Discrete Energy Laws for the First-Order System Least-Squares Finite-Element Approach

Transcription

1 Discrete Energy Laws for te First-Order System Least-Sqares Finite-Element Approac J. H. Adler (B),I.Lask, S. P. MacLaclan, and L. T. Zikatanov 3 Department of Matematics, Tfts University, Medford, MA 55, USA james.adler@tfts.ed, ilya.lask@gmail.com Department of Matematics and Statistics, Memorial University of Newfondland, St. Jon s, NL AC 5S7, Canada smaclaclan@mn.ca 3 Department of Matematics, Penn State, University Park, PA 68, USA ldmil@ps.ed Abstract. Tis paper analyzes te discrete energy laws associated wit first-order system least-sqares (FOSLS) discretizations of timedependent partial differential eqations. Using te eat eqation and te time-dependent Stokes eqation as examples, we discss ow accrately a FOSLS finite-element formlation aderes to te nderlying energy law associated wit te pysical system. Using reglarity argments involving te initial condition of te system, we are able to give bonds on te convergence of te discrete energy law to its expected vale (zero in te examples presented ere). Nmerical experiments are performed, sowing tat te discrete energy laws old wit order O ( p),were is te mes spacing and p is te order of te finite-element space. Ts, te energy law conformance is eld wit a iger order tan te expected, O ( p ), convergence of te finite-element approximation. Finally, we introdce an abstract framework for analyzing te energy laws of general FOSLS discretizations. Introdction First-order system least sqares (FOSLS) is a finite-element metodology tat aims to reformlate a set of partial differential eqations (PDEs) as a system of first-order eqations [8,9]. Te problem is posed as a minimization of a fnctional in wic te first-order differential terms appear qadratically, so tat te fnctional norm is eqivalent to a norm meaningfl for te given problem. In eqations of elliptic type, tis is sally a prodct H norm. Some of te compelling featres of te FOSLS metodology inclde: self-adjoint discrete eqations stemming from te minimization principle; good operator conditioning stemming from te se of first-order formlations of te PDE; and finite-element and mltigrid performance tat is optimal and niform in certain parameters (e.g., Reynolds nmber for te Navier-Stokes eqations), stemming from niform prodct-norm eqivalence. c Springer International Pblising AG 8 I. Lirkov and S. Margenov (Eds.): LSSC 7, LNCS 665, pp. 3, 8. ttps://doi.org/.7/ _

2 4 J.H.Adleretal. Sccessfl FOSLS formlations ave been developed for a variety of applications [5, 7]. One example of a large-scale pysical application is in magnetoydrodynamics (MHD) [ 3]. Tese nmerical metods ave led to sbstantial improvements in MHD simlation tecnology; owever, several important estimates remain to be analyzed to confirm teir qantitive accracy. One of tese is te energy of te system. Using an energetic-variational approac [ 3, 5], energy laws of te MHD system can be derived tat sow tat te total energy sold decay as a direct reslt of te dissipation in te system. Initial comptations sow tat te FOSLS metod indeed captres tis energy law, bt it remains to be sown wy it sold. In tis paper, we describe te discrete energy laws associated wit FOSLS discretizations of time-dependent PDEs, sc as te eat eqation or Stokes eqation, and sow qantitatively ow tey are related to te continos pysical law. Wile we only sow reslts for tese simple linear systems, te reslts appear generalizable to more complicated systems, sc as MHD. Getting te correct energy law is not only important for nmerical stability, bt it is crcial for captring te correct pysics, especially if singlarities or ig contrasts in te soltion are present. Te paper is otlined as follows. In Sect., we discss te energy laws of a given system and describe teir discrete analoges. Section 3 analyzes te energy laws associated wit te FOSLS discretizations of te eat eqation, and te same is done for Stokes eqations in Sect. 4. For bot examples, we present nmerical simlations in Sect. 5. Finally, we discss a generalization of te concepts presented ere in Sect. 6, and some conclding remarks in Sect. 7. Energy Laws Te energetic-variational approac (EVA) [ 3, 5] of ydrodynamic systems in complex flids is based on te second law of termodynamics and relies on te fndamental principle tat te cange in te total energy of a system over time mst eqal te total dissipation of te system. Tis energy principle plays a crcial role in nderstanding te interactions and copling between different scales or pases in a complex flid. In general, any set of eqations tat describe te system can be derived from te nderlying energy laws. Te energetic variational principle is based on te energy dissipation law for te wole copled system: E total = D, () t were E total is te total energy of te system, and D is te dissipation. Simple flids, were we assme no internal (or elastic) energies, can also be described in tis setting and yield te following energy law: ( ) dx = ν dx, () t Ω Ω

3 Energy Laws for FOSLS 5 were represents te flid velocity and ν is te flid viscosity, acconting for te dissipation in te system. Applying te so-called least-action principle reslts in te integral eqation, t + p, y = ν, y, y V, were we assme an incompressible flid, =, and an appropriate Hilbert space, V. Here, we se, to denote te L (Ω) inner prodct. In strong form, we obtain te time-dependent Stokes eqations (assming appropriate bondary conditions): + p ν =, (3) t =. (4) Note tat te energy law can also be derived directly from te PDE itself. First, we consider te weak form of (3) (4), mltiplying (3) by and (4) byp and integrate over Ω. After integration by parts we obtain te following relations: = + p ν, +,p t = t, + p, ν, +,p =, + p, + ν,, p. t Here, we ave assmed tat te bondary conditions are sc tat te bondary terms, reslting from te integration by parts, vanis. Hence, we ave, = ν,. t Tis approac can also be applied to oter PDEs, sc as te eat eqation, to sow similar energy dissipation relations. Let ν be te termal diffsivity of te body Ω, and its temperatre. Ten te PDE describing te temperatre distribtion in Ω is as follows, ν =, on Ω, =, on Ω. (5) t As before, we mltiply (5) by and integrate over Ω to obtain tat = t ν, = t, ν, =, + ν, t Hence,, = ν,, t wic is te scalar version of ().

4 6 J.H.Adleretal. For te remainder of te paper, we analyze (), specifically ow closely te FOSLS metod can approximate te energy law discretely. We will consider bot te scalar (eat eqation) and te vector version (Stokes eqation) in te nmerical reslts, as te form of te energy law is identical. First, we discss ow moving to a finite-dimensional space affects te energy law. 3 Heat Eqation First, we consider te eat eqation, assming a constant diffsion coefficient ν = for simplicity, omogeneos Diriclet bondary conditions, and a given initial condition: (x,t) = Δ(x,t) x Ω, t > (6) t (x,t)= x Ω, t (7) (x, ) = (x) x Ω. (8) To discretize te problem in time, we consider a symplectic, or energy-conserving, time-stepping sceme sc as Crank-Nicolson. Given a time step size,, and time t n = n, we approximate n = (x,t n ) wit te following semi-discrete version of (6), n = Δ + Δ n To simplify te calclations later, we introdce an intermediate approximation,, and re-write te semi-discrete problem as n ( ) = Δ (x) = x Ω, n =,,,... (9) = n Remark. To obtain te semi-discrete energy law for (9), we perform a similar procedre as done in Sect., were we mltiply te first eqation in (9) by and integrate over te domain. After some simple calclations, we obtain te corresponding energy law, sing L norm notation: n = () To se te FOSLS metod, we now pt te operator into a first-order system. Since we ave redced te problem to a reaction-diffsion type problem, we introdce a new vector V =, and se te H -elliptic eqivalent system [8,9]: ( ) L = V V + V V = n. ()

5 Energy Laws for FOSLS 7 Note tat Diriclet bondary conditions on te continos soltion,, gives rise to tangential bondary conditions on V, V n =, were n is te normal vector to te bondary. Next, we consider a finite-dimensional sbspace of a prodct H space, V, and perform te FOSLS minimization of () overv : ( ) ( ), V = arg min (,V ) V L n V, = n. For eac n, te above minimization reslts in te following weak set of eqations: ( ) L V n,l φ = φ V, () were te inner prodcts and norms are all in L (scalar or vector, depending on context), nless oterwise noted. Note, tat wit te introdction of V, te discrete form of te FOSLS energy law can now be written, n V, as. (3) Te goal of te remainder of tis Section is to sow ow well tis energy law is satisfied. To do so, we make se of te following assmption. Assmption. Assme tat te initial condition is smoot enog and te projection onto te finite-element space as te following property, H C p H p+, were p is te order of te finite-element space being considered. Ten, sing standard reglarity estimates we obtain te following Lemma. Lemma. Let { i } i=,,... be a seqence of semi-discrete soltions to (9). Ten, for any sccessive time steps, tere exists a constant C>, sc tat H p C n H p A conseqence of tis reglarity estimate is a bond on te error in te approximation. Lemma. Let f H p H and let te pair (, V ) V solve ( ) ( ) V = arg min (,V ) V L f V.

6 8 J.H.Adleretal. Let û be te exact soltion of te corresponding PDE, i.e., Δû + û = f û = in Ω, on Ω. Ten, û H C()p f H p, were te constant C() may also depend on. Proof. For a fixed, te PDE is a reaction-diffsion eqation. Terefore, standard reslts from te FOSLS discretization of reaction-diffsion can be sed [8,9]. Note tat for a standard FOSLS approac, C() =O ( ), bt a rescaling of te eqations may ameliorate tis worst-case scenario. Next, we make te following observation, wic follows from te wellposedness of te FOSLS formlation [8, 9]. Lemma 3. Let (, V ) V and (, V ) V be two soltions to te following FOSLS weak forms wit different rigt-and sides, ( ) ( ) L F V,L φ =, L F V,L φ = φ V. Ten, H + V V H C() F F. Tis, ten, yields te following reslt. Lemma 4. Given te soltion to te semi-discrete Eq. (9), and te flly discrete soltion, we can bond te error in te L norm: C () p n H p + C () n n. (4) ( ) Proof. Let ũ be te scalar part of te FOSLS soltion ũ, Ṽ of Δ + = n, in Ω, =on Ω, (5) were te exact semi-discrete soltion n, at te previos time step, is sed in te rigt-and side. By te triangle ineqality, ũ + ũ. (6) By Lemma 3, weave ũ ũ V + H Ṽ H C () (7) n n.

7 Energy Laws for FOSLS 9 Te fnctions ũ and are, respectively, FOSLS and exact soltions of te same bondary vale problem (5). Hence, from Lemma, weave ũ C() p n H p Combining (6), (7) and (8), we obtain (4). C() p n H p (8) Finally, we ave te following reslt on te approximation of te exact energy law (3). ( ) Teorem. Let n be te soltion to te FOSLS system, (), attime step n (wit n V n and V defined as before). Tere exists C() > sc tat + V C() n n min φ V V L φ. Proof. To simplify te notation, define te energy law we wis to bond as Note tat and V n = E n := n = n, + V. = n, V, + V, V, were te latter eqation is obtained by integration by parts, continity of te spaces, and appropriate bondary conditions. Ts, En = V + Using (), for any φ V, n ( ) En = L V, = n, + V, V L ( V, V ) n L φ., V.

8 J. H. Adler et al. Next, consider adding and sbtracting te soltions to te semi-discrete, (), and flly discrete, (), FOSLS system from te previos time step, ( ) En = L V n + n n, V L φ n n, V L φ ( ) = L V n, V L φ n n, V L φ ( ) L V n M n + M n n n, were we ave defined Mn := min V φ V L φ. Ten, adding and ( ) sbtracting L yields V En L ( V V ) ( ) + L n V M n + M n Using te continity of L, followed by Lemma 3, gives ( ) En C()M n + V M n C() M n V H n n + M n Combining te two terms completes te proof. n n. n n n n. To provide a better bond for te FOSLS energy law (3), we introdce a measre for te trncation error defined as (v) δ n = max min V (v) v H p+ (Ω) v H p+ φ V Lφ, (9)

9 Energy Laws for FOSLS were (v) andv (v) are te corresponding soltions to te flly discrete problem wit = v as te initial condition. Corollary. Using te same assmptions as Teorem and Assmption, n + V C()δ p H p+, δ = max n δ n. () Proof. Using te definitions of Lemma 4, and, te triangle ineqality, and An indction argment ten gives n n C () p Wit Assmption, n n C () p n j= n j= + n n Using some reglarity argments for eac i, we get, C () p n H p +(C ()+) n n. (C ()+) j n j H p +(C ()+) n. (C ()+) j n j H p +(C ()+) n H p+. n n C(n) p H p+. Ten, wit te definition of δ and te reslt from Teorem, te proof is complete. We note tat te bond in Corollary is a rater pessimistic one. At a fixed time, t, we expect te qality of bot te flly discrete and semi-discrete approximations to te tre soltion to improve as and more time-steps are sed to reac time t; ts, n n sold decrease as forn = t/. Frtermore, for te nforced eat eqation, we expect bot n and n to decrease in magnitde wit n, bt tis is not acconted for in te bond in Corollary. Te bond above worsens wit smaller and bigger n, sowing te limitations of bonding n n by terms depending only on and te finite-element space. Remark. As sown in te nmerical experiments, Sect. 5, te constant δ defined in (9) is of order p for a smoot soltion. Tis indicates tat te energy law (3) olds wit order O ( p). Wile te teoretical jstification of sc statement may be plasible, it is nontrivial as te discrete qantities involved in te definition of δ do not possess enog reglarity (tey are jst finite-element fnctions, only in H ).

10 J. H. Adler et al. 4 Stokes Eqations Next, we retrn to te time-dependent Stokes eqations, (3) (4). For simplicity, we again assme ν =, and rewrite te eqations sing Diriclet bondary conditions for te normal components of te velocity field, and zero-mean average for te pressre field, (x,t) t Δ(x,t)+ p(x,t)= x Ω, t > () Ω (x,t)= x Ω, t > () n (x,t)= x Ω, t (3) (x, ) = g(x) x Ω, (4) p(x,t)dv = t. (5) Using a similar semi-discretization in time wit Crank-Nicolson tat was done in (9) yields n ( ) Δ + p =, =, n (x) = x Ω, p dv = n, Ω = n, p =p p n. (6) To se te FOSLS metod, we pt te operator into a first-order system in a similar fasion to te eat eqation. Least-sqares formlations are well-stdied for Stokes system and we consider a simple, velocity-gradient-pressre formlation, were a new gradient tensor, V =, is sed to obtain an H -elliptic eqivalent system [4, 6, 4]: L V p V + p + = V = V trv n. (7) Appropriate bondary conditions on te continos soltion, sc as n =, gives rise to tangential bondary conditions on V, V n =, were n is te normal vector to te bondary. Ultimately, te corresponding semi-discrete energy law is n = V. (8)

11 Energy Laws for FOSLS 3 Finally, we minimize te residal of (7) over a finite-dimensional sbspace of te prodct H Sobolev space in te L norm obtaining te weak eqations, n L V p,l φ = φ V. (9) Note tat te weak system is similar to () and te energy law is identical to (3) in vector form. Ts, all te above teory still olds sbject to enog reglarity of te soltion to te time-dependent Stokes eqations [, ] and a sitable generalization of te definition of δ. 5 Nmerical Experiments For te nmerical reslts presented ere, we se a C++ implementation of te FOSLS algoritm, sing te modlar finite-element library MFEM [] for managing te discretization, mes, and timestepping. Te linear systems are solved by direct metod sing te UMFPACK package []. 5. Heat Eqation First, we consider te eat Eq. (6), and its discrete FOSLS formlation, (), on a trianglation of Ω =(, ) (, ). Te data is cosen so tat te tre soltion is (x, y, t) =sin(πx)sin(πy)e πt. Note tat tis soltion satisfies te bondary conditions and oter assmptions discssed above. Energy Law Error p = p = p =3 convergence 4 convergence 6 convergence 4 6 # Refinement Levels, Fig.. Energy law error, (3), vs. nmber of mes refinements, l ( = ), for te l FOSLS discretization of te eat eqation, (), sing varios orders of te finiteelement space (p = - linear; p = - qadratic; and p = 3 - cbic). One time step is performed wit =.5.

12 4 J. H. Adler et al. Figre displays te convergence of te energy law to zero as te mes is refined for a fixed time step. Te convergence is O ( p), were p is te order of te finite-element space being considered, confirming Teorem. It also sggests tat te constant δ is O ( p ), as is remarked above. Energy Law Error 4 8 Energy Law Error 4 7 p = p = p = # Time Steps, n Time Step Size, (a) (b) Fig.. Energy law error, (3), vs. (a) nmber of time steps, n (wit fixed =.5), and (b) time step size,, for te FOSLS discretization of te eat eqation, (), sing varios orders of te finite-element space (p = - linear; p = - qadratic; and p = 3 - cbic). Mes spacing is = 3. Figre indicates ow te timestepping affects te convergence of te energy law. As discssed above, taking more time steps decreases te error in te energy law, sowing tat we can improve te reslts on te bond, n n. Onte oter and, if only one time step is taken, te convergence sligtly worsens for small, wic is consistent wit te constants fond in Teorem and Corollary. 5. Stokes Eqations Next, we consider Stokes Eqations, (), and te FOSLS discretization described above, (9). Te same domain, Ω =(, ) (, ), is sed, and we assme data tat yields te exact soltion, ( ) sin(πx)cos(πy) (x,t)= e πt, cos(πx)sin(πy) p(x,t)=. Tis prodces a C soltion tat satisfies te appropriate bondary conditions and reglarity argments needed for te bonds on te energy law described above.

13 Energy Laws for FOSLS 5 Similarly to te eat eqation, Fig. 3 compares te convergence of te energy law to zero as te mes is refined for a fixed time step. Again, we see tat te convergence is O ( p), were p is te order of te finite-element space being considered, confirming tat Teorem can also be applied to te time-dependent Stokes eqations. Ts, te FOSLS discretization can adere to te energy law for flid-type systems, and as te potential for captring te relevant pysics of oter complex flids. Energy Law Error p = p = p =3 convergence 4 convergence 6 convergence 4 6 # Refinement Levels, Fig. 3. Energy law error, (8), vs. nmber of mes refinements, l ( = ), for te l FOSLS discretization of te Stokes eqation, (7), sing varios orders of te finiteelement space (p = - linear; p = - qadratic; and p = 3 - cbic). One time step is performed wit =.5. Energy Law Error 4 8 Energy Law Error 4 7 p = p = p = # Time Steps, n Time Step Size, (a) (b) Fig. 4. Energy law error, (8), vs. (a) nmber of time steps, n (wit fixed =.5), and (b) time step size,, for te FOSLS discretization of te Stokes eqation, (7), sing varios orders of te finite-element space (p = - linear; p = - qadratic; and p = 3 - cbic). Mes spacing is = 3.

14 6 J. H. Adler et al. Figre 4 again confirms ow we expect te timestepping to affect te convergence of te energy law. Taking more time steps decreases te error in te energy law, wile te convergence sligtly worsens for small. Tese reslts also igligt te similarities between te energy laws of te eat eqation and te time-dependent Stokes eqations. Since bot ave nderlying energy laws tat are similar, te FOSLS discretization is capable of captring bot wit ig accracy. 6 Discssion: General Discrete Energy Laws Te above reslts sow tat FOSLS discretizations of two specific PDEs yield iger-order approximation of teir nderlying energy laws. In tis section, we give a more general reslt, wic sggests ideas for extending tis teory for oter discrete energy laws sing FOSLS discretizations. 6. FOSLS Discrete Energy Laws As encontered earlier, an energy law is an integral relation of te form: L, =, for (x, ) = (x), (3) were L : Ṽ Ṽ is a linear operator (tat involves bondary conditions), Ṽ is a fnction space, and Ṽ is te soltion to L =, (x, ) =, for example: L = t Δ. (3) To matc te time-dependent problems considered in earlier sections, Ṽ corresponds to a comptational domain tat involves bot space and time, or as is often dbbed, a space-time domain: Ω = Ω [,T]. Frter, we define a finite-dimensional space, Ṽ on Ω corresponding to a trianglation of tis spacetime domain, as well as a stationary finite-dimensional space, V,fort =. Regarding sc space-time discrete spaces and te related constrctions, we refer te reader to te classical works by Jonson et al. [6,7], to [9] for space-time least sqares formlations, and to [8] for space-time iso-geometric analysis and a compreensive literatre review. To present te FOSLS discretization in an abstract setting, we define an extension of to te wole of Ω. Witot loss of generality, we assme tat te initial condition is a piecewise polynomial and, more precisely, V. Hence, we define te extension w Ṽ of so tat w (x, ) = (x). Tis gives a non-omogenos problem wit zero initial gess, wic is eqivalent to (3). Its weak form is: Find Ṽ sc tat for all v Ṽo tere olds = ϕ + w, were Lϕ, v = Lw,v, (3)

15 Energy Laws for FOSLS 7 Here, te space, Ṽo, is te sbspace of Ṽ of fnctions wit vanising trace at t = (zero initial condition). In a typical FOSLS setting, for te eat eqation, is a vector-valed fnction and te extension w needs to be modified accordingly. We ten ave te following space-time FOSLS discrete problem: Find Ṽ sc tat for all v Ṽ,o tere olds = w + ϕ, were, Lϕ, Lv = Lw, Lv. (33) Restricting Ṽ to a finite-element space-time space, Ṽ Ṽ, reslts in a restriction of L on Ṽ, wic is often called te discrete operator. In te following, we keep L, in all estimates allowing for a nonomogenos rigt-and side in (3). We now estimate te error in te energy law, namely te difference L, L,. Teorem. If Ṽ is te FOSLS soltion of (33). Ten, te following estimate olds: L, L, C p H p+. (34) Proof. For te left side of (34) weave L, L, = L, L, + L( ), = L, + L( ),. Using te continity of L and te standard error estimates for te FOSLS discretization, L, L, L, + L( ), C H ( H + H ) C p H p+. Tis concldes te proof. 6. Exact Discrete Energy Law Next, we provide a necessary and sfficient condition for te FOSLS discretization to exactly satisfy an energy law, namely conditions nder wic we ave L, = L,. Recall te assmption tat V. Consider two standard projections on te finite-element space, Ṽ,o: () te Galerkin projection Π : Ṽ Ṽ,o; and () te L ( Ω)-ortogonal projection, Q : L ( Ω) Ṽ,o. Tese operators are defined in a standard fasion: LΠ, v := L, v, for all v Ṽ,o and Ṽ, Q, v :=, v, for all v Ṽ,o and L ( Ω). Consider a well-known identity (see for example [3] for te case of symmetric L) relating Π and Q, wic is sed in te later proof of Teorem 3.

16 8 J. H. Adler et al. Lemma 5. Te projections Q and Π satisfy te relation L Π = Q L, (35) were L : Ṽ Ṽ is te restriction of L on Ṽ, namely, L v,w = Lv,w, for all v,w Ṽ. Proof. Te reslt easily follows from te definitions of Q, Π, L, and te fact tat L Π v Ṽ. Forv Ṽ, andw Ṽ we ave L Π v, w = L Π v, Q w = LΠ v, Q w = Lv, Q w = Q Lv, Q w = Q Lv, w. Tis completes te proof. Note tat we se Q χ = Π χ = χ for all χ Ṽ,o. In general, sc an identity is not tre for χ Ṽ. However, we can relate te soltion to (33) toa discrete analoge of te energy law (3) sing Lemma 5. Frter, notice tat te FOSLS soltion,, satisfies L, Lχ = only for χ Ṽ,o corresponding to a zero initial gess. Ts, it is not obvios ow to estimate L, L,. Teorem 3. Te soltion of (33) satisfies te discrete energy law L, = L, if and only if tere exists a w Ṽ satisfying te initial condition w (x, ) = (x) and if L,w = L,. Proof. Let w Ṽ be any extension of V in Ω, tatis,w satisfies te initial condition. Te following relations follow directly from te definitions given earlier, Eq. (33), and Lemma 5. L, = L, ( w ) + L,w }{{} Ṽ,o = L,Q ( w ) + L,w = L,Q LL ( w ) + L,w = L, L Π L ( w ) }{{} v Ṽ,o + L,w = L,w. In te last identity, we se te fact tat v = Π L ( w ) is an element of Ṽ,o and te first term on te rigt side vanises (by Eq. (33)). As a reslt, we ave L, L, = L, L,w. wic gives te desired necessary and sfficient condition.

17 Energy Laws for FOSLS 9 From te proof, we immediately obtain te following relation, L, L, =inf w { L, L,w,w (, ) = }. (36) In addition to te estimate in Teorem, it is plasible tat one can se te rigt side of (36) to obtain a sarper reslt. Wile tis is beyond te scope of tis paper, some comments are in order. Te difficlties associated wit eac particlar case in and (eat eqation, Stokes eqation, etc.) amont to estimating te qantity on te rigt side of (36) and sc estimates depend on te spaces cosen for discretization and ow well te timestepping approximates te space-time formlation. Sarper estimates on te error in discrete energy law, wic ses (36), can lead to sarper bonds on te constant defined in (9). 7 Conclsions In tis work, we ave sown nmerically tat convergence of te discrete energy law is of order iger tan te finite-element approximation order for two typical transient problems. Ts, wile it is known tat te FOSLS metod may ave isses wit aderence to some conservation laws (i.e., mass conservation), energy conservation is not sc an isse, and can be satisfied wit ig accracy. Te rigoros teoretical jstification of sc claims are topics of crrent and ftre researc. Acknowledgements. Te work of J. H. Adler was spported in part by NSF DMS I. V. Lask was spported in part by NSF DMS-697 (Tfts University) and DMS (Penn State). S. P. MacLaclan was partially spported by an NSERC Discovery Grant. Te researc of L. T. Zikatanov was spported in part by NSF DMS-74 and te Department of Matematics at Tfts University. References. Adler, J.H., Manteffel, T.A., McCormick, S.F., Nolting, J.W., Rge, J.W., Tang, L.: Efficiency based adaptive local refinement for first-order system least-sqares formlations. SIAM J. Sci. Compt. 33(), 4 (). ttp://dx.doi.org/.37/ Adler, J.H., Manteffel, T.A., McCormick, S.F., Rge, J.W.: First-order system least sqares for incompressible resistive magnetoydrodynamics. SIAM J. Sci. Compt. 3(), 9 48 (). ttp://dx.doi.org/.37/ Adler, J.H., Manteffel, T.A., McCormick, S.F., Rge, J.W., Sanders, G.D.: Nested iteration and first-order system least sqares for incompressible, resistive magnetoydrodynamics. SIAM J. Sci. Compt. 3(3), (). ttp://dx.doi.org/.37/ Bocev, P., Cai, Z., Manteffel, T.A., McCormick, S.F.: Analysis of velocity-flx first-order system least-sqares principles for te Navier- Stokes eqations. I. SIAM J. Nmer. Anal. 35(3), 99 9 (998). ttp://dx.doi.org/.37/s Bocev, P., Gnzbrger, M.: Analysis of least-sqares finite element metods for te Stokes eqations. Mat. Compt. 63(8), (994)

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