A SADDLE POINT LEAST SQUARES APPROACH TO MIXED METHODS

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1 A SADDLE POINT LEAST SQUARES APPROACH TO MIXED METHODS CONSTANTIN BACUTA AND KLAJDI QIRKO Abstract. We investigate new PDE discretization approaces for solving variational formulations wit different types of trial and test spaces. Te general mixed formulation we consider assumes a stability LBB condition and a data compatibility condition at te continuous level. We expand on te Bramble-Pasciak s least square formulation for solving suc problems by providing new ways to coose approximation spaces and new iterative processes to solve te discrete formulations. Our proposed metod as te advantage tat a discrete inf sup condition is automatically satisfied by natural coices of test spaces (first) and corresponding trial spaces (second). In addition, for te proposed iterative solver, a nodal basis for te trial space is not required. Applications of te new approac include discretization of first order systems of PDEs, suc as div curl systems and time-amonic Maxwell equations. 1. Introduction Over te last two decades, tere ave been many advances in applying finite element least squares metods to approximate first order systems of PDEs, [6, 7, 8, 14, 15, 16, 17, 18, 26, 27, 28]. However, wen compared to te more establised field of finite element metods for elliptic problems, a unified teoretical framework for least squares approximation of solutions of first order systems of PDEs is missing. Our proposed framework provides powerful preconditioning tecniques, as efficient error estimators, is suitable to multilevel tecniques, and leads to robust and easy to implement solvers. We combine known teory and discretization tecniques for approximating elliptic problems and for symmetric saddle point problems (see [1, 5, 11, 12, 13, 23, 29, 31, 32, 36]) to obtain a unified framework for discretizing variational formulations wit different types of test and trial spaces. In particular, te framework can be applied to least square approximation for a large class of first order systems of PDEs. For te applications we consider, te solution spaces are L 2 type spaces, and te data can reside in weak negative norm spaces. We require tat te test spaces be H 1 type spaces wit suitable boundary conditions, and 2000 Matematics Subject Classification. 74S05, 74B05, 65N22, 65N55. Key words and prases. least squares, saddle point systems, mixed metods, multilevel metods, Uzawa type algoritms, conjugate gradient, cascadic algoritm, dual DPG. Te work was supported by NSF, DMS

2 2 CONSTANTIN BACUTA AND KLAJDI QIRKO te discrete test spaces be conforming finite element spaces built using te action of te continuous differential operator associated wit a given problem. Among te advantages of te metod are te following: te discretization leads to saddle point variational formulation wit automatic discrete inf sup condition, and assembly of stiffness matrices for te trial spaces is avoided. Te general abstract problem tat we plan to discretize using a Saddle Point Least Squares Metod (SPLS) is: Find p Q suc tat (1.1) b(v, p) = f, v, for all v V or B p = f, were V and Q are infinite dimensional Hilbert spaces and b(, ) is a continuous bilinear form on V Q, tat satisfies a standard inf sup condition, and f V belongs to te range of B. In te special case wen te operator B associated wit te form b(, ) is injective, our metod can be viewed as a conforming Petrov-Galerkin metod. From te point of view of coosing te discrete spaces, our metod can be caracterized as te dual of te Discontinuous Petrov-Galerkin (DPG) metod introduced by Demkowicz and Gopalakrisnan in [20, 21]. Wile bot metods ave strong connections wit te least squares and minimum residual tecniques, te proposed discretization process stands apart from te DPG approac because of te opposite order and different ways in wic te trial and test spaces are cosen. We propose te following main steps of our saddle point least square discretization metod: Step 1 Reduce te general problem (1.1) to a saddle point least square formulation, using te natural inner product a 0 (, ) on V V as te (1, 1) form of te saddle point system (see problem (2.12)). Step 2 Coose a standard conforming approximation space V for te variational space V. Step 3 Construct a discrete trial space M Q using te operator B associated wit te form b(, ). For example, take M := C 1 BV, or M := Q C 1 BV, were C 1 is te Riesz representation operator for te space Q and Q is a projection from Q to a subspace M. Te pair (V, M ) will automatically satisfy a discrete inf sup condition. Step 4 Write te discrete version of SPLS formulation on (V, M ), see (3.3), and replace a 0 (, ) by an equivalent form a prec (, ) on V V. Step 5 Solve te new discrete SPLS problem using an Uzawa type iterative process tat requires only te action of te preconditioner associated wit a prec (, ) and te action of C 1 B or Q C 1 B. Te paper is organized as follows. In Section 2, we introduce notation and review basic abstract results needed to describe te metod. We also include ere te first step of reduction of a mixed problem to a Saddle Point Problem (SPP). In Section 3, we present te discretization part of te metod (Step 2 and Step 3) and discuss te coice of discrete spaces

3 SADDLE POINT LEAST SQUARES 3 and teir approximability. Uzawa type Iterative solvers witout trial space bases are presented in Section 4. Te special coice of spaces wit discrete inf sup condition and teir approximability properties are presented in Section 5. In Section 6 we present an example of SPLS discretization for a div curl system. Te Appendix (Section 8) contains some important functional analysis results needed for te proofs of te paper. 2. Notation and Background In tis section, we start wit a review of te notation of te classical SPP teory and introduce te spaces, te operators and te norms for te general abstract case. We let V and Q be two Hilbert spaces wit inner products a 0 (, ) and (, ) respectively, wit te corresponding induced norms V = = a 0 (, ) 1/2 and Q = = (, ) 1/2. Te dual pairings on V V and Q Q are denoted by,. Here, V and Q denote te duals of V and Q, respectively. Wit te inner products a 0 (, ) and (, ), we associate te operators A : V V and C : Q Q defined by and Au, v = a 0 (u, v) for all u, v V Cp, q = (p, q) for all p, q Q. Te operators A 1 : V V and C 1 : Q Q are called te Rieszcanonical isometries and satisfy te following propperties (2.1) a 0 (A 1 u, v) = u, v, A 1 u V = u V, u V, v V, (2.2) (C 1 p, q) = p, q, C 1 p = p Q, p Q, q Q. Next, we suppose tat b(, ) is a continuous bilinear form on V Q, satisfying te inf-sup condition. More precisely, we assume tat (2.3) inf p Q sup v V and (2.4) sup p Q sup v V b(v, p) p v = m > 0 b(v, p) p v = M <. Here, and trougout tis paper, te inf and sup are taken over nonzero vectors. Wit te form b, we associate te linear operators B : V Q and B : Q V defined by Bv, q = b(v, q) = B q, v for all v V, q Q. Let V 0 be te kernel of B or C 1 B, i.e., V 0 = Ker(B) = {v V Bv = 0} = {v V C 1 Bv = 0}. Due to (2.4), V 0 is a closed subspace of V.

4 4 CONSTANTIN BACUTA AND KLAJDI QIRKO We notice ere tat C 1 B : V Q and A 1 B : Q V are dual to eac oter in te Hilbert sense, and consequently, te Scur complement S 0 := C 1 BA 1 B is a symmetric operator. In addition, S 0 is a positive definite operator wit σ 0 (S 0 ) [m 2, M 2 ], see Lemma 8.1 or [2] Te general problem. Assume tat b : V Q R is a bilinear form satisfying (2.3) and (2.4), and let f V be given. Many variational formulations of PDEs, including first order systems, can be written in te mixed form: Find p Q suc tat (2.5) b(v, p) = f, v, for all v V. Te operator form of (2.5) is to solve te following equation for p (2.6) B p = f. Te existence and te uniqueness of (2.5) was first studied by Aziz and Babuška in [1] and is known as te Babuška s Lemma. Lemma 2.1. (Babuška) Let b : V Q R be a bilinear form satisfying (2.3) and (2.4), and let f V. Te problem: Find p Q suc tat (2.7) b(v, p) = f, v, for all v V as a unique solution if and only if (2.8) f, v = 0, for all v V 0. If (2.8) olds, and p is te solution of (2.7), ten (2.9) m p p S0 := (S 0 p, p) 1/2 = F V M p. A proof of te Lemma can be found in [1, 2]. To justify our saddle point least squares terminology, we write (2.6) in te equivalent form (2.10) A 1 B p = A 1 f. Using tat te Hilbert transpose of A 1 B : V Q is te operator C 1 B : V Q, te normal equation for solving (2.10) in te least square sense is: (2.11) C 1 BA 1 B p = C 1 BA 1 f. Since S 0 := C 1 BA 1 B is a symmetric positive definite operator, te problem (2.11) as a unique solution p. Ten, we consider te well posed saddle point problem: Find (u, p) (V, Q) suc tat (2.12) a 0 (u, v) + b(v, p) = f, v for all v V, b(u, q) = 0 for all q Q. One can immediately ceck tat p is te unique solution of (2.11) if and only if (u, p) is te unique solution of (2.12), were u = A 1 (B p F ).

5 SADDLE POINT LEAST SQUARES 5 Proposition 2.2. In te presence of te continuous inf sup condition (2.3) and te compatibility condition (2.8), we ave tat p is te unique solution of (2.5) if and only if (u = 0, p) is te unique solution of (2.12). Remark 2.3. Even in te absence of te compatibility condition (2.8), but assuming te continuous inf sup condition (2.3), we ave tat (2.12) as unique solution regardless of te fact tat (2.5) is wellposed or not. Te above arguments justify te name of te system (2.12) as te SPLS variational formulation of (1.1) A note on approximating te solution by Uzawa type algoritms. Given a relaxation parameter α (0, 2/M 2 ), te Uzawa algoritm for approximating te solution (u = 0, p) of (2.12) can be described as follows. Start wit any p 0 Q. For j = 1, 2,... Compute u j V, q j, p j Q by a 0 (u j, v) = f, v b(v, p j 1 ), v V q j = C 1 Bu j, p j = p j 1 + α q j Te convergence of te algoritm is discussed in many publications, see e.g., [5, 13, 22, 23, 30]. It is easy to ceck tat (2.13) p p j = (I αc 1 BA 1 B )(p p j 1 ) = (I αs 0 )(p p j 1 ). By using te spectral bounds for S 0, we obtain a convergence rate γ := I αs 0 = max{ 1 αm 2, 1 αm 2 } < 1, provided tat α ( ) 2 0, M. 2 It is easy to ceck tat for j 1, we also ave (2.14) u j+1 = (I αa 1 B C 1 B)u j. Due to te compatibility assumption (2.8), eac residual representation u j in te Uzawa algoritm, belongs to V0. Using te spectral properties of S 1 := A 1 B C 1 B : V0 V0 (see Lemma (8.1) (v)) and (2.14), we ave tat te series u j is absolutely convergent. Consequently, if we j=1 start te algoritm wit p 0 = 0, ten p k = αc 1 B( k u j ) and p k p = C 1 B w wit w = α j=1 u j V0. Tus, te Uzawa algoritm not only approximates te solution of (2.7), but also finds a special representation of te solution: p = C 1 B w wit w V0. Similar remarks old for Uzawa Gradient (UG) and Uzawa Conjugate Gradient (UCG) algoritms. 3. SPLS discretization In ligt of Propsition 2.2, we are led to te corresponding SPLS discretization of (1.1). We let V V and M Q be finite dimensional j=1

6 6 CONSTANTIN BACUTA AND KLAJDI QIRKO approximation spaces and consider te restrictions of te forms a 0 (, ) and b(, ) to te discrete spaces V and M. Assume tat te following discrete inf sup condition olds for te pair (V, M ). (3.1) inf p M b(v, p ) sup v V p v = m > 0. We define te corresponding discrete operators A, C, B, and B. For example, A is te discrete version of A, and is defined by A u, v = a 0 (u, v ), for all u V, v V. We define V,0 to be te kernel of B V,0 := {v V b(v, q ) = 0, for all q M }, V,1 := V,0 = A 1 B (M ), and S,0 := C 1 B A 1 B. Remark 3.1. If V,0 V 0, ten te compatibility condition (2.8) implies a discrete compatibility condition. Consequently, under te discrete stability assumption (3.1), te problem: Find p M suc tat as unique solution. b(v, p ) = f, v, for all v V, In general, V,0 is not contained in V 0. Tus, te compatibility condition (2.8) does not imply a discrete compatibility condition, and te problem: Find p M suc tat (3.2) b(v, p ) = f, v, for all v V, or B p = f, migt not be well posed. Neverteless, under te assumption (3.1), te following saddle point discrete variational problem : Find (u, p ) V M suc tat (3.3) a 0 (u, v ) + b(v, p ) = f, v for all v V, b(u, q ) = 0 for all q M, always as a unique solution. Using te corresponding discrete operators, te problem (3.3) is equivalent to: Find (u, p ) V M suc tat u (3.4) = A 1 (f B T p ) C 1 B A 1 B p = C 1 B A 1 f. Since te operator S,0 = C 1 B A 1 B is a symmetric positive definite on M, it is invertible, and te second equation of (3.4) as te unique solution p. Tus, one way to solve te system (3.3), would be to solve te second equation of (3.4) for p, and ten find u from te first equation of (3.4). Since C 1 B and A 1 B are dual to eac oter in te Hilbert sense and te problem (3.2) is also equivalent wit A 1 B p = A 1 f, we ave tat te solution of te second equation of (3.4) is in fact te least squares solution of (3.2). Te component u of te solution of (3.3) or (3.4) becomes te

7 SADDLE POINT LEAST SQUARES 7 representation on V of te residual associated wit te least square solution of (3.2). Anoter reason for wic te saddle point least squares discretrization of (1.1) is te variational formulation (3.3) is tat (3.3) is te natural discretization of te SPLS formulation (2.12) of (1.1). Using tat (3.3) is te discrete variational formulation of (2.12), and based on te classical error analysis for SPP teory, we can find standard estimates for 0 u and p p, see [9, 11, 32, 36]. If we assume discrete stability, ten te second component p of te solution of (3.3) provides a good approximation to te least square solution p of (1.1) even in te absence of te compatibility condition (2.8). In te case wen te compatibility condition (2.8) olds, we are dealing wit a special saddle point problem, and a sarp error estimate for p p can be proved using te Xu-Zikatanov argument in [36]. Teorem 3.2. Let b : V Q R satisfy (2.3) and (2.4), and assume tat f V is given and satisfies (2.8). Assume tat p is te solution of (1.1) and V V, M Q are cosen suc tat te discrete inf sup condition (3.1) olds. Ten, if (u, p ) is te solution of (3.3), te following error estimate olds: (3.5) 1 M u p p M m inf p q. q M Te proof of te estimate can be found in te Appendix. Te rigt part of te estimate (3.5) is an improvement of te similar estimate presented in [10] tat provides a bound tat is linear in m 2. Tis improvement is significant if m depends on. Remark 3.3. All te considerations made so far in tis section make sense if te form a 0 (, ), as an inner product on V, is replaced by anoter inner product wic gives rise to an independent of equivalent norm on V. Certainly, te definition of A, S,0, and m, will cange accordingly wit te new inner product, but te error estimate (3.5) remains valid wit different estimating constants tat factor in te norm equivalence constants. Tis observation leads to an efficient preconditioning approac for SPLS discretization. More precisely, if te Uzawa algoritm is involved to solve (3.3), ten te action of A 1 can be replaced by te action of any equivalent preconditioner. From te first equation of (3.3) and (3.5) we get tat te solution (u, p ) satisfies: u = f B p V M 2 inf p q, q M were f is te functional v f, v on V. Tis gives an estimate for te residual associated wit te discrete equation (3.2). Remark 3.4. Te Bramble-Pasciak least squares metod in [10] for discretizing (2.7), focuses on solving only te variational formulation of te m

8 8 CONSTANTIN BACUTA AND KLAJDI QIRKO second equation of (3.4), i.e., Find p M suc tat (3.6) (C 1 B A 1 B p, q ) = f, A 1 B q, for all q M, wic can be reformulated as: Find p M suc tat (3.7) b(a 1 B q, p ) = f, A 1 B q, for all q M, Te approac in [10] to numerically solve (3.7) is based on using families of stable pairs {(V, M )}, bases for bot V and M, and on replacing te action of A 1 on V by an approximate inverse. Our approac is based on iteratively solving te coupled saddle point system (3.3) wose p component of te solution is te least squares solution of (3.2) and on using bases and matrix assembly only for te test space V. Remark 3.5. Te DPG metod, [20, 21], for discretizing (2.7), also reduces to a saddle point formulation tat is similar to (3.3). Te details are presented in Teorem 2.4 of [20] and a proof can be found in [24]. For te DPG approac, te trial spaces Q or M are sougt as a product between an interior component and and an interface compomonent at bot continuous and discrete levels. Once a finite dimensional subspace V r of V (wit certain properties) is cosen, te discrete test space, using our notation, is V = A 1 r B M, were A 1 r is te Riesz representation operator for te Hilbert space (V r, a 0 (, )). In our approac, we coose te discrete trial space V V first, a finite dimensional subspace M Q, and ten define M := Q C 1 BV, were C 1 is te Riesz representation operator for te space Q and Q is a projection from Q to M, (see Section 5 for details). Given te connection between approximating te solution of te mixed variational problem (1.1) and solving te discrete problem (3.3), one can just rely on te available tecniques and solvers for symmetric saddle point problems of type (2.12) on stable families of spaces {(V, M )} in order to solve (1.1). Neverteless, if it is difficult to find families of stable pairs or tey exist but are difficult to implement, ten we present next alternative ways of approximating te solution of (1.1). 4. Iterative solvers witout basis for te trial space Te computational callenge we face wen solving (3.3) on (V, M ) is tat one migt not be able to find stable pairs (V, M ). Even in te case tat stable pairs are available, a global linear system corresponding to (3.3) migt be difficult to assemble and solve. It is possible to solve (3.3) witout even aving explicit bases for M by using te Uzawa (U), Uzawa Gradient (UG), or Uzawa Conjugate Gradient (UCG) algoritms. Following [3], we ave tat te standard U and UG algoritms can be rewritten suc tat tey differ only by te way te parameter α is cosen. For te Uzawa ( ) algoritm, 2 we ave to coose a fixed number α = α 0 in te interval 0,. For te M 2 UG algoritm, te parameter α is cosen to impose te ortogonality of

9 SADDLE POINT LEAST SQUARES 9 consecutive residuals associated wit te second equation in (3.3). Te first step for Uzawa is identical wit te first step of UG. We combine te two algoritms in: Algoritm 4.1. (U-UG) Algoritms Step 1: Set u 0 = 0 V, p 0 M, compute u 1 V, q 1 M by a 0 (u 1, v) = f, v b(v, p 0 ), for all v V (q 1, q) = b(u 1, q), for all q M. Step 2 : For j = 1, 2,..., compute j, α j, p j, u j+1, q j+1 by (U UG1) a 0 ( j, v) = b(v, q j ), v V (Uα) α j =α 0 for te Uzawa algoritm or (UGα) α j = (q j, q j ) for te UG algoritm b( j, q j ) (U UG2) p j =p j 1 + α j q j (U UG3) u j+1 =u j + α j j (U UG4) (q j+1, q) = b(u j+1, q), for all q M. Here, f is te restriction of f to V. To obtain te UCG algoritm, te UG algoritm is modified as in [9, 35] as follows: First, we define d 1 := q 1 in Step 1, and ten modify Step 2 by replacing b(, q j ) wit b(, d j ), were {d j } is a sequence of conjugate directions: Algoritm 4.2. (UCG) Algoritm Step 1: Set u 0 = 0 V, p 0 M. Compute u 1 V, q 1, d 1 M by a 0 (u 1, v) = f, v b(v, p 0 ), v V (q 1, q) = b(u 1, q), for all q M, d 1 := q 1. Step 2 For j = 1, 2,..., compute j, α j, p j, u j+1, q j+1, β j, d j+1 by (UCG1) a 0 ( j, v) = b(v, d j ), v V (UCGα) α j = (q j, q j ) b( j, q j ) (UCG2) p j =p j 1 + α j d j (UCG3) u j+1 =u j + α j j (UCG4) (q j+1, q) = b(u j+1, q), for all q M (UCGβ) β j = (q j+1, q j+1 ) (q j, q j ) (UCG6) d j+1 =q j+1 + β j d j

10 10 CONSTANTIN BACUTA AND KLAJDI QIRKO Eac one of te described algoritms converges to te discrete solution (u, p ) of (3.3). In addition, te following sarp error estimation result was proved by one of te autors in a sligtly more general context in [3]. Teorem 4.3. If (u, p ) is te discrete solution of (3.3), and (u j+1, p j ) is te j t iteration for U, UG, or UCG, ten (u j+1, p j ) (u, p ) and (4.1) 1 M 2 q j+1 p j p 1 m 2 q j+1, m M 2 q j+1 u j+1 u M m 2 q j+1. Besides convergence of te iteration processes, te result entitles q j+1 as a computable, robust, efficient, and uniform-modulo m estimator for all tree algoritms. Under stability presence, we can use te Teorem 3.2 and te estimates (4.1) to build adaptive or multilevel algoritms for SPLS discretization. A cascadic algoritm for solving symmetric SPPs was introduced by one of te autors in [3]. One major advantage of solving te system (3.3) as te SPLS discretization of (1.1), using one of tese tree algoritms is tat (4.1) remains valid, if te rigt correction of constant factors is made, wen a 0 (, ) is replaced by a uniform equivalent form a prec (, ). Similar to te continuous case in Section 2.2, an important observation ere is tat if te starting initial guess is p 0 = 0, ten te p j - iterates remain in te space C 1 B V and tat {p j } approximates and represent te solution p in te form C 1 B w, wit w V,0. In addition, as presented in te next section, for certain spaces M a basis is not needed for solving te variational problems associated wit q j, and at eac step of te U-type iterative processes, only te action of A 1 or a preconditioner requires an inversion process. 5. Construction of special discrete spaces Next, we will present a metod to address te lack of stability of te approximation spaces. Let V be a subset of V, and let M be a finite dimensional subspace of Q tat as good approximability properties. Typical examples of spaces M are te spaces of piecewise polynomials. We equip M wit an inner product tat could differ from te restriction of te Q inner product on M, but induces an equivalent norm (independent of ). For convenience, we denote te inner product on M by (, ). If Q : Q M is te ortogonal projection onto M, we simply define te space M by M := Q C 1 BV. We consider te restriction of te form a 0 (, ) to V V and te restriction of b(, ) to V M, and define te discrete operators A, C, B, and B

11 SADDLE POINT LEAST SQUARES 11 for te pair (V, M ). For any q M, v V, we ave b(v, q ) = B v, q = (C 1 B v, q ). On te oter and, since q M Q, and v V V, we ave b(v, q ) = (C 1 Bv, q ) = ( Q C 1 Bv, q ). From te above identities, we get (5.1) C 1 B v = Q C 1 Bv for all v V. Tis implies tat C 1 B is onto M and, using tat V and M are finite dimensional spaces, te discrete inf-sup condition (3.1) olds for some m > 0 tat migt depend on. Tus, te problem (3.3) as unique solution (u, p ), and any of te tree Uzawa type algoritms presented in Section 4 can be applied to approximate it. In addition, in ligt of 5.1, we ave tat te residual q j for eac Uzawa type algoritm satisfies q j = C 1 B u j = Q C 1 Bu j. Consequently, te computation of q j involves te computation Bu j, te action of C 1 at te continuous level (often just a multiplication operator) to find C 1 Bu j, and a projection onto te space M - tat is usually a standard finite element space. Tus, for eac of te Uzawa type algoritms, a basis for M is not needed for solving te variational problems associated wit q j Approximability of te space M. Due to Teorem 3.2, in order to expect small discretization error p p, besides stability, one needs to investigate te minimization problem inf p q in te special case q M wen M is a proper subspace of M. Te continuous inf sup condition (2.3) tat we assume, guaranties tat A 1 B is injective and as closed range, see Lemma (8.1) (iii). Tus, te dual operator C 1 B is onto Q, and for any p Q, tere exists u V suc tat C 1 Bu = p. For p = C 1 Bu Q, and any q = Q C 1 Bv M, we ave tat (5.2) p q = C 1 Bu Q C 1 Bv C 1 Bu C 1 Bv + Q C 1 Bv C 1 Bv M u v + Q C 1 Bv C 1 Bv. In order to get good SPLS approximation properties for te solution p of (1.1), it would be enoug to ask for some regularity of a solution u of C 1 Bu = p, for approximation properties of V, and for approximation properties of te projection Q : C 1 BV M. Note tat te argument remains valid regardless of te discrete stability presence.

12 12 CONSTANTIN BACUTA AND KLAJDI QIRKO 5.2. Te special projection case. If Q C 1 Bv = 0 implies C 1 Bv = 0, for any v V, (i.e., Q is injective on C 1 BV ), ten V,0 V 0 and, see Remark 3, te variational formulation (3.2) is well posed, as unique solution p M, and using Proposition 2.2 for te discrete pair (V, M ), we ave tat (u = 0, p ) is te solution of (3.3). Tis will always be te case if we coose M = C 1 BV, and Q to be te identity operator on C 1 BV. In tis special case we do ave M = M = C 1 BV and, if p is te solution of (1.1) and p is te solution of (3.2), from (1.1) and (3.2) we obtain tat 0 = b(v, p p ) = (C 1 Bv, p p ), for all v V. Tus, we simply ave tat p is te ortogonal projection of p onto M, and consequently, using te representation p = C 1 Bw for some w V, we ave (5.3) p p = inf p q = inf C 1 Bw C 1 Bv q M v V M inf w v. v V Compared wit (3.5), te discretization error estimate (5.3) as te advantage tat is independent of te stability constant m and, in addition, it reduces to approximability of functions in V by discrete functions in V. If one of te Uzawa type iterative metods of Section 4 is applied to find te solution (0, p ) of (3.3) ten, according to Teorem (4.3), te iteration error satifies p j p 1 m 2 q j+1. If te discretization error order is available, say p p = O( α ), and an estimate about m is also available, te iteration error can matc te discretization error by imposing te stopping criteria (5.4) q j+1 c 0 m 2 α, were c 0 is a constant independent of. Consequently, using one of te Uzawa algoritms, we can approximate te solution p by an iterate p j C 1 BV up to optimal discretization error order, regardless of (discrete B-) stability presence. From te numerical experiments we performed so far for concrete SPLS discretization problems, projecting on smoot spaces M could lead to better approximation of te continuous solution p, and projecting on coarser spaces MH could lead to stability. It is wort noticing ere, tat if one cooses M suc tat Q q c q, for all q C 1 BV, ten stability of ( V, C 1 ) ( BV implies stability for te pair V, Q C 1 BV ). A typical coice for te space M used to define te projection Q is te space of continuous piecewise polynomials of degree m wit respect

13 SADDLE POINT LEAST SQUARES 13 to a mes T. On M we can coose te standard inner product or we can consider te lumping inner product introduced in [4]. To define te lumping inner product, assume tat {φ 1, φ 2,..., φ m } is a nodal basis for M. We can define (, ) l by (5.5) (φ i, φ j ) l := δ j i (1, φ i), i, j = 1, 2,..., m, and extend it to M M by m m (5.6) α i φ i, β j φ j := i=1 j=1 l m α i β i (1, φ i ). i=1 Using te identity (5.1), see [4], te computation of C 1 m m C 1 B v = i=1 (C 1 Bv, φ i ) φ i = (1, φ i ) i=1 B v becomes b(v, φ i ) (1, φ i ) φ i. We note ere tat by using te lumping inner product, one avoids mass matrix inversion at eac iterative step. All of te above provides a more detailed picture of te five step SPLS discretization summarized at te end of introduction. A concrete example of ow to apply te metod is presented next. 6. SPLS discretization of a div curl system Here, we apply te SPLS discretization metod for a model div curl problem on a polyedral domain Ω R 3. For a given data, we are looking to find te vector function L 2 (Ω) suc tat (6.1) = j in Ω (µ) = g in Ω (µ) n = σ on Γ := Ω, were µ is a given parametric scalar L 2 function tat is strictly positive on Ω. Te variational formulation we adopt for (6.1) is similar to te approac of [10]. We multiply te first equation in (6.1) by w H 1 0 (Ω), multiply te second equation by ϕ H 1 (Ω)/R and after we integrate by parts, we obtain (6.2) (, w) = (j, w) for all w H 1 0 (Ω) (, µ ϕ) = G, ϕ := ( g, ϕ) + (σ, ϕ) Γ for all ϕ H 1 (Ω)/R, were (, ) denotes te standard L 2 type inner product. If we define V := H 1 0 (Ω) H1 (Ω)/R, Q := L 2 (Ω), and te form b(v, p) on (V, Q), by b((w, ϕ), ) := (, w + ϕ), for all (w, ϕ) V, Q, te variational formulation for (6.1) becomes: Find Q suc tat (6.3) b((w, ϕ), ) = F, (w, ϕ) := (j, w) + G, ϕ, for all (w, ϕ) V. Te coice of inner products on te spaces V and Q is essential in building robust solvers wit respect to te function µ. By coosing te weigted

14 14 CONSTANTIN BACUTA AND KLAJDI QIRKO (, ) µ inner product on Q = L 2 (Ω) and te weigted inner product on V := H 1 0 (Ω) H1 (Ω)/R induced by te norm (w, ϕ) 2 V := a µ 1(w, w) + a µ (ϕ, ϕ) := µ 1 w 2 + µ ϕ 2, Ω Ω it can be proved tat, in tis case, te form b(, ) satisfies a continuous inf sup condition wit a constant independent of µ, see [10]. Te C 1 B operator tat appears in te SPLS discretization is (6.4) C 1 B (w, ϕ) = µ 1 curl w + ϕ. By using a Helmoltz decomposition, it is easy to ceck tat C 1 B is onto Q = L 2 (Ω). If te data (j, g, σ) is suc tat te compatibility condition (2.8) is satisfied, ten (6.2) is a well-posed problem and our SPLS discretization can be applied. For SPLS discretization, we can cose V V to be te standard vector space of continuous piecewise functions of degree m wit te appropriate boundary conditions for eac component of V. For M we consider two coices: Case 1) M := C 1 BV, and Case 2) M := Q (C 1 BV ), were Q is te ortogonal projection onto M - te continuous piecewise polynomials of degree m. On M we can consider te standard inner product or te lumping inner product defined in (5.6). In bot cases, we automatically ave a positive discrete inf sup constant m. Sufficient conditions tat can establis stability for tis problem remain to be investigated for various coices of discrete spaces. For te 2D version of (6.1), it is easy to ceck tat we do ave (free) stability for bot types of discretization. Tis is simply due to te fact tat te (curl, curl) form is equivalent wit te Diriclet form (, ) on te space H0 1(Ω). In general, establising stability for te family {(V, M = C 1 BV )} migt not be an easy problem, even if we focus on V to be a standard conforming space of piecewise polynomial of degree k. For example, if we consider te simple case of te operator B being te div : H 1 0 (Ω) L2 (Ω) and C 1 to be just te identity operator, ten te problem of discrete B-stability is equivalent wit establising te classical Stokes stability for (V, div V ). From te works of Scott and Vogelius on divergence stability, [33, 34], for te case of general quasi-uniform meses, te divergence stability remains a 30 year old open problem. For many problems of interest, te B-stability can be establised based on paricular properties of te operator B. Preliminary numerical results for div curl systems and te Maxwell equations, suggest te presence of B-stability for particular coices of test spaces and projection operators. A more detailed investigation of B stability for te 3D div curl or Maxwell problem remains to be investigated in te near future Numerical results. We performed numerical experiments for approximating (6.1) wit Uzawa Conjugate Gradient (UCG) and V te conforming P 1 elements. For M we consider bot cases described above in tis

15 SADDLE POINT LEAST SQUARES 15 section. In Case 2) we project on M te vector space of continuous P 1 functions, wit te componentwise inner product given by (5.6). Te domain Ω was cosen te unit cube, and te function µ was cosen to be te restriction to Ω of { 1, if x < 1 µ = 2 µ 0, if x 1 2, for various values of te constant µ 0. First we consider te case wen µ 0 = 1 (constant µ) and data cosen suc tat te exact solution is = (x(1 x)yz, xy(1 y)z, xyz(1 z)). For Case 1), using a power metod for te discrete Scur complement S,0 we estimated m and imposed te stopping criterion given by (5.4) wit c 0 = 0.4 and α = 1. For Case 2) we just imposed a fixed number of iterations on eac level. Te numerical results are presented below in Table 1 and Table 2. Level µ = 1 P 1 C 1 BP 1 comp Order # of it. m Table 1: UCG for SPLS-P 1 -discretization Case 1) Level µ = 1 P 1 Q C 1 B(P 1 ) comp Order # of it Table 2: UCG for SPLS-P 1 -discretization Case 2) Next, we consider te case of discontinuous µ (µ 0 = 50) and data cosen suc tat te exact solution is = (x(1 x)( 1 2 x)yz, x(1 2 x)y(1 y)z, x(1 x)yz(1 z)). 2 Te numerical results are presented below in Table 3 and Table 4. For Case 1), wit µ aving 2D ceckerboard jump discontinuities (1 on wite squares, µ 0 on te black squares and constant 1 in te z- direction), we obtained similar results. Te estimates for te discrete inf sup

16 16 CONSTANTIN BACUTA AND KLAJDI QIRKO Level µ 0 = 50 P 1 C 1 BP 1 comp Order # of it. m Table 3: UCG for SPLS-P 1 -discretization Case 1) Level µ 0 = 50 P 1 Q C 1 B(P 1 ) comp Order # of it Table 4: UCG for SPLS-P 1 -discretization Case 2) constant m are presented in Table 5. Wile we can notice instability wit respect to (or Level) we can also notice numerical stability wit respect to µ 0. Level µ Table 5: Estimates for m for different values, µ 0 Remark 6.1. We note tat for Case 1), te discretization error satisfies (5.3) and for te conforming coice P 1 for V we ave = O(). Ten, since comp + comp, and te iteration error comp matces te discretization error (see Teorem 4.3) we also expect O() for comp. Tis is numerically observed in Table 1 and Table 3. For Case 2), te discretization error satisfies an estimate of type (5.2). In tis case, our estimate for te discretization error is not optimal, because te functions C 1 Bv for v V are piece wise constants, and te most we can expect from Q C 1 Bv C 1 Bv is O( 1/2 ). Neverteless, te computation we performed in tis case, (see Table 2 and Table 4), sow

17 SADDLE POINT LEAST SQUARES 17 tat te comp is at least O(). Te supper convergence beavior in tis case remains to be teoretically and numerically investigated. 7. Conclusions We presented a saddle point least squares metod to discretize variational formulations wit different types of trial and test spaces. To te autors knowledge, te proposed SPLS approac is different from te DPG formulations as presented in [20, 21, 19], were a trial space is cosen firstly, and a (close to optimal) test space tat provides stability of te pairs is cosen secondly. In te first order systems case, te essential differences between te proposed SPLS discretization and te classical least squares finite element metod approac, as described in [6, 7, 8, 16, 17] or te FOSL & FOSLL approaces in [14, 15, 18, 26, 27, 28], are tat for SPLS discretization te test spaces and te trial spaces are of a different nature. Tis makes SPLS discretization more general, but requires special attention in finding stable spaces for discretization or in finding efficient iterative solvers. Tanks: Te autors would like to tank to Jay Gopalakrsnan for valuable discussions tat relate DPG and te SPLS and to Francisco Sayas for valuable software and programming support for obtaining te numerical solvers. 8. Appendix 8.1. Functional analysis results. For a bounded linear operator T : X Y between two Hilbert spaces X and Y, we denote by T t te Hilbert transpose of T. If X = Y, we say tat T is symmetric if T = T t. For a bounded linear operator T : X X, we denote te spectrum of te operator T by σ 0 (T ). Te following lemma provides important properties of norms and operators to be used in tis paper. A proof of it can be found in [2]. Lemma 8.1. Let A, C, B, and B be te operators associated wit te spaces V, Q and te connecting form b(, ). Assume tat (2.3) and (2.4) are satisfied. i) Te operators C 1 B : V Q and A 1 B : Q V are symmetric to eac oter, i.e., (8.1) (C 1 Bv, q) = a 0 (v, A 1 B q), v V, q Q, consequently, (C 1 B) t = A 1 B and (A 1 B ) t = C 1 B. ii) Te Scur complement on Q is te operator S 0 := C 1 BA 1 B : Q Q. Te operator S 0 is symmetric and positive definite on Q, satisfying (8.2) (S 0 p, p) = sup v V b(v, p) 2 v 2.

18 18 CONSTANTIN BACUTA AND KLAJDI QIRKO Consequently, m 2, M 2 σ 0 (S 0 ) and (8.3) σ 0 (S 0 ) [m 2, M 2 ]. iii) Te following estimate olds (8.4) p S0 := (S 0 p, p) 1/2 = A 1 B p V m p for all p Q. Consequently, A 1 B : Q V as closed range, V 1 := A 1 B (Q) is a closed subspace of V and A 1 B : Q V 1 is an isomorpism. iv) Te Scur complement on V is defined as te operator S := A 1 B C 1 B : V V. Te operator S is symmetric and non-negative definite on V, wit Ker(S) = V 0, S(V) = V 1, and satisfies (8.5) a 0 (Su, v) = (C 1 Bu, C 1 Bv), u, v V. v) Te Scur complement on V 1 = V 0 is te restriction of S to V 1, i.e., S 1 := A 1 B C 1 B : V 1 V 1. Te operator S 1 is symmetric and positive definite on V 1, satisfying (8.6) σ 0 (S 1 ) = σ 0 (S 0 ) [m 2, M 2 ] Proof of Teorem 3.2. Proof. Let p be te solution of (1.1) and assume tat(u, p ) is te solution of (3.3). First, we notice tat te operator T : Q Q defined by T p = p is linear and idempotent. To prove tat T is idempotent we consider te problem: Find (ũ, p ) V M suc tat (8.7) a 0 (ũ, v ) + b(v, p ) = b(v, p ) for all v V, b(ũ, q ) = 0 for all q M. Due to te assumption (3.1), we get tat (8.7) as unique solution and by noticing tat (ũ, p ) = (0, p ) solves te problem, we can conclude tat T p = p and consequently, T 2 = T. According to Kato [25], and Xu and Zikatanov [36], we ave Next, for any q M we ave (8.8) I T L(Q,Q) = T L(Q,Q). p p = (I T )p = (I T )(p q ) I T p q To estimate T we use (3.1): = T p q. T p 1 b(v, T p) sup = 1 b(v, p ) sup. m v V v m v V,1 v Solving for b(v, p ) from te first equation of (3.3), and using te fact tat f, v = b(v, p) for all v V, we furter get

19 SADDLE POINT LEAST SQUARES 19 (8.9) T p 1 m M m p. b(v, p) a 0 (u, v ) sup = 1 b(v, p) sup v V,1 v m v V,1 v Tus, T M m, and from (8.8) we obtain te rigt side of (3.5). To prove te left side of (3.5), from (1.1) and te first equation of (3.3) we get Ten, b(v, p p ) = a 0 (u, v ), for all v V. a 0 (u, v ) b(v, p p ) u = sup = sup M p p, v V v v V v wic concludes te validity of te estimate (3.5). References [1] A. Aziz and I. Babuška. Survey lectures on matematical foundations of te finite element metod. Te Matematical Foundations of te Finite Element Metod wit Applications to Partial Differential Equations, A. Aziz, editor, [2] C. Bacuta. Scur complements on Hilbert spaces and saddle point systems. J. Comput. Appl. Mat., 225(2): , [3] C Bacuta. Cascadic multilevel algoritms for symmetric saddle point systems. Comput. Mat. Appl., 67(10): , [4] C. Bacuta, B. McCracken, and L. Su. Residual reduction algoritms for nonsymmetric saddle point problems. J. Comput. Appl. Mat., 235(6): , [5] M. Benzi, G. Golub, and J. Liesen. Numerical solutions of saddle point problems. Acta Numerica, pages 1 137, [6] P. Bocev, Z. Cai, T. A. Manteuffel, and S. F. McCormick. Analysis of velocity-flux first-order system least-squares principles for te Navier-Stokes equations. I. SIAM J. Numer. Anal., 35(3): , [7] P. Bocev and M.D. Gunzburger. Least-squares finite element metods. In International Congress of Matematicians. Vol. III, pages Eur. Mat. Soc., Züric, [8] P. Bocev and M.D. Gunzburger. Least-squares finite element metods, volume 166 of Applied Matematical Sciences. Springer, New York, [9] D. Braess. Finite Elements. Teory, Fast Solvers, and Applications in Solid Mecanics. Cambridge University Press, Cambridge, [10] J. H. Bramble and J. Pasciak. A new approximation tecnique for div-curl systems. Mat. Comp., 73: , [11] S. Brenner and L.R. Scott. Te Matematical Teory of Finite Element Metods. Springer-Verlag, New York, [12] F. Brezzi. On te existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. Rev. Française Automat. Informat. Recerce Opérationnelle Sér. Rouge, 8(R-2): , [13] F. Brezzi and M. Fortin. Mixed and Hybrid Finite Element Metods. Springer-Verlag, New York, [14] Z. Cai, C.-O. Lee, T. A. Manteuffel, and S. F. McCormick. First-order system least squares for te Stokes and linear elasticity equations: furter results. Iterative metods for solving systems of algebraic equations.

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21 SADDLE POINT LEAST SQUARES 21 University of Delaware, Department of Matematics, 501 Ewing Hall address: University of Delaware, Department of Matematics, 501 Ewing Hall address: