INRIA. B.P. 105 Le Chesnay Cedex France. Moulay D. Tidriri. NASA Langley Research Center. Hampton VA Abstract

Transcription

1 Convergence Analysis of Domain Decomposition Algorithms with Full Overlapping for the Advection-Diusion Problems. Patrick LeTallec INRIA Domaine de Voluceau Rocquencourt B.P. 05 Le Chesnay Cedex France Moulay D. Tidriri Institute for Computer Applications in Science and Engineering NASA Langley Research Center Hampton VA Abstract The aim of this paper is to study the convergence properties of a Time Marching Algorithm solving Advection-Diusion problems on two domains using incompatible discretizations. The basic algorithm is rst presented, and theoretical or numerical results illustrate its convergence properties. This work has been supported by the Hermes Research program under grant number RDAN 6./3. The author was also supported by the National Aeronautics and Space Administration under NASA contract NAS-940 while he was in residence at the Institute for Computer Applications in Science and Engineering. i

2 Introduction Domain decomposition methods have become an ecient strategy for solving large scale problems on parallel computers ([], [2], [3], [4], [5], [6]). Nevertheless, they can also be used in order to couple dierent models [], [], [9] and [2]. In this paper we will examine a domain decomposition strategy which can be applied to such situations. This approach was introduced in order to solve several diculties that occur in uid mechanics. In particular, our aim is to introduce several subdomains in order to ally introduce an enriched model next to a domain boundary. For this purpose, we propose to fully overlap the subdomains and to couple the solutions through natural \friction" (Neumann) forces acting on the internal boundary of the domain, these friction forces being updated inside the time marching algorithm used for the solution of the initial problem. The theoretical study of our method will be done on an Advection- Diusion problem, which will serve as our model problem from now on. The analysis will be made at the continuous level, independently of any discretization strategy, which means that the derived results will be mesh independent. The use of friction (Neumann) coupling boundary conditions makes the convergence analysis somewhat dierent of the analysis done in Kuznetsov [22] or Rannacher [23] in their study of explicit Schwarz additive methods for time evolution parabolic problems. In the next section we will describe this model problem. In the third section we will present our algorithm for some basic cases. The fourth section will treat the one-dimensional stationary problem. We will show also that the convergence of this method can be improved by introducing a relaxation parameter [5]. The fth section will focus on the linear convergence of the implicit version of the coupling algorithm in the general multidimensional case. In the last section we study the numerical stability of the explicit algorithm. Practical applications of the proposed algorithm to real life CFD problems can be found in [4], [9], [20], and [2]. 2 The Model Problem Consider a bounded domain, of R n such that its is lipschitzian and a connected domain of R n with (g. ). The

3 Ω Γi Γ Γb Ω Figure : Description of the computational domain. boundaries of the two subdomains are dened as follows: b ; (internal boundary) i \ ; (interface) b : (fareld boundary) We denote by n the external unit normal vector or@. We will make use of the following notation kvk 0;O = kvk L2 (O) kvk s;o = kvk H s (O) jvj ;O = krvk L2 (O) where O is an open bounded domain of R n. Let v be the veity eld inside a given incompressible ow such that: < : divv = 0 in ; v:n = 0 on b: () We consider the following convection-diusion model problem: Find ', a real valued function, dened on and satisfying 2

4 div(v') ' = 0in; ' = ' on ; ' = 0on b : Above v is the owveity and is the diusion coecient. Problems of this form typically occur in uid mechanics, gas dynamics or wave propagation. Most CFD algorithms will in fact consider the solution of this problem as the stationary solution of the evolution problem (3) described below : Find :(0;T)!R such + div(v) = 0 in = on (0;T); = 0 on b (0;T); (0) = 0 in : The general CFD algorithm consists then in integrating (3) in time until reaching a stationary solution. 3 General Algorithm 3. Time Continuous Case Let us introduce the al subdomain (see g. ) which has as external boundary i, and let us consider the trace of on the subdomain, as an independent variable, to which we associate an arbitrary independent initial value ol 6= 0 j.wenow replace the evolution problem (3) by the following evolution system : Find (resp. ):!R(resp.! R) + div(v) = 0 in = @n on b (0;T); 3 (2) (3) (4)

5 + div(v ) = 0 in (0;T); = 0 on b (0;T); = on i (0;T); (5) (0) = 0 in ; (0) = ol in : (6) Remark 3. The global problem (4) with the initial condition (6) has no no-slip boundary condition. This suppresses the boundary layer which appears at low viscosity and facilitates the numerical solution of this problem. The boundary layers are modeled by the al problems (5)-(6) which are only to be solved on a small domain, with a very ne discretisation if needed. The two problems are only coupled by their boundary conditions and not by volumic interpolation. 3.2 Time Discrete Case The general algorithm that we propose for the solution of our model problem (2) is as usual to integrate in time the evolution problem (4)-(5)-(6) until we reach a stationary solution. This integration in time is then achieved by the following uncoupled semi-explicit algorithm, where the operators are treated implicitly inside each subdomain and where one of the coupling boundary conditions is treated explicitly and the other is treated implicitly: set 0 = ol and 0 = 0 ; then, for n 0; n and n being known, solve successively n+ n+ t t n n + div(v n+ ) n+ = 0 in ; n+ = n on i; n+ = 0 on b; + div(v n+ ) n+ = 0 in ; n+ = on ; @n on b: 4

6 Remark 3.2 We have a full decoupling between (7) and (). They can (and actually will) be discretized and solved by two independent solution techniques. Remark 3.3 The fully implicit version of this method consists in replacing the condition : n+ = n on i by the condition : n+ = n+ on i: The two subproblems are then coupled at each time step. Remark 3.4 If we replace in()by E dened as follows : E =n ; and b by i, and if we set t =, we obtain a nonoverlapping version of our strategy, which is a standard Dirichlet-Neumann algorithm [6], [7] and therefore requires a relaxation strategy to converge. Remark 3.5 The initial condition ol is not assumed tobeequal to 0 on the al subdomain because in most cases this condition is impossible to impose at the discrete level since the grid used on will be in general dierent from the grid used on. In addition, even if we assume ol = 0, we will not have n = n on unless we use the fully implicit algorithm on compatible grids. 4 Stationary one-dimensional case For t =+, the above algorithm can be written : set 0 = 0 and 0 = 0, then, for n 0; n and n being known, solve div(v n+ ) n+ = 0 in ; n+ = n on i; n+ = 0 on b; (9) 5

7 div(v n+ ) n+ = 0 in ; n+ = @n on b: (0) In one space dimension, we take the global domain to be the interval ]0; [ of R decomposed into two fully overlapping subdomains =]0; [ and =]h 2 ; [ with 0 <h 2 <: () We then consider the following one dimensional problem Find ', a real valued function, dened on and satisfying v' 0 ' 00 = 0on; '(0) = a; '() = b (2) with a constantveity v. In this one dimensional case, the above algorithm corresponds to: v' (n)0 2 ' (n)00 2 = 0 on ]h 2 ; [; ' (n) (n ) 2 (h 2) = ' (h 2 ); ' (n) 2 () = b; v' (n)0 ' (n)00 = 0 on ]0; [; ' (n) (0) = a; ' (n)0 () = ' (n)0 2 (): (3) (4) By introducing two relaxation parameters and 2,we can also introduce the following variant of the above algorithm : 6

8 v' (n)0 2 ' (n)00 2 = 0 on ]h 2 ; [; ' (n) 2 () = b; ' (n) (n ) (n ) 2 (h 2) = 2 ' (h 2 )+( 2 )' 2 (h 2 ); v' (n)0 ' (n)00 = 0 on ]0; [; ' (n) (0) = a; (n )0 ' (n)0 () = ' (n)0 2 () + ( )' (): (5) (6) We shall now exhibit the conditions under which the algorithm (5)-(6) converges, and those for which this convergence is optimal. For this purpose, we write the interface solution under the form ' (n)0 () = ' 0 () + n ; (7) ' (n) 2 (h 2)='(h 2 )+ n ; () where ' is the solution of the initial problem (2). Using the analytical solutions of the problems (5) and (6), we obtain the following induction formula!! n (n ) n = M IN (n ) : (9) with M IN = 0 ( 2 )( v ) e ( v )(h 2 ) 2 2 v e ( v ) (e v h 2 ) e ( v ) 2 (e ( v )h 2 ) (e ( v )(h 2 ) ) +( ) C A (20) This iterative process converges if the spectral radius of the matrix M IN is less than. A direct but tedious calculation then yields: Lemma 4. The spectral radius of the transfer matrix of the algorithm (5)-(6) is: 7

9 (M IN )=max[ 2 jd p D 2 4Rj] (2) with D =2 ( + 2 )+ 2 e ( v=) (e (v=)h 2 ) e v= e v(h 2=) (22) R =( )( 2 ): (23) From this calculation we obtain the following results: i) When h 2 goes to (nonoverlapping), D goes to +, and then, (M IN ) goes to +. There is no-convergence at this limit. ii) The optimal convergence is obtained in the case where all the eigenvalues of the matrix M IN are zero, i.e., when : D = 0 and R = 0. The latter conditions imply in particular = or 2 =: If we choose, in addition, = 2, the condition D = 0 implies h 2 =0. In this case the subdomain is equal to the whole domain, and the associated algorithm is no-longer of interest. iii) The convergence of the method depends symmetrically on both relaxation parameters. According to ii) it is reasonable to take one of the i equal to and call the other. By setting: e A = ( v=) (e (v=)h 2 ) (24) e (v=)(h 2 ) we then have In this case, setting (M IN )=j Aj: (25) e opt = f ( v=) (e (v=)h 2 ) g e (v=)(h ; (26) 2 ) which is <, we get the following convergence results:

10 Theorem 4. ) The convergence is optimal (convergence in iteration) if = opt : (27) 2) The algorithm converges for all in ]0; 2 A [. Corollary 4. ) The case without relaxation ( =)converges only if : 2 A ; i.e., by setting d = h 2 (overlapping length), only if : d v Log 2 ( + e v= ) (stability condition). 2) When v goes to zero, we must have d 2. Remark 4. This theorem states that the application of the algorithm (5)- (6) to the time-independent problem (2) converges only if the overlapping d is suciently large. In the same situation, we will see that if the problem (2) can be regarded as the steady solution of a time-dependent problem and we apply our strategy to this evolution problem, the resulting algorithm will converge to the same steady solution but with less restrictions on d. This motivates the introduction of the time marching algorithm of section 3. Moreover, this time marching technique is well adapted to nonlinear problems such as those encountered in uid mechanics (see [4], [9], [20], and [2]). 5 Implicit Time Discretization 5. The General Algorithm This section deals with the convergence analysis of the proposed algorithm in multiple dimensions when one uses the fully implicit version of our strategy (4)-(6) : Set 0 = ol and 0 = 0 ; then, for n 0; n and n being known, solve 9

11 n+ t n + div(v n+ ) n+ = 0 in ; n+ = @n on b; (2) n+ t n 5.2 Convergence Analysis + div(v n+ ) n+ = 0 in ; n+ = n+ on i; n+ = 0 on b: (29) Before establishing the convergence result we shall state the preliminary results that are central to the proof of the convergence of our algorithm. The rst result states the basic L 2 and H al estimates. Lemma 5. We have the following estimates: 2t kn+ n+ k2 0; + j n+ n+ k n+ n+ k2 0; j2 ; 2t kn n k 2 0; ; (30) +2tc kn n k 2 0; ; (3) k n+ n+ k2 0; ( +2tc )n+ k 0 0 k 2 0; ; (32) k n+ n+ k2 0; + 2t nx i=p where c is the Poincare constant on subdomain. j i+ i+ j2 ; k p p k2 0; p n; (33) 0

12 Proof of lemma 5. Substracting (29) from (2), multiplying the result by n+ n+ and integrating by parts over,we obtain the classical following relation: Z t (n+ n+ )2 Z + Z ) t (n n )( n+ n+ jr( n+ n+ )j2 =0: (34) By using the Cauchy-Schwarz inequality, we obtain the estimate (30). The second estimate (3) follows by using the Poincare inequality with c the Poincare constant bounding the squared H seminorm of any function v of H ( ) with zero trace on i by its squared L 2 norm. By induction we also obtain the basic L 2 estimate (32). And nally, we obtain the estimate (33) by summing (30). The above lemma states that the restriction of n+ n+ to converges to 0 in both L 2 and H norms. We shall establish now other L 2 and H al estimates. Let x n be dened by x n = (n+ n+ ) (n t n ) ; (35) and let G be dened by G(n) = kvk2 2 2 kn n k2 0; + j n n j2 ; : (36) Lemma 5.2 We have the following estimates: kx n k 2 0; (G(n) G(n + )) (37) t G(n +) ( 2t +kvk2 2 2 )kp p k2 0; ; p n: (3) Proof of lemma 5.2 Substracting the two rst equations in (2) and (29), multiplying the result by x n and integrating over we obtain

13 0 = Z jx n j Z Z Z div(v( n+ n+ ))xn r( n+ @n (n+ n+ )xn : Using the boundary conditions in (2) and (29), and the Cauchy-Schwarz inequality we obtain kx n k 2 0; 2 kvk2 jn+ n+ j2 ; + 2 kxn k 2 0; + 2t jn n j 2 ; 2t jn+ n+ j2 ; : (39) Using now the relation (30) (lemma 5.) leads to the rst estimate of our lemma. In fact, this estimate implies that G is a decreasing function. This property then yields (n +2 p)g(n+) n+ X i=p n+ X i=p G(i) j i i j 2 ; + kvk2 2 2 n+ X i=p k i i k 2 0; (40) Using again the relations (30) and (3) (lemma 5.) yields the second estimate (3). And the lemma is proved. We shall establish now the global L 2 and H estimates. Let n+ be dened as follows n+ = < : n+ ' in n ; n+ ' in ; (4) with ' the solution of the stationary problem (2). By construction, n+ satises the following equations: 2

14 n+ t n + div(v n+ ) n+ = 0 in [ (n ); n+ = 0 n+ continuous across i: (42) Let A, B, and B 2 be dened by the following relations A = +(c 2c 2 )t; B = t c 2 A; B 2 =t( kvk2 c 2 +)A; where c is the Poincare constant and c > 0 is an arbitrary constant as will be seen in the proof of the following lemma. Lemma 5.3 We have the following estimates ( 2t c 2 )k n+ k 2 0; + 2 j n+ j 2 ; 2t k n k 2 0; + 2c 2 kx n k 2 0; + ( kvk2 2c )jn+ n+ j 2 ; (43) k n+ k 2 0; Ak n k 2 0; + B kx n k 2 0; + B 2 j n+ n+ j 2 ; (44) Proof of lemma 5.3 Multiplying the equation (42) by n+ and integrating by parts over and n, and taking into account the boundary conditions in (42) we obtain the following relation: Z n+ Z n n+ + jr n+ j (n+ n+ ) n+ =0: i 3 Z (45)

15 On, n+ n+ satises the following equation ( n+ n+ ) ( n n ) + div[v( n+ n+ )] ( n+ n+ )=0: t (46) Therefore, multiplying the above equation by n+,integrating by parts and using the Cauchy-Schwarz inequality we obtain Z (n+ n+ ) n+ j 2c 2 kx n k 2 0; + 2 c2 k n+ k 2 0; + kvk2 2c 2 j n+ n+ j 2 ; + 2 c2 k n+ k 2 0; + 2 jn+ n+ j 2 ; + 2 j n+ j 2 ; ; with c > 0 arbitrary. Combining the above inequality with (45), bounding the al norm jfj i; by jfj i; and using the Cauchy-Schwarz inequality we obtain the estimate (43). The estimate (44) results immediately from the estimate (43) by applying the Poincare inequality on with c the Poincare constant. And the lemma is proved. Finally, we are in a position to state the main result of this section. Theorem 5. The solution of the algorithm (2)-(29) converges linearly in H () to the solution of the stationary problem (2), for all values of t and all choices of. Proof of theorem 5. Let c be chosen such that c 5.3) we obtain by induction 2c 2 > 0. Using the relation (44) (lemma k n+ k 2 0; A p k n+ p k 2 0; + P p i=0 Ai (B kx n i k 2 0; + i B 2 j n+ n+ i j 2 ; ): (47) Since A<by assumption on c, this implies 4

16 k n+ k 2 0; Ap k n+ p k 2 0; + A(B B 2 n X i=n+ p nx i=n+ j i+ i+ j 2 ; ): p kx i k 2 0; + (4) Now, using (37) (lemma 5.2) and (33) (lemma 5.) we obtain k n+ k 2 0; A p k n+ p k 2 0; + A(B (G(n + p) G(n+ ))+ t B p 2 n+ p k 2 0; ): 2t kn+ The same relation written between 0 and n + p yields (49) k n+ p k 2 0; An+ p k o k 2 0; + A(B (G(0) G(n p + )) t +B 2 2t k0 0 k2 0; ): By combining this relation with (49), we nally obtain k n+ k 2 0; A n+ k 0 k 2 0; + A p+ (B t G(0) + B 2 2t k0 0 k2 0; ) + A(B t G(n + p)+ B 2 2t kn+ p n+ p k 2 0; ): Choosing p such that n =2p+q; q and using (3) (lemma 5.2) we conclude that k n+ k 2 0; An+ C 2 + A p+ C 3 + C 4 k p p k 2 0; ; (50) which, from (32) (lemma 5.), implies the linear convergence of k n+ k 2 0; to 0. On the other hand by combining (37) (lemma 5.2) and (43) (lemma 5.3) we obtain 5

17 ( c 2 2t )k n+ k 2 0; + 2 j n+ j 2 ; 2t k n k 2 0; + 2c 2t(G(n) G(n + )) + (kvk2 + 2c 2 2 )jn+ n+ j 2 ; : (5) Therefore by using (30) we obtain ( 2t c 2 )k n+ k 2 0; + 2 j n+ j 2 ; 2t k n k 2 0; + 2c 2 t(g(n) G(n + )) + (kvk2 2c )(kn n k 2 0; ): 2t (52) Our result follows then from (3) (lemma 5.2), (32) (lemma 5.), and the linear convergence of k n k 0;. 5.3 Convergence of a Fixed Point Method for the Implicit Scheme The implicit scheme proposed in this section couples the global and the al problem. To uncouple them, it is advisable to use the xed point algorithm below : set 0 ;0 = ol and 0 = 0, then for k 0; n+ kj i solve being known, n+ ;k+ t n + div(v n+ ;k+ ) n+ ;k+ = 0 in ; n+ ;k+ = n+ k on i; n+ ;k+ = 0 on b; (53) 6

18 n+ k+ n t + div(v n+ k+ ) n+ k+ = 0 in ; n+ k+ = on n+ k+ =@n ;k+ =@n on b: (54) We will study now the algorithm (53)-(54). By setting ;k;q = n+ ;k+ n+ ;q+ ; (55) k;q =( n+ k n+ q ); (56) we observe that ;k;q and k;q verify the following equations : ;k;q=t + div(v ;k;q ) ;k;q = 0 in ; ;k;q = k ;q on i; ;k;q = 0 on b; k;q=t + div(v k;q ) k;q = 0 in k;q = 0 on ; on b: (57) (5) If t is suciently small, we prove in [5] that k;q and ;k;q converge linearly to zero. Hence the sequences n+ k and n+ ;k are Cauchy sequences which converge linearly towards the unique solutions n+ and n+ of the implicit scheme. This guarantees the convergence of the above xed point algorithm. 6 Numerical Analysis of the Stability of the Algorithm (7)-() In this section we focus on the application of the explicit time marching algorithm (7)-() studied in the previous sections to the numerical solution 7

19 of the steady problem (2). We rst assume that the boundary condition on b in () is @n ) so that the resulting algorithm is parallel (Jacobi type). Here, denotes the domain surrounding the obstacle (an ellipse in our numerical study) as described in Figure. The global and al domains are discretized by fully overlapping compatible nite element grids. The global mesh contains 37 nodes and 2662 elements (see gure 7). Further the time marching algorithm is being initialized by setting 0 to zero. In a rst step, the veity eld is obtained by solving the following inviscid incompressible ow problem: divv = 0; curl v = 0 v = (; 0); v:n = 0 on the body b with a rst order nite element method using the same global mesh. If we set v = 0, the algorithm may ormay not converge depending on the values of t. More precisely, we observe that the algorithm converges linearly when t < 0 and is divergent otherwise. This is graphically shown in the gures (4-7) where the values of kn+ n k tk 0 k are plotted versus the iteration count n for t equal respectively to 0 6,0,, and 0. Further, when the veity is taken suciently large, the algorithm becomes unconditionally stable. In particular, the initialization of our algorithm by 0 = 0 with kv k =,=0:and t = 00 leads to a converging algorithm (g. ). By intuition such abehavior seems natural. An overestimation of the solution n at the interface i implies an overestimation of the friction forces on b. For suciently small time steps, this overestimation will not aect the value of n+ on i and can therefore be ignored at the next time step. If the Reynolds is suciently large, this error will only aect the wake region but will not have any inuence at the interface i. To the contrary, for large t and, this error does aect the value of n+ on i. The inuence of the error on n+ may be amplied throughout the iteration process. Another variant of the algorithm consists of replacing the explicit Dirichlet condition

20 n+ = n on i in the algorithm (7)-() by the following semi-implicit condition n+ = n+ on i: In fact, this implies replacing the previously parallel algorithm (Jacobi like ) by the sequential algorithm (Gauss-Seidel like ). When we solve the pure diusion problem (i.e. with ow veity v =0) with = and t = (respectively t = 2)we obtain a better convergence history : the speed of the new algorithm is linear and clearly faster than the parallel algorithm. the domain of convergence is moderately larger (see table ). To study experimentally in more details the convergence behavior of both algorithms we assume that we have a linear behavior of our algorithm, and hence that the error at the iteration n will satisfy the following inequality k n+ n k K n k 0 k : The algorithm converges if K<. An estimate for K can be found by considering as in table the ratio n logkn+ n k k 0 k = log K which is displayed as a function of (t) for n = 4 and dierent values of V = v. A negative value of this ratio means divergence of the algorithm. As expected, this ratio is positive for suciently small values of t and converges to zero as t goes to zero. In this table, we observe that for V =0;t< 0 2, the algorithm converges. However the convergence is slow since the minimal contraction constant K min (for the optimal value of t) is close to one (see table 2). For V = 0, the algorithm converges for a much larger range of values of t and the optimal contraction constant ismuch smaller. This is summarized on table 2 where we have displayed the best possible contraction constants for each of the coupling algorithms and for dierent values of the Reynolds V = v. 9

21 t /000 /0 / Gauss-Seidel V =0 Jacobi V =0 Gauss-Seidel V =0 Jacobi V =0 Jacobi V = 000 Table : Contraction constant (in fact minus its logarithm) in function of t for the explicit (Jacobi) and semi-explicit (Gauss-Seidel) version of our coupling algorithm. We observe divergence for V = 0 and t >2 and convergence otherwise. Jacobi (parallel) Gauss-Seidel (sequential) V K min V K min 0 0:5 0 0:6 0 0:50 0 0: 0 3 0:4 Table 2: Minimal contraction constant versus the Reynolds V for both sequential and parallel versions of the algorithm. 20

22 k n+ Figure 2: Convergence of the Time Marching Algorithm: tk 0 k are plotted versus the iteration count n for t =0 6,v= 0 (Jacobi). Observe the very slow convergence. n k 2

23 k n+ n k tk 0 k Figure 3: Convergence of the Time Marching Algorithm: plotted versus the iteration count n for t =0,v= 0 (Jacobi). are 22

24 Figure 4: Convergence of the Time Marching Algorithm: plotted versus the iteration count n for t =,v= 0 (Jacobi). k n+ n k tk 0 k are 23

25 n k tk 0 k are plot- Figure 5: Divergence of the Time Marching Algorithm: kn+ ted versus the iteration count n for t = 0, v = 0 (Jacobi). 24

26 k n+ Figure 6: Convergence of the Time Marching Algorithm: tk 0 k are plotted versus the iteration count n for t = 0 and the ow veity is equal to (Jacobi). n k 25

27 7 Conclusion We have analysed the convergence properties of a standard time marching algorithm for solving a domain decomposed advection-diusion problem with full overlapping and coupling by friction. We were able to prove theoretically the unconditional stability and linear convergence of the fully implicit algorithm (x5). When using the uncoupled semi-explicit algorithm in the general case, numerical evidence indicate that this algorithm is unstable for large values of t and small overlapping, and that it becomes linearly convergent when t is below a Reynolds dependent threshold (x7). This conditional stability is not a real issue for practical CFD problems because most solvers already require to use small time steps inside each domain. Nevertheless, it would be nicer to derive an uncoupled unconditionally stable version of the present time marching algorithm. References [] R. Glowinski, G. Golub and J. Periaux (eds), Proceedings of the First International Symposium on Domain Decomposition Methods for Partial Dierential Equations, Paris, France, January 7-9, 97, (SIAM, Philadelphia, 9). [2] T. Chan, R. Glowinski, J. Periaux and O. Widlund (eds), Proceedings of the Second International Symposium on Domain Decomposition Methods for Partial Dierential Equations, Los Angeles, California, January 9, (SIAM, Philadelphia, 99). [3] T. Chan and R. Glowinski (eds), Proceedings of the Third International Symposium on Domain Decomposition Methods for Partial Dierential Equations, Houston, Texas, March 20-22, 99, (SIAM, Philadelphia, 990). [4] R. Glowinski, Y. Kuznetsov, G. Meurant, J. Periaux and O. Widlund (eds), Proceedings of the Fourth International Symposium on Domain Decomposition Methods for Partial Dierential Equations, Moscow, June 990, (SIAM, Philadelphia, 99). 26

28 [5] T. Chan, D. Keyes, G. Meurant, S. Scroggs and R. Voigt (eds), Proceedings of the Fifth International Symposium on Domain Decomposition Methods for Partial Dierential Equations, Norfolk, May 99, (SIAM, Philadelphia, 992). [6] A. Quarteroni (ed), Proceedings of the Sixth International Symposium on Domain Decomposition Methods for Partial Dierential Equations, Como, June 992, (AMS, Providence, 994). [7] Y. Achdou and O. Pironneau, A fast solver for Navier-Stokes equations in the laminar regime using mortar nite element and boundary element methods, Technical Report (Centre de Mathematiques Appliquees, Ecole Polytechnique, Paris, 993). [] A. D. Aleksandrov, Majoration of solutions of second order linear equations, Vestnik Leningrad Univ. 2, 5-25(966) English translation in Amer. Math. Soc. Transl. (2) 6, 20-43(96). [9] I. Ya. Bakel'man, Theory of quasilinear elliptic equations. Siberian Math. J. 2, 79-6(96). [0] J. Bramble, R. Ewing, R. Parashkevov and J. Pasciak, Domain decomposition methods for problems with partial renement SIAM J. Sci. Stat. Comp {40 (992). [] C. Canuto and A. Russo, On the Elliptic-Hyperbolic Coupling. I: the Advection Diusion Equation via the -formulation, Math. Models and Meth. Appl. Sciences (to appear) (993). [2] D. Gilbarg and N. S. Trudinger, Elliptic partial dierential equations of second order. Berlin-Heidelberg-New York, Springer Verlag 977. [3] W. Gropp and D. Keyes, Domain Decomposition Methods in Computational Fluid Dynamics. Int. J. Num. Meth. Fluids (992). [4] P. Le Tallec and M. D. Tidriri, Coupling Navier-Stokes and Boltzmann. Submitted to J. Comp. Phy. [5] P. Le Tallec and M. D. Tidriri, Analysis of the explicit time marching algorithm. ICASE Report No

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30 Figure 7: Description of the nite element mesh and of the al subdomain. 29