LEAST-SQUARES FINITE ELEMENT APPROXIMATIONS TO SOLUTIONS OF INTERFACE PROBLEMS

Transcription

1 SIAM J. NUMER. ANAL. c 998 Society for Industrial Applied Matematics Vol. 35, No., pp , February LEAST-SQUARES FINITE ELEMENT APPROXIMATIONS TO SOLUTIONS OF INTERFACE PROBLEMS YANZHAO CAO AND MAX D. GUNZBURGER Abstract. A least-squares finite element metod for second-order elliptic boundary value problems aving interfaces due to discontinuous media properties is proposed analyzed. Bot Diriclet Neumann boundary data are treated. Te boundary value problems are recast into a firstorder formulation to wic a suitable least-squares principle is applied. Among te advantages of te metod are tat nonconforming, wit respect to te interface, approximating subspaces may be used. Moreover, te grids used on eac side of an interface need not coincide along te interface. Error estimates are derived tat improve on oter treatments of interface problems a numerical example is provided to illustrate te metod te analyses. Key words. least-squares finite element metods, interface problems AMS subject classification. 65N30 PII. S Introduction. Least-squares finite element metods are te subject of muc current interest; a small sample of te recent literature is given by [], [4], [5], [6], [7], [8], [9], [0], [], [], [3], [4]. Te obvious advantages of tis class of metods is tat te discrete problems one must solve are symmetric positive definite. However, te practicality of tese metods is still not fully documented due to a lack of study of te beavior of te metods in te presence of difficulties arising from, for example, te use of low-order piecewise polynomial spaces, te application of mixed Diriclet Neumann boundary conditions, te discretization of nonconvex polygonal domains, te need to conserve some global quantity suc as mass. Some of tese issues were addressed from a computational point of view in []. Te purpose of tis paper is to address anoter difficulty by defining analyzing a least-squares finite element metod for second-order elliptic equations wit discontinuous coefficients; more specifically, we consider interface problems. One of te first finite element metods not of least-squares type) treating interface problems was proposed in [], a survey of finite element metods for suc problems can be found in [3]. In [], a least-squares metod for te interface problem of Poisson equations is introduced after a general teory of te least-squares metod as been developed. Te autors of [] were well aware tat proving error estimates for te metod is difficult, terefore te weigts tey use in te terms related to te interface conditions cannot be rigorously justified. In tis paper, following te approac of [8] [3], we formulate te problem as a first-order system ten apply least-squares principles to tis system. Te teory of [8] can also be applied to te interface problem. However, te error estimate tere requires tat solutions be sufficiently smoot, wic may not be true for interface problems. Least-squares finite element metods for interface problems are also considered in [5]. Received by te editors May 8, 996; accepted for publication in revised form) October 3, 996. ttp:// Department of Matematics, Virginia Tec, Blacksburg, VA yanzao@mat.vt.edu). Tis researc was supported by Air Force Office of Scientific Researc grant AFOSR Department of Matematics, Iowa State University, Ames IA gunzburg@ iastate.edu.edu/gunzburg). Tis researc was supported by Air Force Office of Scientific Researc grant AFOSR

2 394 YANZHAO CAO AND MAX GUNZBURGER To avoid global regularity requirements, we introduce two terms in te leastsquares functional tat are related to te conditions on te interface. Our error analysis sows tat te metod as nearly optimal order of accuracy wit respect to an appropriately defined norm. Te weigts used for tese terms are justified by te error estimate are supported by our numerical experiments. Te paper is organized as follows. In te next section we introduce te problem some necessary notations. An existence uniqueness teorem is stated. Ten, in section 3 we define analyze te least-squares finite element metod for te case of Diriclet boundary conditions. A coercive property for te least-squares functional is proved error estimates are obtained. In section 4 we extend te analyses to problems wit inomogeneous Neumann boundary conditions. Finally, in section 5, a computational example is presented. In order to keep te exposition simple, our discussion is in te context of a single interface separating two subdomains in eac of wic te coefficients of te partial differential equations are smoot. However, our algoritms results extend in an obvious manner to problems wit multiple interfaces domains, so long as te assumed regularity results witin te subdomains separated by te interfaces remain valid. In particular, we will assume tat eac of te subdomains as a smoot boundary or, in te very special situations for wic tis can be arranged, eac as a convex boundary.. Statement of te problem. Assume tat Ω is an open bounded domain in R n, n = or 3, wit smoot boundary. Ω Ω are two open subsets of Ω suc tat Ω=Ω Ω,Ω Ω =. Let 0 = Ω, = Ω Ω, = Ω Ω, = Ω Ω. Here, is referred to as te interface. Trougout, we assume tat te subdomains Ω Ω bot ave smoot or in very special situtations, convex) boundaries. Smoot boundaries can occur, for example, if Ω Ω, Ω =Ω Ω, Ω as a smoot boundary. Convex subdomains result, for example, if one subdivides a rectangle into smaller rectangles. Consider te following elliptic boundary value problem on Ω:.) div A i u i )+c i u i =f i in Ω i,,,.) u i i =0, i =,,.3) u = u, A u n ) + A u n ) =0, were c i c>0 A i =a i lk ), i =,, l, k =,...,n, are n n positive definite matrices so tat, if λ i j, j =,...,n, denote te eigenvalues of A i, ten tere exist two constants C a C b suc tat 0 <C a λ i j C b, i =,,j=,...,n. Te cases for wic c = 0 /or c = 0 may also be treated at te expense of greatly complicating te analyses. Te constants appearing in our estimates will, in general, depend on C a C b, in particular, on te ratio C b /C a. In.3), n i denotes te unit outer normal vector on Ω i, i =,. For k 0, we denote by H k D) te stard Sobolev space consisting of functions defined over te domain D aving square integrable derivatives of order up to k. For negative values of k, tese spaces are also defined in te usual manner as appropriate dual spaces. In particular, H D) is te dual space of H 0 D), were te

3 LEAST-SQUARES METHODS FOR INTERFACE PROBLEMS 395 latter is te space of functions aving one square integrable derivative wit respect to D tat vanis on te boundary of tat domain. Also, H k D) =H k D)) n denotes te space of vector-valued functions, eac of wose components belongs to H k D). Te stard Sobolev norm for functions belonging to H k D) H k D) =H k D)) n is denoted by k,d. For k 0, define te Banac spaces Ḣ k Ω) = {u =u,u ) u i =u Ωi H k Ω i ), i =,} wit norm u k = u k,ω + u k,ω V k Ω) = {v =v,v ) v i =v Ωi H k Ω i ), i =,} wit norm v k = v k,ω + v k,ω. We may extend tese definitions to k = ; for example, Ḣ Ω) = {u =u,u ) u i =u Ωi H i Ω i ), i =,}, were H i Ω i ) denotes te dual space of H i Ω i )={u H Ω i ) u = 0 on i }, i =,. Note tat, generally, Ḣ k Ω) H k Ω) V k Ω) V k Ω). In particular, we will work wit te space Ḣ Ω), wic is generally not a subspace of H Ω), so tat approximations for {u u Ωi = u i,,}will be nonconforming in te sense tat tese approximations need not belong to H Ω). Let.4) H = Ḣ Ω), V = V Ω), H 0 = {u H u 0 =0}. Along te interface, let [u] = u u [v n] = v n + v n. Let.5) H, Ω) = {u H [u] =0}, H 0, Ω) = H, Ω) H 0,.6) V, Ω) = {v V [v n] =0}. Note tat H, Ω) H Ω). Also, define te Hilbert spaces Vdiv, Ω i )={v L Ω i )=L Ω i )) n div v L Ω i )} wit norm v div,ωi = v 0,Ωi + div v 0,Ωi Vdiv) = {v =v,v ) v i Vdiv, Ω i )} wit norm v div = v div,ω + v div,ω. Concerning te problem.).3), we ave te following result. THEOREM.. Assume, for k, tat f Ḣk Ω), a i lj Hk Ω i ), i =,, l, j =,...,n, c i H k Ω i ) for i =,. Ten, tere exists a unique solution u Ḣk Ω) for.).3). Proof. See [6].

4 396 YANZHAO CAO AND MAX GUNZBURGER 3. Least-squares finite element approximations. We rewrite.).3) as a system of first-order differential equations: 3.) div v i )+c i u i =f in Ω i, i =,, 3.) A i u i v i = 0 in Ω i, i =,, 3.3) u 0 =0, [u] =0, [v n] =0. We introduce subspaces H H 0 V V parameterized by, usually cosen to be some measure of te grid size suc as te largest diameter of te triangles in a triangulation of Ω. Note tat H need not be a subset of H Ω) so tat in tis sense our metod is nonconforming. We assume tat te subspaces H V possess te approximation properties inf u H u u k,ωi C s k u s,ωi u Ḣs Ω),u H,, inf v v k,ωi s k v s,ωi v V s Ω), v V,,, v V were 0 <k<s. As a result, we ave tat 3.4) inf u H u u k C s k u s u Ḣ s Ω),u H 3.5) inf v V v v k s k v s v V s Ω), v V, were 0 <k<s. We also assume tat te following inverse inequality olds in H : tere exists a constant C suc tat for u =u,u ) H, 3.6) u u /, C / u u 0,. Note tat if te restrictions to of te approximating spaces in Ω Ω coincide, ten te inverse property 3.6) is simply te inverse property in te usual sense. 3.. Te least-squares functional. We define a functional on H 0 V as follows. For u H 0 v V, let 3.7) J u, v; f) = div v i + c i u i f 0 + A i u i v i 0) + [u] +ɛ0 d+ [v n] d, ɛ were f Ḣ0 Ω) ɛ 0,ɛ > 0. Note tat if u H 0, Ω) v V, Ω), ten te last two terms in 3.7) vanis. Also, note tat if u v satisfy 3.) 3.3), ten J u, v; f) =0. Te functional J, ; ) satisfies te following coercivity property.

5 LEAST-SQUARES METHODS FOR INTERFACE PROBLEMS 397 PROPOSITION 3.. Let u =u,u ) H, v =v,v ) V,u=u,u ) H 0, Ω), v =v,v ) V, Ω). Ten, for sufficiently small, tere exists a constant C > 0independent of suc tat 3.8) J u u, v v ;0) C u u + v v div). Proof. Let Ĵ u, v) = c / i div v i + c / i u i 0 + A i u i A i v i 0) + [u] +ɛ0 d+ [v n] d. ɛ It is easy to see tat J, ; 0) Ĵ, ) are equivalent, i.e., tat tere exist two positive constants C C suc tat C J u, v;0) Ĵu, v) C J u, v;0) for all u H v V. Tus, it suffices to prove tat Ĵ u u, v v ) C u u + v v ) div for some constant C>0. Now, since u H 0, Ω) v V, Ω), we ave tat Ĵ u u, v v ) = c / i div v i vi )+c / i u i u i) 0 ) + A / i u i u i) A / i v i vi) 0 + u +ɛ0 u d+ v ɛ v ) n) d. Integrating by parts, one obtains, for i =,, c / i div v i vi )+c / i u i u i) 0+ A / i u i u i) A / i v i vi) 0 = c / i div v i vi ) 0 + c / i u i u i ) 0 div v i vi )u i u i ) dω Ω i + A / i u i u i ) 0 + A / i v i vi ) 0 u i u i ) v i vi ) dω Ω i = c / i div v i vi ) 0 + c / i u i u i ) 0 + A / i u i u i ) 0 + A / i v i vi ) 0 u i u i )v i vi ) n d. Hence, for some constant C 3 > 0, Ĵ u u, v v ) C 3 div vi vi ) 0 + u i u i 0 + u i u i ) 0 + v i vi 0) + + +ɛ0 u u )v v ) n d u u ) d+ ɛ u u )v v ) n d v v ) n ) d.

6 398 YANZHAO CAO AND MAX GUNZBURGER By te definition of H 0, Ω) V, Ω), trace teorems, te inverse property 3.6) on H, we ave, for some constant C 4 > 0, u u )v v ) n d u u )v v ) n d = u u )v v ) n d+ u u )v v ) n d u u /, v v) n /, + u u 0, v v) n 0, C 4 ɛ v v div + ɛ u u 0, + ) ɛ v v 0, + ɛ u u. Hence, Ĵ u u, v v ) C 3 C 4 ɛ) u u + v v ) div ) ) + C 4 +ɛ0 ɛ u u d+ C 4 ɛ ɛ v v ) n ) d. We first coose ɛ small enoug so tat C 3 C 4 ɛ>0. Ten, for tis fixed ɛ, we coose 0 sufficiently small so tat C 4 ɛ0 0 ɛ C 4 ɛ 0 ɛ. Tus, for 0 << 0,we ave tat Ĵ u u, v v ) C u u + v v div). Proposition 3. sows a certain coercive property about te functional J. If we coose u = 0 v = 0 in 3.8), ten we see tat te coercive property is true on te finite-dimensional subspace H V of H V, i.e., J u, v ;0) C u + v Hdiv) ) for u, v ) H V. However, tis does not old for all elements of H V. Nevertless, Proposition 3. suffices for us to obtain an error estimate for te least-squares finite element approximations of te solution of 3.) 3.3). 3.. Finite element approximations. We define u, v ) to be te solution of te following problem: 3.9) Ju, v ; f) = min u H,v V Ju, v ; f). We ten ave te following error estimate. THEOREM 3.. Let s>0. Assume tat te solution u, v) of 3.) 3.3) satisfies u Ḣs+ Ω) H 0, Ω) v V s+ Ω) V, Ω). Ten, for sufficiently small for 0 <ɛ any δ>ɛ 0 >0, tere exists a constant C>0suc tat 3.0) u u + v v div C s δ u s+ + v s+ ). Proof. By te approximation properties 3.4) 3.5), tere exist û H v V suc tat 3.) u û C s u s+ 3.) v v C s v s+.

7 LEAST-SQUARES METHODS FOR INTERFACE PROBLEMS 399 By Proposition 3. te definition of u v, u u + v v div CJ u u, v v ;0) =CJu,v ;f) CJû, v ;f)=cju û,v v ;0) 3.3) u û + v v div + û +ɛ0 û ) d+ v ɛ v )n ) d. Using trace teorems, we ave tat û +ɛ0 û ) d+ v ɛ v ) n) d = û +ɛ0 u + u û ) d + v v + v v ) n ) d 3.4) +ɛ0 ɛ + ɛ ) û u) d û u) d+ v v) n ) d+ v v) n ) d ) u û +ɛ0 /)+δ ɛ + 0) v v ɛ ɛ s δ u s+ + ɛ s+ ) v ɛ s+ C s δ) u s+ + v s+) for sufficiently small. Combining 3.) 3.4) yields 3.0). Remark. Te conclusion of Teorem 3. is also valid for problems wit omogeneous Newmann boundary conditions mixed omogeneous boundary conditions. Remark. Teorem 3. is a generalization of Teorem 5. of [8]. We merely require tat u Ḣs+ Ω) v V s+ Ω), i.e., regularity witin eac subdomain not across interfaces. Furtermore, we allow for te use of nonconforming elements in te sense tat te finite element functions u H need not belong to H Ω). 4. Inomogeneous Neumann boundary conditions. We now consider problem 3.) wit te omogeneous Dirclet boundary condition replaced by an inomogeneous Neumann boundary condition; i.e., we consider te problem 4.) div v i )+c i u i =f in Ω i, i =,, 4.) A i u i v i = 0 in Ω i, i =,, 4.3) v n 0 = g, [u] =0, [v n] =0. Define te functional Ku, v; f,g) onh V as follows. For u H v V, 4.4) Ku, v;f, g) = + +ɛ0 divv i )+c i u i f 0+ A i u i v i 0) [u] d+ [v n] d.+ v n g) d. ɛ ɛ 0

8 400 YANZHAO CAO AND MAX GUNZBURGER First we prove a coercivity property for K; te result its proof are similar to tat of Proposition 3.. Let H H V V be finite-dimensional subspaces satisfying te approximation properties 3.4) 3.5). PROPOSITION 4.. Let u, v ) H V wit u =u,u ) v =v,v ) u, v) H, Ω) V, Ω) wit u =u,u ) v =v,v ). Ten, for sufficiently small, tere is a constant C > 0independent of suc tat Proof. Define Ku u, v v ;0,0) C u u + v v div). 4.5) Ku, v) = c / i div v i )+c / i u i 0+ A / i u i A / i v i 0 + +ɛ0 [u] d+ [v n] d+ v n) d. ɛ ɛ 0 It is easy to see tat K ;0,0) K ) are equivalent, i.e., tat tere exist two positive constants C C suc tat C Ku, v;0,0) Ku, v) C Ku, v;0,0) for all u H v V. Tus, it suffices to prove tat Ku u, v v ) C u u + v v div) for some constant C>0 all u H, Ω) v V, Ω). Now, by te definition of H, Ω) V, Ω), Ku u, v v ) = c / i div v i vi )+c / i u i u i) 0+ A / i u i u i) A / i v i vi) 0) + +ɛ0 u u d+ v ɛ v ) n) d+ v n v n) d. ɛ 0 Integrating by parts, one as c / i div v i vi )+c / i u i u i) 0+ A / i u i u i) A / i v i vi) 0 = c / i div v i vi ) 0 + c / i u i u i ) 0 divv i vi )u i u i ) dω Ω i + A / i u i u i ) 0 + A / i v i vi ) 0 u i u i ) v i vi ) dω Ω i = div v i vi ) 0 + u i u i 0 + A / i u i u i ) 0 + A / i v i vi ) 0 u i u i )v i vi ) n d + u u )v v ) n d. 0

9 LEAST-SQUARES METHODS FOR INTERFACE PROBLEMS 40 Hence, Ku u, v v ) C div vi vi ) 0 + u i u i 0 + u i u i ) 0 + v i vi 0) + u u )v v ) n d u u )v v ) n d 4.6) + u u )v v ) n d+ u 0 +ɛ0 u ) d + v ɛ v ) n) d+ v n v n) d ɛ 0 =I+II + III + IV, were I =C div v i vi ) 0 + u i u i 0 + u u ) 0 + v i vi 0), ) II = u u )v v ) n d u u )v v ) n d, III = u u )v v ) n d, 0 IV = u +ɛ0 u ) d + v ɛ v ) n) d+ v n v n) d. ɛ 0 By te proof of Teorem 3. we ave tat for ɛ>0, II Cɛ v v div + ɛ u u 0, + ɛ v v 0, + ɛ u u. Using te Scwartz inequality we ave tat III = u u )v n v n) d ɛ 3 u u ) d+ v n v n) d 0 0 ɛ 3 0 for ɛ 3 > 0. Hence Ku u, v v ) C ɛ ɛ 3 ) u u +C ɛ) v v div + ) ) u +ɛ0 u ɛ d+ v ɛ v ) n) d ɛ + ) v n v n) d. ɛ ɛ 3 0 We first coose ɛ ɛ 3 small enoug so tat C i ɛ ɛ 3 > 0. Ten, for tis fixed ɛ, we let 0 be sufficiently small so tat +ɛ0 0 ɛ, ɛ 0 <ɛ, ɛ 0 <ɛ 3. Tus, for 0 >> 0, we obtain Ku u, v v ) C u u + v v div). Te proof is complete.

10 40 YANZHAO CAO AND MAX GUNZBURGER Assume tat u, v ) is te solution of te following problem: Ku, v ; f,g) = min Ku, v ; f,g). u H,v V We ave te following error estimate. THEOREM 4.. Let s>0. Assume tat te solutions u v of 4.) 4.3) satisfy u Ḣs+ Ω) v V s+ Ω). Ten, for sufficiently small for 0 <ɛ,ɛ, any δ>ɛ 0 >0, tere exists a constant C>0suc tat 4.7) u u + v v div C s δ u s+ + v s+ ). Proof. By te approximation properties 3.4) 3.5), tere exist û H v V suc tat u û C s u s+ v v C s v s+. By Proposition 4. te definition of u v we ave tat 4.8) u u + v v div CKu u, v v ;0,0) = CKu, v ; f,g) CKû, v ;f,g) =CKu û,v v ;0,0) u û + v v div + û +ɛ0 û ) d + v ɛ v ) n) d+ v v) n) d. ɛ 0 From te proof of Teorem 3., we ave tat 4.9) +ɛ0 û û ) d+ v ɛ v ) n) d s δ u s+ + ɛ s+ ) v ɛ s+ C s δ) u s+ + v s+). Using trace teorems we ave tat 4.0) ɛ 0 v n v n) d ɛ v v ɛ ɛ s+ ) v ɛ s+ = C s v s+ for sufficiently small. Substituting 4.9) 4.0) into 4.8), we obtain 4.7).

11 LEAST-SQUARES METHODS FOR INTERFACE PROBLEMS 403 FIG..Interface problem used in computational example. 5. Numerical results. In tis section we report te results of computations wic illustrate our metod error analysis. We take for te domain te rectangle Ω=0,) 0, ). Te interface occurs at x = so tat Ω =0,) 0, ) Ω =,) 0, ). In.).3), A i = diag a i,a i ) wit a = a =/ c = c = ; see Figure. For te exact solution, we coose u x, y) = sinπx) sinπy), x, y) Ω =0,) 0, ) u x, y) = sinπx) sinπy), x, y) Ω =,) 0, ). Te rigt- sides f f in.) are ten determined from tis coice for A,c,u ) A,c,u ), respectively. Note tat te global solution merely belongs to H Ω). We coose ɛ 0 =/3 ɛ =3/4 in te functional 3.4). Stard tecniques of te calculus of variations may be used to deduce tat any solution u, v ) of 3.8) necessarily satisfies te variational problem: find u, v ) H V suc tat 5.) B u, v ), ũ, ṽ ) ) = F ũ, ṽ ) ) ũ, ṽ ) H V, were, for u =u,u ), v =v,v ), ũ =ũ,ũ ), v =ṽ,ṽ ), we ave B u, v ),ũ, ṽ ) ) = div v i )+c i u i,div ṽi )+c i ũ ) i Ω i + ) A i u i vi,a i ũ i ṽi )Ω i + u +ɛ0 u, ũ ũ ) + v ɛ v ) n, ṽ ṽ ) n ) F ũ, ṽ ) ) = f i, div ṽi )+ũ i) Ωi. Here,, ) Ωi, ) denote te L Ω i ) L ) inner products, respectively.

12 404 YANZHAO CAO AND MAX GUNZBURGER 9 L error for u 7 L error of V=v,w) H error of u L error for divv) FIG..Negative of logaritm of error vs. log). TABLE Rates of convergence. Function L error H error u v.3.4 w div v.898 For our numerical results, globally continuous piecewise quadratic finite element functions based on uniform triangulations of Ω i, i =,, were used for all unknowns, i.e., u i te components of v i, i =,. Te nodes of te triangulations of Ω Ω coincide on te interface. Hence, we expect tat convergence rates will be determined according to 3.9) wit s = 3. Figure displays te L error of te approximate solutions for u = u,u ), v =A u,a u ), te error of u in te H seminorm, te L norm error of div v. In Table, we list te rates of convergence estimated by linear regression. Tese convergence rates matc our error estimates in section 3. Remark. From Table, we see tat te L error in te approximation to u is one order iger tan tat for its derivative. Remark. As sown in [9] [3], if curl v is added to te stard leastsquares functional, ten te optimal error estimate in te H -norm for v may be acieved. Also, te error for v in te L norm of is one order iger. Our numerical experience see Figure 3) indicates tat if we add curl v to te functional 3.4), ten te H error in te approximation v is seemingly better, but te L error is not improved.

13 LEAST-SQUARES METHODS FOR INTERFACE PROBLEMS L error for V,noncurl/curl H error for V, noncurl/curl FIG. 3. Negative of logritm of error vs. log). Solid line: witout curl term in functional; dased line: wit curl term in functional. Acknowledgment. Te autors tank Steve McCormick for te many suggestions tat materially improved tis paper. REFERENCES [] A. AZIZ, R. KELLOG, AND A. STEPHENS, Least-squares metods for elliptic systems, Mat. Comp., ), pp [] I. BABUSKA, Te finite element metod for elliptic equations, Computing, 970), pp [3] I. BABUSKA AND A. AZIZ, Survey lectures on te matematical foundations of te finite element metod, in Te Matematical Foundations of te Finite Element Metod wit Applications to Partial Differential Equations, A. Aziz, ed., Academic, New York, 973, pp [4] P. BOCHEV, Z. CAI, T. MANTEUFFEL, AND S. MCCORMICK, First-order system least-squares for te Navier-Stokes equations, in Proc. 7t Copper Mountain Conf. on Multigrid Metods, to appear. [5] P. BOCHEV AND M. GUNZBURGER, Accuracy of least-squares metods for te Navier-Stokes equations, Comput. Fluids, 993), pp [6] P. BOCHEV AND M. GUNZBURGER, Analysis of least-squares finite element metods for te Stokes equations, Mat. Comp., ), pp [7] P. BOCHEV AND M. GUNZBURGER, Least squares metods for te velocity-pressure-stress formulation of te stokes equations, Comp. Met. Appl. Mec. Engrg., 6 995), pp [8] Z. CAI, R. LAZAROV, T. MANTEUFFEL, AND S. MCCORMICK, First-order system least-squares for partial differential equations: Part I, SIAM J. Numer. Anal., 3 994), pp [9] Z. CAI, T. MANTEUFFEL, AND S. MCCORMICK, First-order system least-squares for partial differential equations: Part II, SIAM J. Numer. Anal., ), pp [0] Z. CAI, T. MANTEUFFEL, AND S. MCCORMICK, First-order system least-squares for te Stokes equations wit application to linear elasticity, SIAM J. Numer. Anal., ), pp [] C.-L. CHANG, An error estimate of te least squares finite element metod for te Stokes problem in tree dimensions, Mat. Comp., ), pp [] J. DEANG AND M. GUNZBURGER, Issues related to least-squares finite element metods for te Stokes equations, SIAM J. Sci. Comp., to appear. [3] B.-N. JIANG AND L. POVINELLI, Optimal least-squares finite element metod for elliptic problems, Comput. Metods Appl. Mec. Engrg., 0 993), pp. 99. [4] D. LEFEVRE, J. PERAIRE, AND K. MORGAN, Least-squares finite element solution of compressible incompressible flows, Internat. J. Numer. Metods Heat Fluid Flow, 99), pp [5] T. MANTEUFFEL, S. MCCORMICK, AND G. STARKE, First-order system of least-squares for second order elliptic problems wit discontinuous coefficients, to appear. [6] Z. SEFTEL, A general teory of boundary value problems for elliptic systems wit discontinuous doefficients, Ukrainian Mat. Z., 8 966), pp