all 2 H:

Proof. The estimates (3.13) and (3.14) can be found in the paper by Johnson and Nedelec [39]. Note that we need the inverse assumption for He h associated with fSh gh>0 for the proofs of (3.13), (3.14). The proof of (3.12) is due to the fact that on the skin the inequality kvh kW 1;2 (!1h [!Ph ) ch 12 kvh kW 1;2 ( ) for all vh 2 Vh holds. This inequality can be found in [32, Lemma 3.3.12], however for a slightly dierent extension operator. But the proof in our case is completely analogous to the one by Feistauer and Zenizek in [32]. Thus we obtain

ja(uh juh ; vh ) ? ah (uh juh ; vh )j

Z

[

(jruh j2)jruh j jrvh j dx

h !h !1 P

0 kuh kW 1;2 (!1h [!Ph ) kvh kW 1;2 (!1h [!Ph ) chkuh kW 1;2 ( ) kvh kW 1;2 ( ) : The functions (r ~nh ) ?h 1 and ?h 1 are piecewise uniformly Lipschitz conh tinuous and their derivatives are uniformly bounded in !1 . Hence we get the

estimates (3.16) ( (r ~nh ) ?h 1 ? r ~n (x; ) (c1 + c2j j)h ?1 ? (x; ) (c + c j j)h for all x 2 ?1 : 1 2 h Thus, with (3.14) and (3.16) we obtain `h (v; ) ? ` (v; ) = jh[(r ~n ) ?1 ? (r ~n)]; vij 1 1 h 1 h (c1 + c2 j j)hkvkL1(?1 ) (c1 + c2 j j)hkvkW 1;2 ( ) for all v 2 V ; the rst of the desired estimates (3.15). Similarly, for the second estimate of (3.15) we nd with(3.14)


`2 ( ; ) ? `2 ( ; ) h ?1 ? ; i + d ( ?1; ) ? d( ; ) (c1 + c2j j)hk k ?1=2 (?1 ) + d ( ?1 ? ; ) + jd ( ; ) ? d( ; )j h

h

h

1421

h

H

h

h

(c1 + c2j j)hk kH ?1=2 (?1 ) :

h

For later use we collect the following nite element approximation results. Lemma 3.2. (a) Let us denote by Ph : H ! Hh the L2{projection de ned by (3.17) hPh '; h i := h'; h i for all h 2 Hh : Then we have for every ' 2 H (3.18) k' ? Ph 'kH ? 21 (?1 ) ! 0 for h ! 0 :

(b) Further, we de ne the Ritz projections Reh : V ! Veh resp. Rh : V ! Vh by (3.19)

Z

h

Z

rRe hv revh dx = rv rvh dx for all veh 2 Veh

where vh 2 Vh is the extension of evh to . Then

lim

v ? Re hv

W 1;2 ( ) = 0 for all v 2 V h!0

and

Re v

c kvkW 1;2 ( ) for all v 2 V : ( h ) (c) There exist families fThgh>0 of triangulations having additional properties which h

W 1;2

imply that the Ritz projection de ned in (3.19) satis es the stability estimate

(3.20)

rRe v h

L1

( h )

c krvkL1 ( ) for all v 2 W 1;1 ( )

where the constant c 1 is independent of v; vh and h. Proof. (a) and (b) are well known properties, see Babuska and Aziz [3]. The esti-

mate (3.20) in (c) is e. g. a consequence of the quasiuniformity assumption, i. e. the angle property is satis ed and each triangle Ti 2 Th contains a circle of radius ch where the constant c > 0 does not depend on Ti or h, as was proved by Rannacher and Scott [52]. For the Ritz projection associated with homogeneous Dirichlet conditions, Crouzeix and Thomee [25] prove (3.20) for a much wider class of families fTh gh>0 which includes grids generated by most adaptive methods. Under our assumptions on ?P ; and ?1 , their proof can be modi ed so that (c) also holds for the Ritz projection (3.19).

4. The discrete minimization problem The goal of this section is the formulation of problem (3.7) as a discrete minimization problem. The underlying idea goes back to Glowinski and Pironneau [36], and has since been further developed, see Bristeau et al. [15] or Berger et al. [9]. For

1422


transonic ow, the hyperbolic character of the supersonic region creates additional diculties. As pointed out in Section 2, we must take into account an additional selection principle. This will be done by a penalization due to Glowinski and Pironneau [36]. To this end, we de ne the following functional Jh : Veh \ Ke s0 ! R by (4.1) 81 R a2 0 (subsonic ow) , > if globally s0 < 2+1 < 2 h jrh(h)j2 dx Jh (h ) := > 1 R : 2 jrh(h)j2 dx + Ph(h) for s0 < 2?a201 (transonic ow) ,

h

where the penalty functional Ph : Veh \ Ke s0 ! R is given by

02 Z 3+12 Z X 1 @4 5A: (4.2) P ( ) := 2 A ? r rw i dx ? B w i dx N

h

h

i=1 pi 62[?1

i

h

h

h

h

h

By []+ we denote the nonnegative part of the quantity in brackets. Here, > 0, B > 0 and 2 > > 1 are constants, which do not depend on h. These constants can be chosen according to numerical experiments depending on the speci c pro le, the travelling velocity ~v1 etc., but then they are xed for mesh re nement. The function h (h ) 2 Veh0 is the solution of the following state equation

Z

(4.3)

h

rh (h ) rvh dx

= ah (h jh ; vh ) ?hh (h ; h ); vh ih ? èh1 (vh ; h ) for all vh 2 Veh0 : This state equation is the nite element approximation of the Neumann problem for the Poisson equation with given Neumann data on ?P , see (1.15), and on ?1 where h is given. On the other hand, h ('h ) 2 He h is to be determined by the Galerkin discretization of the boundary integral equation (1.28), i.e. (4.4) ebh(h (h; h ); h) = èh2 ( h ; h) ? hh ; hih + deh(h; h) for all h 2 Heh : Instead of solving the discrete equations (3.7), we will now solve the following:

Discrete minimization problem es0 Heh I such that Find (uh ; h ; h ) 2 K Jh (uh ) := min (4.5) Jh (h ) f 2

h Ks0

under the constraints (2.4) and F ( h ) = 0 where I R is an appropriately xed nite interval.

Jh (h ) is de ned via (4.1) and

Remark. To simplify the notation, we are not using the e {sign for nite element

functions as previously. Since h is the circulation of uh , it is sought only in a bounded interval I . The solution of problem (4.5) subject to F ( h ) = 0, to (4.3) and (4.4) exists, because we


1423

minimize a dierentiable functional over a bounded, convex and nonempty subset of a nite{dimensional space. The nonemptyness is due to the fact that the zero function lies in Ke s0 and satis es F (0) = 0. Note that the minimization of the rst term of (4.1) in view of (4.3) is equivalent to the minimization of the W 1;2( ){ seminorm of the Riesz representation of the residual of (1.3) in the least squares sense. The above method, hence, can be considered as a least squares method. The solution uh 2 Ke s0 is not necessarily unique; nevertheless, any solution uh de nes a corresponding ux h (uh ) 2 He h as the unique solution of (4.4). Thus we may assume for a given sequence of meshsizes the existence of a sequence f(uh ; h ; h )gh>0, where uh 2 Ke s0 is a solution of (4.5) and h 2 He h is the corresponding solution of (4.4) with the corresponding h . If Jh (uh ) = 0, then (uh ; h ; h ) is a solution of (3.7). The discrete minimization problem (4.5) can be solved by a Polak-Ribiere type conjugate gradient algorithm, which takes into account the constraint F ( ) = 0 and the weak coupling equation (4.4). The method we used is described in detail in [10]. 5. On the convergence of the minimization method The goal of this section is the convergence proof for the sequence f(ueh ; ~ h ; h )gh>0 of solutions of the minimization problems (4.5). Under the assumption of existence and uniqueness for the solution to problem (1.37) we will show that the sequence f(ueh ; ~h ; h )gh>0 2 Ke s0 He h I converges to this solution. For the case of subsonic ow, the proof of this assertion is straightforward. The case of transonic ow will be more dicult. For the proof we need a compactness result from Mandel and Necas [44] and Murat [49]. For this case, a discrete entropy condition will play the crucial role and will enforce the convergence. 5.1. The case of subsonic ow. For this case, we need rst a preliminary result which involves standard results of approximation theory.

Lemma 5.1. Let u 2 V \ Ks0 with s0
0: To the function u there corresponds a 2 R which satis es F ( ) = 0. Then for suciently small meshsizes h, there exists a sequence of functions ueIh h>0 with corresponding eh satisfying 2 ueIh 2 Veh \ Ke se0 with s0 < se0 < 2+a01 ; 2 2 (5.2) F ( fh ) := rueIh jT+ ~t+ ? rueIh jT? ~t? = 0 ;

I

!0 0 for every p 2 [1; 1] : u ? uh W 1;p ( ) h?!

(5.1)

1424


Figure 2. The triangular grid involving the Kutta{Joukowski condition

Here uIh 2 Vh is the extension of ueIh 2 Veh to all of as de ned in Section 3. ~t+ ; ~t? are the unit tangential vectors at the trailing edge pointing in the direction of TE . Let T+ and T? denote the respective triangles adjacent to TE as indicated in Figure 2.

Proof. Let us denote by e hu the interpolant of u in h . Since u 2 V we have

e hu 2 Veh . By hu we denote the extension of e hu into Vh . Using (5.1), standard approximation results for " 2 N0, see Ciarlet [18, p. 123], interpolating these inequalities for noninteger " and nally using the property that the width of the skin is of order h2 , we obtain (5.3) ku ? hukW 1;1 ( ) ch" kukW 2+";2 ( ) :

The de nition of hu implies that on the equation (hu)+ ? (hu)? = still holds. Note that e hu does not satisfy the Kutta{Joukowski condition exactly, but only approximately; with (5.3), however, we obtain

2 2 re u j + ~t+ ? re u j ? ~t? = O(h ): We shall now modify the interpolant e u along the slit by changing the function (5.4)

h

h

T

T

"

h

values only in the respective upper and lower points p+j and p?j on as described in [10]. The modi ed function will be denoted by ueIh 2 Veh and it satis es F ( eh ) = 0 with a corresponding eh 2 R. From the construction of ueIh it follows that the inequalities (5.5) j ? eh j ch1+" and

(5.6)

e hu ? ueIh

W 1;1 ( h) ch"=2 are valid. Thus, combining (5.3) and (5.6), we obtain

u?u

I h W 1;p

h!0 ( ) ?! 0 for all p 2 [1; 1] :


1425

a0 a0 Now since s0 < 2+1 , for suciently small h we can nd a constant se0 < 2+1 such that jruIhj2 es0 holds a.e. in .

2

2

We shall now prove the convergence in V H I of the sequence f(ueh ; eh ; h )gh>0 of solutions of the minimization problems (4.5). Again, we shall denote by f(uh ; h ; h )gh>0 the corresponding sequence in (Vh Hh I ). a0 , 2 H and 2 I be the unique Theorem 5.2. Let u 2 V \ Ks0 with s0 < 2+1

2

solution of the coupled ow problem (1.37). Moreover let this solution possess the2 a0 regularity u 2 W 2+";2( ) for some " > 0. Then there exists a constant es0 < 2+1 such that the sequence f(ueh ; eh ; h )gh>0 ; ueh 2 Ke se0 ; eh 2 He h ; h 2 I of solutions of the minimization problem (4.5) converges in V H I to the solution (u; ; ) of problem (1.37). Proof. The proof of Theorem 5.2 will be split into several steps.

a2 0 and a function Step 1: Lemma 5.1 implies the existence of a constant es0 < 2+1 ueIh 2 Veh \ Ke se0 , satisfying (5.2). For this interpolant ueIh we take 2 R corresponding

to the exact solution and determine Ih by solving equation (4.4) with these ueIh and . By (4.3) we then determine the corresponding eh . Thus, using the de nition of f(ueh ; eh ; h )gh>0 as the solution of minimization problem (4.5) subject to F ( h ) = 0, we obtain 0 Jh (ueh ) = e min Jh (eh ) Jh (ueIh ) : f

(5.7)

2

h Kse0

We will now consider the expression Jh (ueIh ) as h tends to zero. With de nition (4.3), properties (1.31), (5.1), (5.3) and the interpolation property (5.4) we obtain 2Jh (ueIh ) =

(5.8)

Z ? 2 re ue dx

h

h

I h

= ah (ueIh j ueIh ; eh ) ? hIh ; eh ih ? èh1 (eh ; ) ? = ah uIh j uIh ; h ? Ih ; h ? `h1 (h ; ) = ah (uIh j uIh ; h ) ? a(uIh j uIh ; h ) + a(uIh j uIh ; h ) ? a(u j u; h ) +h ? Ih ; h i + `1 (h ; ) ? `h1 ( h ; ) ch uIh W 1;2 ( ) kh kW 1;2 ( ) + c u ? uIh W 1;2 ( ) kh kW 1;2 ( )

+c ? Ih H ? 21 (?1 ) kh kH 21 (?1) + (c1 + c2j j)h kh kW 1;2 ( )

n

c h( uIh W 1;2 ( ) + 1 + j j) o

+ u ? uIh W 1;2 ( ) + ? Ih H ? 21 (?1 ) kh kW 1;2 ( ) :

Using the L2 {projection Ph de ned in Lemma 3.2, inequality (1.34), equations (1.37) and (3.11), the continuity of the form d which follows from (1.29), together

1426


with the inequalities (1.33), (3.13), (3.14) and (3.15), we obtain the estimates

1 ? Ih 2H ? 12 (?1 ) b( ? Ih ; ? Ih ) = b( ? Ih ; ? Ph ) + b(; Ph ? Ih ) ? bh (Ih ; Ph ? Ih ) + bh (Ih ; Ph ? Ih ) ? b(Ih ; Ph ? Ih )

c ? Ih H ? 21 (?1) kPh ? kH ? 12 (?1 ) ? hu; Ph ? Ih i + d(u; Ph ? Ih ) + `2(Ph ? Ih ; ) ? d(uIh ; Ph ? Ih ) + d(uIh ; Ph ? Ih ) + huIh ; Ph ? Ih i ? dh (uIh ; Ph ? Ih ) ? `h2 (Ph ? Ih ; )

+ ch Ih H ? 12 (?1) Ph ? Ih H ? 21 (?1 )

c ? Ih H ? 21 (?1) k ? Ph kH ? 12 (?1 )

+ c u ? uIh H 21 (?1 ) Ph ? Ih H ? 12 (?1 )

+ ch 32 uIh W 1;2 ( ) Ph ? Ih H ? 12 (?)

+ (c1 + c2 j j)h Ph ? Ih H ? 21 (?1)

+ ch Ih H ? 12 (?1) Ph ? Ih H ? 21 (?1 ) :

We use Young's inequality ab "a2 + 41" b2 and the triangle inequality to obtain with the appropriate choice of " the consistency estimate k ? Ih kH ? 21 (?1 ) (5.9) cfk ? Ph kH ? 21 (?1 ) + ku ? uIh kW 1;2 ( ) + hkIh kH ? 12 (?1 ) g : We remark that this estimate is also valid for boundary element collocation instead of the Galerkin method, see [2]. Then together with (5.8) and (5.9) we have n

Jh (ueIh ) c h uIh W 1;2 ( ) + Ih H ? 12 (?1) + 1 + j j (5.10) o

+ k ? Ph kH ? 21 (?1 ) + u ? uIh W 1;2 ( ) kh kW 1;2 ( ) :

The sequence fuIh gh>0 is uniformly bounded in W 1;2( ) and hence by (3.11) and (1.34) we can show that fIh gh>0 is also uniformly bounded in H . The construction of h 2 Vh0 from eh 2 Veh0 implies together with (4.1) the estimate (5.11)

q

kh kW 1;2 ( ) c

eh

W 1;2 ( ) c

reh

L2 ( ) c Jh (ueIh ) : h

h

Combining (5.7), (5.10), and (5.11), the boundedness of fuIh gh>0 and fIh gh>0 and the approximation properties of uIh and Ph , we obtain (5.12) 0 Jh (ueh ) Jh (ueIh ) ! 0 for h ! 0 : 2. Step: Since the sequence fuhgh>0 Kse0 , it follows that the sequence is bounded in V . Again we can use (3.11) and the coercivity (1.34) to show that the sequence fh gh>0 is bounded in H . Therefore, the sequence f(uh ; h ; h )gh>0 is


1427

bounded in (V \ Kse0 ) H I . Since V \ Kse0 is a closed convex subset of V with respect to the topology in V , we may extract a subsequence f(uh ; h ; h )gh>0 and nd an element (û; ^; ^) 2 (V \ Kse0 ) H I such that uh * u^ weakly in W 1;p( ) for every p 2 (2; 1) ; (5.13) h * ^ weakly in H ; h ! ^ in R : We shall now show that f(uh ; h ; h )gh>0 converges strongly to (û; ^; ^). 3. Step: We combine the inequalities (1.34) and (1.32) and obtain

min( 1 ; 2 ) ku^ ? uh

(5.14)

k2

W

2

^ 1;2 ( ) + ? ? 1 2 (?1 ) h

H

a(ûju^; u^ ? uh ) ? a(uh juh; u^ ? uh ) +b(^; ^ ? h ) ? b(h ; ^ ? h ) = a(ûju^; u^ ? uh ) + b(^; ^ ? h ) ?a(uh j uh; u^ ? hu^) ? b(h ; ^ ? Ph ^) ?a(uh uh ; h ub ? uh ) + ah (uh uh ; h ub ? uh ) ?b(h ; Ph^ ? h ) + bh (h ; Ph ^ ? h ) ?ah (uh juh ; hu^ ? uh ) ? bh (h ; Ph^ ? h ):

The weak convergence (5.13) implies !0 0: (5.15) a(ûju^; u^ ? uh ) + b(^ ; ^ ? h ) h?! Using the boundedness of f(uh ; h ; h )gh>0 and the approximation properties of h and Ph due to Lemma 3.2, we obtain !0 0: (5.16) a(uh juh ; u^ ? hu^) + b(h ; ^ ? Ph ^) h?! The inequalities (3.12) and (3.13) imply ?a(uh juh ; hu^ ? uh ) + ah (uh juh; hu^ ? uh ) (5.17) !0 0: ?b(h ; Ph^ ? h ) + bh (h ; Ph ^ ? h ) h?! In order to prove that f(uh ; h ; h )gh>0 converges strongly to (û; ^; ^), it suces in view of (5.14){(5.17) to show that !0 0 (5.18) ah (uh juh ; hu^ ? uh ) + bh (h ; Ph ^ ? h ) h?! holds. In order to use the equality (4.3) it is necessary to show that hu^ ? uh 2 Vh0 . Usually this will not be the case. But since uh and h u^ are uniformly bounded in W 1;p( ), we modify the interpolant hu^ into ubh only along the slit , such that (5.19) ubh ? uh 2 Vh0 : To this end we use a xed sequence of nite element functions h 2 Vh with a jump of constant height 1 along satisfying kh kW 1;2 ( ) c and de ne ubh := h ub + ( h ? b)h :

1428


Then still

kubh ? ubkW 1;2 ( ) ! 0 for h ! 0 :

(5.20)

Now because of (5.19) the equalities (4.3), (4.4) with the test function ubh ? uh and, together with the de nitions (3.10), we obtain

ah (uh juh; ubh ? uh ) + bh (h ; Ph b ? h ) = `h1 (ubh ? uh ; h ) + hh ; ubh ? uh i +`Zh2 (Ph b ? h ; h ) ? huh ; Phb ? h i + dh (uh ; Ph b ? h ) + reh (ueh ) r(ubh ? ueh ) dx

h

(5.21) = `1 (ubh ? uh ; h ) ? `1(ubh ? uh ; b) +`h2 (Ph b ? h ; h ) ? `2 (Ph b ? h ; b) +dh (uh ; Phb ? h ) ? d(uh ; Ph b ? h ) +`1 (ubh ? ub; b) + `1 (ub ? uh ; b) + `2 (Ph b ? b; b) + `2 (b ? h ; b) +d(uh ; Ph b ? b) + d(uh ; b ? h ) +hh ; ubh ? ubi + hh ? b; ubi ?hZuh ; Phb ? bi ? huh ? ub; bi + h re(ueh ) r(ubh ? ueh ) dx : h

Using the result (5.12) and the boundedness of ubh ? ueh in V , we have (5.22)

Z

h

8 Z 9 12 0 yields (5.23)

d(uh ; b ? h ) = huh ; K0(b ? h )i ! 0 for h ! 0 :

Thus (5.22), (5.23) imply together with (5.21), (3.14), (3.15), the weak convergence (5.13) and the approximation property of h and Ph the result (5.19). Hence we have shown that (5.24)

uh ! ub strongly in V ; h ! b strongly in H ; h ! b in R for h ! 0 :


1429

4. Step: In the last step we show that (ub; b; b) must coincide with the solution (u; ; ) of (1.37). Using (4.3) for an arbitrary but xed v 2 V , we obtain the following inequality

a(ubjub; v) ? hb; vi ? `1(v; b) n ja(ubjub; v) ? a (ubjub; v)j + jh ? b; vij +ja (ubjub; v) ? a (u ju ; v)j + j`1 (v; o ) ? `1 (v; b)j h

h

h

h

h

h

h

+ jah (uh juh ; v) ? hh ; vi ? `1 (v; h )j

h

h

(5.25)

8 < :ja(ubjub; v) ? a (ubjub; v)j + jh ? b; vij +ja (ubjub; v) ? a (u ju ; v)j + j`1 (v; ) ? `1 (v; b)j +ja (u ju ; v ? R v) ? h 9; v ? R vi ? `1 (v ? R v; )j Z = e e + r (ue ) rR v dx ; : h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

Here Rhv 2 Vh is the Ritz projection of v de ned by (3.19). We obtain from (5.25), using Lemma 3.2, (3.12), (3.15), (5.11), (5.12) and (5.24), (5.26)

a(ubjub; v) ? hb; vi ? `1 (v; b) = 0 for all v 2 V :

In a similar way we obtain from (4.4) the inequality

b(b; ) + hub; i ? d(ub; ) ? `2( ; b) n jb(b; ? P )j + jb(b ? ; P )j + jhub ? u ; ij +jb( ; P ) ? b ( ; P )j + j`2( ; b) ? `2 ( ; )j +jd(ub ? u ; )j + jd(u ; ? P )j +jd(u ; P ) ? d (u ; P )j o h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

+ j`2 ( ? Ph ; h )j + jhuh ; ? Ph ij : h

Using the same arguments as for the proof of (5.26), we obtain the equality (5.27)

b(b; ) + hub; i ? d(ub; ) ? `2 ( ; b) = 0 for all 2 H:

Since ub 2 V \ Kse0 de nes a subsonic ow, Theorem 1.3 implies the validity of the Kutta{Joukowski condition at TE . Thus we have proven that (ub; b; b) is a subsonic ow solution of (1.37). The assumption on the uniqueness of such a solution implies that (ub; b; b) = (u; ; ) holds and moreover, that the whole sequence of solutions f(uh ; h ; h )gh>0 of the minimization problem (4.5) converges to the solution.

1430


Note that due to the uniform L1 {bound for the gradients of the sequence fuh gh>0 , the latter converges in the W 1;p ( ){norms for p 2 [1; 1). 5.2. The case of transonic ow. The purpose of this section is a convergence proof for the sequence f(ueh ; eh ; h )gh>0 of solutions of problem (4.5). The main underlying ideas for this proof go back to Berger [6], who has proved the convergence of the interior problem with a simpli ed boundary condition on polygonal domains. An extension of this work to domains with arbitrary curved boundaries can be found in Berger and Feistauer [7]. Therefore we shall now state the corresponding main results, which enable us to show the convergence of the exterior coupled problem.

Lemma 5.3. Let fThgh>0 be a family of triangulations satisfying the uniform angle property. This implies that the supports of the hat functions whi contain at most a certain nite number of triangles independent of h. This in turn implies that there is a constant c > 0 such that Ai ch2 for i = 1; : : : ; N and all 0 < h 1 :

(5.28)

Then a family of conforming solutions uh to (4.5), resp. (3.11), satis es the estimate

(5.29)

Z

Z

? ruh rvh dx B vh dx + ch"?1kvh kL1( ) for all vh 2 Eh

with 2 > " > 1 and B as in (4.2) where

Eh := vh 2 Vh vh 0 and vh = 0 on ?1 [ : Remark. Note that this result is slightly stronger than Berger's in [6, Theorem

4.1].

P w with 0 aci =1 cording to the correspondence between v and ev in Section 3. By using kve k1 =

Proof. The nonnegativity of vh 2 Eh implies veh = h

maxi=1;::: ;N i , this gives

Z

N i

i

Z

h

0 Z h 1 Z X @ = ? rue rw i dx ? B w i dxA i=1

h

h pi 62[?1 2 Z 3+ Z X 4 ? rue rw i dx ? B w i dx5 : kve k1 N

i

h

h

h

N

h

i=1 pi 62[?1

h

h

h

j

h

? rueh revh dx ? B veh dx

(5.30)

h

h

h

h


1431

The Cauchy{Schwarz inequality gives with (4.2)

2 3+ Z X 4 Z ? rue rw i dx ? B w i dx5 N

i=1 pi 62[?1

h

h

h

h

h

1 8 02 Z 3+ 129 2 1 > ! > Z 2 = < X 1 @4 X ? rue rw i dx ? B w i dx5 A > A > A ; : i=1 =1

h

h pi 62[?1 1 X !2 r 2 A = P (ue ) =1 21 X ! 12 r 2 =1max A ?1 A P (ue ) : N

N

" i

N

h

" i

i

" i

h

h

h

h

i

i

;::: ;N

N

" i

i

=1

h

h

i

By our assumptions on the mesh we have

Ai Ch2 and

X N

=1

Z

Ai C dx = C j hj :

h

i

By (4.5) we have the uniform bound

0 Jh (ueh ) Jh (0) = 21 krh (0)k2 + Ph (0) :

Therefore (5.30) implies with the above estimates the inequality

Z

Z

h

h

? rueh revh dx ? B veh dx ch"?1kevh k1 = ch"?1 kvh k1 ; since kevh k1 = kvh k1. Further, we have

Z Z Z Z ve dx ? v dx v dx + ev dx ch2kv k 1

h

h n

n h h

h

h

h

h

due to [32]. Correspondingly, there holds

Z ru rv dx ? Z rue rev dx

h Z Z rue rev dx ru rv dx + hn

n h ? se0 krv k 1( n h ) + krev k 1 ( h n ) (j n j + j n j) se0 ckv k1 h h

h

h

h

h L

h

h

h

h L

h

h

h

h

due to the inverse estimate [18, Theorem 3.2.6]. Since 1 < " < 2, we have h h1?" for 0 < h 1 and (5.29) follows.

1432


Lemma 5.4. Under the assumptions of Lemma 5.3 let fueh gh>0 be a sequence of solutions to problem (4.5) with h ! 0 and let fuh gh>0 be the corresponding sequence of conforming approximations de ned in Section 3. Then there exists a subsequence fueh gh>0 and a function u 2 V \ Ks0 such that the strong convergence !0 0 for all p 2 [1; 1) ku ? uh kW 1;p ( ) h?!

(5.31)

and the entropy inequality

(5.32)

?

Z

ru rv dx B 0

Z

v dx for all v 2 E

hold for an appropriate B 0 > 0 where

E := v 2 W 1;2( )j v 0 and v = 0 on ?1 [ : Proof. The sequence fuh gh>0 satis es (5.29). The Sobolev embedding theorem applied to the case W 1;q ( ) L1 ( ) for q > 2 gives for all vh 2 Eh

Z

Z

GhB (vh ) := ruh rvh dx + B vh dx ?ch"?1kvh k1;q : Choosing B 0 large enough there exists an h0 > 0 such that for any h with 0 < h < h0 we obtain for B 0 > ch0"?1 + B the estimate

GhB0 (vh ) 0 : Due to kruh k j j 21 kruhk1 j j 21 s0 we have with an appropriate constant c>0 jGhB0 (vh )j kruhk krvh k + B j j 21 kvh k ckvh k1;2 : This implies GhB0 2 [W 1;2( )]0 and thereby GhB0 2 [W 1;q ( )]0 for q 2 since is a

bounded domain. The sequence is bounded and therefore has a weakly convergent subsequence. The Corollary to Lemma 3.1 in Mandel and Necas [44] gives GhB0 ! G in [W 1;q ( )]0 for q > 2. As in the rst part of the proof of Feistauer and Necas [31, Theorem 4.23] this implies the strong convergence uh ! u in W 1;2( ). Since kruhk1 is bounded by s0 this implies convergence in W 1;p( ) for p 2 [1; 1). B0 The inequality Gh (vh ) 0 implies (5.32) in the limit when taking vh ! v in W 1;2( ). For the following convergence proof we have to modify and extend the density in order to obtain a function e: [0; 12) ! [0; 1) such that e(s) c for some constant c > 0. We choose s 2 s0 ; 2?a01 , set e(s) = (s) for s 2 [0; s ] and use the extension described in Feistauer and Necas [31] whereby e(s) = c for large s. Note that the set of solutions to problem (1.37) is not changed by this modi cation.

Theorem 5.5. Suppose that the following assumptions hold: (a) Let the family of triangulations be quasiuniform.


1433

(b) The problem (1.37) has exactly one solution (u; ; ) 2 (V \ Ks0 ) H I , which satis es the entropy condition (5.32). (c) The solution of (1.37) satis es the subsonic ow condition (1.38) near the

trailing edge. Then there exists a constant se0 s0 such that a sequence of solutions f(ueh ; eh ; h )gh>0 of the modi ed minimization problem

Jh (ueh ) =

(5.33)

min

~2 \

eh Ks~ 'h V 0

Jh ('eh )

subject to (4.3), (4.4) and F ( h ) = 0 converges to the unique solution (u; ; ) of the variational problem (1.37). Proof. We shall split the proof into three steps.

Step 1: The assumption (3.20) implies that Re hu 2 Veh \ Kc s0 . We modify

Rehu along the slit such that the modi ed function ueIh satis es the Kutta{Joukowski condition. This modi cation is the same as the one described in the proof of Lemma 5.1 and in [10] Since Rehu is uniformly bounded in W 1;1( h ), the modi ed function will also be uniformly bounded in W 1;1 ( h ). Therefore, we can nd a constant es0 c s0, such that

rue

es 1 0

(5.34)

I h

holds. Moreover, the modi ed function has the same approximation properties as Rehu, which implies (5.35)

u ? u

I h W 1;2

( ) ! 0 for h ! 0 :

Step 2: Since the density is now modi ed, the functional Jh may be applied to ueIh 2 Veh \ Kse0 ; and we obtain from (5.34) the inequality (5.36)

Z 2 0 Jh (ueh ) Jh (ueIh ) = 12 reh (ueIh ) dx + Ph (ueIh ) :

h

By using the same arguments as in the proof of (5.12) we get with the help of (5.35) (5.37)

1 Z re (ueI ) 2 dx ! 0 for h ! 0 : 2 h h

h

Note that our function ueIh coincides with Rehu in the points pi for i = 2L +1; : : : ; N . Since our penalty functional has no contributions from nodes belonging to we have Ph (ueIh ) = Ph (Reh u). Further, we have by the de nition of the Ritz projection (3.19), by the fact that u satis es the entropy inequality (5.32), and by the nonnegativity

1434


of wehi , that 0 Ph (Reh u)

2

3+

N X 1 4? Z rRe u rwe dx ? B Z we dx5 = 2 h hi hi (Ai )" i=1

h

pi 62[?1

h

2

3+ 2 Z Z X 1 4? ru rw dx ? B w dx5 2 2 i i (A ) i=1

pi 62[?1 2Z 3+ Z X 1 4 B 5 2 +2 w w e i dx ? i dx (A ) i=1

h pi 62[?1 32 2 X 1 6Z B w i dx75 = 2 (A ) 4 N

i

h

h

"

h

i "

h

N

i=1 pi 62[?1

i "

h

n h

N X 1 h 2 B 2 i=1 A"i (meas supp whi ) \ !P :

pi 62[?1

Since the grid satis es the uniform angle property and P is piecewise smooth we have meas(supp whi ) \ !Ph ch3i and ch2i Ai . Hence, we obtain 0 Ph (Reh u) c

X N

i=1 pi 62[?1

h2i h4i ?2" c0h2(2?") :

If we combine this with (5.36) and (5.37) we obtain the result (5.38) Jh (ueh ) ! 0 for h ! 0 : Step 3: We now apply Lemmata 5.3 and 5.4. We get the existence of a subsequence fueh gh>0 and a function ub 2 Kse0 such that !0 0 ; kub ? uh kW 1;p ( ) h?! (5.39) for all p 2 [1; 1) ; (5.40)

Z

Z

? rub rv dx B v dx

for all v 2 E :

The convergent subsequence fuh gh>0 is bounded. Then (3.11) holds. Taking

h = h and using the coercivity (1.34) gives the boundedness of the corresponding sequence fh gh>0 in H . We may derive the analogue of (5.9) for b ? h . Then the strong convergence (5.39) together with the boundedness of fh gh>0 in H implies

(5.41)

b

? h

H? 2

1

(?1 )

!0 0 ; ?!

h

(5.42) b(b; ) + hub; i ? d(ub; ) = `2( ; b) for all 2 H : Using the same arguments as in Step 4 of Theorem 5.2 together with (5.39){(5.41) and assumptions (a), (b) we obtain the result.


1435

Remark. The result of Theorem 5.5 is a generalization of the results proved in the

papers by Berger in [6] and [5]. 6. Numerical results We present some results of our numerical computations made in 1989. Here, we compare three dierent treatments of the far eld boundary condition. The rst condition corresponds to the condition (1.8) which is just the parallel

ow at in nity. In the second case we rst compute the FEM{BEM approximation to the harmonic solution U of the exterior incompressible ow problem, U = 0 in [ c ; @n U = 0 on ?p and rU ! ~v1 at in nity ; U + ? U ? = e and @nU + ? @nU ? = 0 on subject to F ( e) = 0 : Note, that our algorithm for the full coupling procedure needs only a slight modi cation to obtain U . With U , we used as a second boundary condition (jruj2)@n u = (j~v1 j2)@n U on ?1 : The third case shows the results of the complete coupling described in Section 4. For better comparison we give results for two standard test cases of ows around the NACA{0012 pro le. Two dierent sized C{grids were used, see Figure 3. Here ?1 has two corner points where the C 1 assumption used in the foregoing analysis is violated; however, due to the remark in Section 1.2, our convergence result can be extended to this simple Lipschitz curve. A large computational domain with 115 by 15 nodes, outer boundary 6 chord lengths from the pro le; and a smaller

Figure 3. Large computational domain, containing the small subdomain

1436


subdomain with 111 by 13 nodes, outer boundary 3 chord lengths from the pro le. We give the lift coecients ca calculated from the pressure distribution along the pro le. Test Case 1: (purely subsonic ow) M1 = 0:63 = 2:0 ca large domain ca small domain 1st case 0.3470 0.3684 2nd case 0.3395 0.3481 full coupling 0.3371 0.3391 In a research report Kroll and Jain [41] give a lift coecient of ca = 0:3333. They used an O{grid with 256 by 64 nodes, outer boundary 50 chord lengths from the pro le. Their calculation was done with a potential and with an Euler code. Test Case 2: (transonic ow) M1 = 0:8 = 1:250 ca large domain ca small domain 1st case 0.4853 0.5852 2nd case 0.4705 0.5257 full coupling 0.4209 0.4459

Figure 4. Pressure distributions for the transonic test case 2 on

the large domain


1437

Figure 5. Pressure distributions for the transonic test case 2 on

the small domain

In Rizzi and Viviant [53] a number of solutions for this test case is given. These were obtained by nite dierences and nite volume discretizations of the full potential equation. The calculated lift coecients vary between 0.5 and 1.1. In the AGARD Report Nr. 211 [1] newer results for the Euler equations were published. There the lift coecients vary between 0.35 and 0.37. The improvement in the lift coecients corresponds to a movement of the shock location to an upstream position and a slight reduction in shock strength. This is shown in Figures 4 and 5 for the above cases. Acknowledgement

The authors would like to thank the referee for his critical and helpful remarks. References [1] AGARD Advisory Report No. 211, Test Cases for Inviscid Flow Field Methods, Report of Fluid Dynamics Panel Working Group 07, 1985. [2] D. N. Arnold and W.L. Wendland, "The convergenceof spline collocationfor strongly elliptic equations on curves", Numer. Math. 47, 317-341 (1985). MR 87f:65142 [3] I. Babuska and A. K. Aziz, "Survey lectures on the mathematical foundations of the nite element method", in The Mathematical Foundation of the Finite Element Method with Applications to Partial Dierential Equations, A.K. Aziz (ed.), Academic Press, New York 1-359 (1972). MR 54:9111

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E-mail address :

[email protected]

Mathematisches Institut A, Universitat Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany

E-mail address :

[email protected]