J.H. Bramble, R.D. Lazarov, and J.E. Pasciak. A least-squares approach based on a discrete minus one inner product for first order systems. Math. Comp., 66:935--955, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....[8] and references therein) For completeness, we study an inverse norm least squares functional and show that its homogeneous form is equivalent to the L norm for the stress and the displacement. This functional can be used to develop a discrete inverse norm least squares method (see, e.g. [7]) For some applications, it is convenient to impose boundary conditions weakly by adding boundary functionals. Such a functional is also studied in this paper. See [21] for how to use these types of functionals to develop a computationally feasible numerical method. An outline of the paper is as ....

....over VB . In this paper, we concentrate on the least squares problem based on the L norm functional in (3.8) nd ( u) 2 VB such that G( u ; f ) inf ( v)2VB Note that the inverse norm functional in (3. 7) can be used to develop a discrete inverse norm least squares method (see [7]) as well. Remark 3.1. Since the minimum of the quadratic functional G( u ; f ) is zero, by (3.6) the symmetry of the stress tensor is guaranteed by the rst term of the functional, i.e. the constitutive equation. Remark 3.2. The least squares functionals de ned in (3.7) and (3.8) di er from ....

J. H. Bramble, R. D. Lazarov, and J. E. Pasciak, A least-squares approach based on a discrete minus one inner product for rst order systems, Math. Comp., 66 (1997), pp. 935{ 955.

....ellipticity of augmented functional G 2 in (3.2) This somewhat restrictive assumption is not necessary for functional G 1 in (3.1) which supports an efficient practical algorithm (the H norm in (3. 1) can be replaced by a discrete inverse norm or a simpler mesh weighted norm; see [5] and [8] for analogous inverse norm algorithms) and which has the weaker norm equivalence assured by Theorem 3.1. Nevertheless, the principal result of this paper is Theorem 3.2, which establishes full H product ellipticity of least squares functional G 2 for the generalized Stokes system. Since we have ....

J. H. Bramble, R. D. Lazarov, and J. E. Pasciak, A least-squares approach based on a discrete minus one inner product for first order system, manuscript.

....conditions. Next, we turn our attention to a least squares functional in which the residual of the momentum equation is measured in the norm of the negative order Sobolev space H 1 . For earlier work on H 1 norm functionals, we refer the reader to papers by Bramble, Lazarov, and Pasciak [4], 5] and Cai, Manteu#el, and McCormick [6] In the present paper, our analysis again combines previous results on analogous operators for the Stokes equations with the abstract framework outlined above. Although in both cases we deal with similar abstract formulations of the least squares ....

....nonlinear theory to the new H 1 functional. Let H 1 denote the dual of H 1 0(##2 Using the equivalence of the seminorm and norm on H 1 0(##1 we equip H 1 with the norm f 1 = sup ##H 1 0(## (f, #) # 1 #f # H 1 , 51) for which the following representation result holds (cf. [4]) LEMMA 6. For all f # H 1 (## , we have f 1 = Sf, f) 52) and #Sf# 1 # C f 1 , 53) where S : H 1 (## ## H 1 0 is the solution operator of the Dirichlet problem #u = f in# , 54) u = 0 on #; 55) that is, u = Sf is the solution of (54) 55) 1002 P. BOCHEV, Z. ....

J. BRAMBLE, R. LAZAROV, AND J. PASCIAK, A Least Squares Approach Based on a Discrete Minus One Inner Product for First Order Systems, Tech. report 94-32, Mathematical Science Institute, Cornell University, Ithaca, NY, 1994.

....COROLLARY 5.1. We have that 1 C #V# 2 # 2 #V 1 # 2 # G 1 (QV; 0) # C #V# 2 # 2 #V 1 # 2 (5.4) for all V # U . FIRST ORDER SYSTEM LEAST SQUARES FOR ELASTICITY 333 Remark 5.1. Corollary 5. 1 implies that standard discrete H 1 norms (for more details, see [1] and [8] can be used to develop a discretization and solution process that achieves uniform and optimal L 2 approximations to the deformations and stresses, and that displacements can be recovered as in section 4 with uniform and optimal H 1 performance. Let U h and Z h be ....

....2 # C 1 C 0 C h 2l 2 #u# 2 l #p# 2 l 1 , 5.9) ##u #u h # # #U U h # #U h #u h #, 5.10) ##u #u h # 2 # C G 1,h (U h ; f ) #U h #u h # 2 , 5.11) ##u #u h # # C h l 1 (#u# l #p# l 1 ) 5.12) Proof. An analysis similar to that in [1] (see also [8] shows that (5.4) is valid for G 1,h (QV; f ) and for any V # U h . Thus, these estimates follow directly from the usual FOSLS a posteriori error bounds and approximation estimates (4.10) and (4.11) Remark 5.2. The theory here assures that the discrete functional G 1,h ....

J. H. BRAMBLE, R. D. LAZAROV, AND J. E. PASCIAK, A least-squares approach based on a discrete minus one inner product for first order system, manuscript.

....formulations of operator equations. The possibility of using adaptive wavelet techniques for certain least squares formulations of rst order elliptic systems has been indicated in [78] In the nite element context least squares formulations are currently a very active eld of research, see e.g. [20]. An important issue in this context is to identify appropriate least squares functionals which, in particular, do not pre impose too strong regularity properties of the solution. The problem is that usually these functionals involve Sobolev norms of negative or fractional index which are hard to ....

J.H. Bramble, R.D. Lazarov, and J.E. Pasciak, A least-squares approach based on a discrete minus one inner product for rst order systems, Math. Comput. 66 (1997), 935-955.

....[12] As mentioned earlier, we need to replace the H 1 inner product in (3.4) by a computationally feasible discrete inner product that ensures the equivalence on V h between the standard norm in V and that induced by the discrete bilinear form. A discrete H 1 approach was introduced in [4] for scalar elliptic equations and was extended to the Stokes problem in [11] in the context of rstorder system least squares methods. So, let A h : H 1( d U h be the discrete solution operator = A h v 2 U h for the Poisson problem (2.3) de ned by the relations Z (r r ) ....

J. H. Bramble, R. D. Lazarov, and J. E. Pasciak, A least-squares approach based on a discrete minus one inner product for a rst order system, manuscript.

....Indeed, FOSLL appears to exhibit the generality of the inverse norm FOSLS approach while retaining the full e#ciency of the L 2 norm approach. 1. Introduction. First order system least squares (FOSLS) was developed for numerical solution of a wide range of partial di#erential equations (see [1, 2, 3, 4, 7, 9, 10, 11, 12, 13, 14, 19] and references therein) The basic idea behind the standard FOSLS approach is that it recasts the original system as an expanded first order system to which a least L 2 norm principle is applied. The central aim is to reformulate the original system as the minimization of a functional whose ....

....by standard multigrid. One limitation of L 2 norm FOSLS is that product H 1 equivalence can usually be confirmed only under su#cient smoothness assumptions on the original problem (e.g. the domain, coe#cients, and data) Inverse norm versions of FOSLS can overcome this limitation (cf. [4, 9, 12]) but at the expense of rather awkward norm evaluation requirements and the attendant loss of full e#ciency. In fact, because the inverse norm usually does not take the underlying problem into account (e.g. varying coe#cients) the constants in the inverse norm continuity and coercivity bounds ....

J. H. Bramble, R. D. Lazarov, and J. E. Pasciak, A least--squares approach based on a discrete minus one inner product for first order system, Math. Comp., 66 (1997), pp. 935--955.

....(or the multigrid solver, had they analyzed it) In fact, except for [21] and [23] which is the basis of the work presented in the present paper, none of the methods cited above were shown to achieve optimal discretization accuracy and multigrid convergence. For general literature on FOSLS, see [8], 11] 12] 21] and [22] which treat least squares functionals based on L 2 and L 2 H 1 (that is, a combination of L 2 and H 1 ) inner products. The least squares functionals described in the next section are also based on these norms. In fact, the analytical techniques used in ....

....solution processes. For brevity, the equivalence results of the next two sections focus on G; we will state but not prove the analogous H 1,k equivalence results for F (cf. 21] for detailed proofs of similar results) 3. Nonuniform coercivity: W(##3 A compactness argument (cf. 7] and [8]) is used here to establish equivalence between the homogeneous part of the L 2 H 1 least squares functional, G(u, p; 0) and the square of the product norm [##] 2 ## 1,k . In what follows, C denotes a generic constant that may change meaning at each occurrence but is independent of k. ....

J.H. Bramble, R.D. Lazarov, and J.E. Pasciak, A least-squares approach based on a discrete minus one inner product for first order systems, Math. Comp., 66 (1997), pp 935--955.

....grant DMS 8704169, and the Department of Energy under grant DE FG03 93ER25165. 1125 1126 P. BOCHEV, T. A. MANTEUFFEL, AND S. F. McCORMICK We recall that f 1 = Sf, f ) where S denotes the solution operator for the Poisson equation with homogeneous Dirichlet boundary condition (see [2] [3]) Since the exact evaluation of this norm is not computationally feasible, direct minimization of (5) is not practical. However, theoretical results from Part I will be used here as a vehicle for establishing optimal error estimates for a practical counterpart of (5) based on a computable ....

....feasible, direct minimization of (5) is not practical. However, theoretical results from Part I will be used here as a vehicle for establishing optimal error estimates for a practical counterpart of (5) based on a computable discrete negative norm. Such negative norms were first proposed in [3] [4] In [4] and [5] discrete negative norms were used to develop least squares methods for a perturbed form of the velocity vorticity pressure Stokes equations, which include as a particular case the equations of linear elasticity. A negative norm least squares method for the Stokes problem in ....

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J. Bramble, R. Lazarov, and J. Pasciak, A least squares approach based on a discrete minus one inner product for first order systems, Technical report 94-32, Mathematical Science Institute, Cornell University, Ithaca, NY, 1994.

....Similar to the results of Theorem 2.4, we have ## AU, # AV# 1 = ## AU, z# = #AU, #z#, 3.42) where z # (H 1 #) 3 satisfies ##z, #w# = ## AV, w# = #AV, #w#, 3.43) for every w # (H 1 #) 3 . Equation (3. 43) can be approximately solved using multilevel techniques (cf. [3]) To approximate the right hand side in (3.41) notice that it is a simple chore to find Y # (H(div) 3 such that # Y = f . Thus, #f , # AV# 1 = #Y, #z#, 3.44) where z satisfies (3.43) Similarly, the second term in (3.41) satisfies ##U, #V# L = ##U, Z# = #U, #Z#, 3.45) ....

J. H. Bramble, R. D. Lazarov, and J. E. Pasciak, A least-squares approach based on a discrete minus one inner product for first order system, Math. Comp., 66 (1997), pp. 935--955.

....in a product norm involving Re and the L 2 and H 1 norms. The H Gamma1 norm in the functional is further replaced by the discrete H Gamma1 norm to make the computation feasible, following the discrete H Gamma1 least squares approach proposed by Bramble, Lazarov, and Pasciak [3] for scalar second order elliptic equations. Such discrete H Gamma1 functionals are shown to be uniformly equivalent to the Sobolev norms weighted by the Reynolds number. This property enables us to show that standard finite element discretization error estimates are optimal with respect to the ....

....functional G(u; p; 0) Bound (3.5) now follows from Theorem 2.1 and approximation properties (3.1) 3.3) Remark 3.1. The above result indicates that the finite element approximation is optimal, both with respect to the order of approximation and the required regularity of the solution (see [3]) More specifically, bound (3.5) holds with d r (u; p) i 2 kuk 2 r 1 kpk 2 r j 1 2 since = r Thetau and kr Thetauk r C kuk r 1 . 4. Solution Method and Discrete H Gamma1 Functional. Theorem 3.1 indicates that the finite element approximation based on functional G is also ....

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J. H. Bramble, R. D. Lazarov, and J. E. Pasciak, A least-squares approach based on a discrete minus one inner product for first order systems, Manuscript.

.... problem by proving that the hypotheses of the Lax Milgram lemma are satisfied on appropriate spaces (see, e.g. 9] and [35] More recently, a least squares functional involving a discrete inner product related to the inner product in the Sobolev space of order Gamma1, was introduced in [5], and an approach more closely coupled to the Galerkin method was studied by the same authors in [6] Following the approach of [5] 6] the design of the least squares method requires the use of some negative and half integer Sobolev norms, such as the norms of H Gamma1 ( Omega Gamma and H ....

.... More recently, a least squares functional involving a discrete inner product related to the inner product in the Sobolev space of order Gamma1, was introduced in [5] and an approach more closely coupled to the Galerkin method was studied by the same authors in [6] Following the approach of [5], 6] the design of the least squares method requires the use of some negative and half integer Sobolev norms, such as the norms of H Gamma1 ( Omega Gamma and H Gamma1=2 ( Gamma) which seem to be difficult to compute in practice. However, due to recent results in multilevel ....

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Bramble, J., Lazarov, R.D. and Pasciak, J.E.: A least-squares approach based on a discrete minus one inner product for first order systems. Mathematics of Computation 66, 219, pp. 935-955, (1997).

....error estimates. 1. Introduction. Recent developments of ecient multilevel preconditioning techniques have revived the interest in least squares discretizations of various variational problems such as mixed formulations of second order boundary value problems or the Stokes problem, see e.g. [ADN, AKS, BLP1, BLP2, CLMM, CMM, St]. The central issues in this context are (i) the identi cation of suitable discrete least squares functionals, and (ii) preconditioning and fast solution of the resulting linear systems. Here suitable refers to the stability of the discretization as well as to the realization of optimal error ....

....is de ned in this case by hw; Lvi = a(v; w) Under the above assumptions one has kvkH1 kLvk H 0 1;0 for v 2 H 1;0 . Thus, 2.1.9) holds for H = H 1;0 , that is, m = 1, and the least squares functional should be equivalent to kLv fk 2 H 0 1;0 involving the familiar H 1 norm. Recall from [BLP1] that the corresponding scalar product ( H 1( can be expressed as (T ; L2( where T is the solution operator of the second order mixed Dirichlet Neumann problem for w w = f . Aside from advantages with respect to symmetry when k is a rst order operator, we have included this example ....

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J.H. Bramble, R.D. Lazarov, J.E. Pasciak, A least-squares approach based on a discrete minus one inner product for rst order systems, Math. Comput. 66, 1997, 935-955.

....error estimates. 1. Introduction. Recent developments of ecient multilevel preconditioning techniques have revived the interest in least squares discretizations of various variational problems such as mixed formulations of second order boundary value problems or the Stokes problem, see e.g. [ADN, AKS, BLP1, BLP2, CLMM, CMM, St]. The central issues in this context are (i) the identi cation of suitable discrete least squares functionals, and (ii) preconditioning and fast solution of the resulting linear systems. Here suitable refers to the stability of the discretization as well as to the realization of optimal error ....

....is de ned in this case by hw; Lvi = a(v; w) Under the above assumptions one has kvkH1 kLvk H 0 1;0 for v 2 H 1;0 . Thus, 2.1.9) holds for H = H 1;0 , that is, m = 1, and the least squares functional should be equivalent to kLv fk 2 H 0 1;0 involving the familiar H 1 norm. Recall from [BLP1] that the corresponding scalar product ( H 1( can be expressed as (T ; L2( where T is the solution operator of the second order mixed Dirichlet Neumann problem for w w = f . Aside from advantages with respect to symmetry when k is a rst order operator, we have included this example ....

[Article contains additional citation context not shown here]

J.H. Bramble, R.D. Lazarov, J.E. Pasciak, A least-squares approach based on a discrete minus one inner product for rst order systems, Math. Comput. 66, 1997, 935-955.

....or the rst order Stokes formulation also avoids squaring the order on the discrete level. We conclude this section with a few remarks on the formulation (7.2) of the original problem (3.1. 6) This is to relate the present approach to recent activities around least squares schemes, see e.g. [BP, BLP1, BLP2, BSch, CLMM, CMM]. Obviously (7.2) is just the (in nite) system of normal equations for the least squares problem min V2 2 (J ) kLV Gk 2 2 (J ) 7.7) 30 which by (6.2.5) possesses a unique solution. Moreover, as pointed out in [DKS] a natural least squares formulation of the operator equation (3.1.9) is ....

....This imposes only the natural minimal regularity requirements on the solution U of (7.8) and equivalently (3.1.6) and gives rise to optimal error estimates in H. The search for suitable least squares functionals is re ected by many investigations mainly in the nite element context, see e.g. [AKS, BLP1, BLP2, CLMM, CMM] and the literature cited in [DKS] While, in principle, the choice (7.8) is at least implicitly treated in many papers but is usually directly intertwined with discretizations and the (often severe) problem of numerically evaluating the dual norms k k H 0 i;0 . In fact, the examples in ....

J.H. Bramble, R.D. Lazarov, and J.E. Pasciak, A least-squares approach based on a discrete minus one inner product for rst order systems, Math. Comput. 66, 1997, 935-955.

....boundary conditions. Next, we turn our attention to a least squares functional in which the residual of the momentum equation is measured in the norm of the negative order Sobolev space H Gamma1 . For earlier work on H Gamma1 norm functionals, we refer the reader to papers by Bramble et al. [3] [4] and Cai, Manteuffel and McCormick [5] In the present paper, our analysis again combines previous results on analogous operators for the Stokes equations with the abstract framework outlined above. Although in both cases we deal with similar abstract formulations of the least squares ....

.... the dual of H 1 0( Omega Gamma8 Using the equivalence of the seminorm and norm on H 1 0( Omega Gamma7 we equip H Gamma1( Omega Gamma with the norm jf j Gamma1 = sup OE2H 1 0 (f; OE) jOEj 1 ; 8f 2 H Gamma1( Omega Gamma ; 51) for which the following representation result holds (cf. [3]) Lemma 6.1. For all f 2 H Gamma1( Omega Gamma , we have jf j Gamma1 = Sf; f) 52) and kSfk 1 Cjf j Gamma1 ; 53) where S : H Gamma1 ( Omega Gamma 7 H 1 0( Omega Gamma is the solution operator of the Dirichlet problem Gamma4u = f in Omega (54) u = 0 on Gamma ; 55) that ....

J. Bramble, R. Lazarov, and J. Pasciak, A least squares approach based on a discrete minus one inner product for first order systems, Technical Report 94-32, Mathematical Science Institute, Cornell University, 1994.

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J. H. Bramble, R. T. Lazarov, and J. E. Pasciak. A least-squares approach based on a discrete minus one inner product for first order systems. Technical report, Brookhaven National Laboratory, 1994.

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J. H. Bramble, R. Lazarov and J. Pasciak. A least-squares approach based on a discrete minus one inner product for first order systems. Math. Comp., 66:935-955, 1997.

....become a popular technique for deriving unconditionally stable approximations including high order schemes. The idea of least squares is quite old (see, e.g. the pioneering work of Neitaanmaki and J. Saranen [55] but only recently a new development in the method has been accomplished (see, e.g. [10, 11, 12, 17, 18, 56]) For a comprehensive review of new recent results in least squares method and their applications to a wide range of problems we refer to the paper by Bochev and Gunzburger, 9] Attractive feature of this approach is that it leads to symmetric and positive definite discrete systems and allows ....

....and studied by Lazarov, Tobiska, and Vassilevski in [48] 4.2. Least squares based on H inner product To overcome some of the deficiencies of the least squares method based on L 2 inner product a more balanced set of norms in (4. 1) has been proposed by Bramble, Lazarov, and Pasciak in [10]. In order to introduce the method we need to define a minus one inner product. First, we define the space H as the set of all functionals q for which the norm kqk H Gamma1 = sup OE2H kOEk H 1 ( Omega Gamma is finite. Here (q; OE) is the value of the functional q at OE. Below we ....

[Article contains additional citation context not shown here]

J. H. Bramble, R. D. Lazarov, and J. E. Pasciak, A least-squares approach based on a discrete minus one inner product for first order systems, Math. Comp. 66, 219 (1997), 935--955.

....1 s . If we assume that the variational solution u belongs to an intermediate space H 1 s (## # H 1 D(##1 s # (0, 1) and u is not in H 2(##3 then it is natural to ask: Does [H 2 #H 1 D , H 1 D(##2 1 s coincide with H 1 s (## #H 1 D This type of question arose in [4] and [5]. The paper will give a positive answer to this question for the special case when# is a polygonal domain in R 2 . The remaining part of the paper is organized as follows. In Section 2 general interpolation results and some notation are presented. The proof of the fact that [H 2 # H 1 D , ....

J. H. Bramble, R. Lazarov and J. Pasciak. A least-squares approach based on a discrete minus one inner product for first order systems. Math. Comp., 66:935-955,

No context found.

J. H. Bramble, R. Lazarov and J. Pasciak. A least-squares approach based on a discrete minus one inner product for first order systems. Math. Comp., 66:935-955, 1997.

....develop a nite element method which is unconditionally stable for problems with traction type of boundary conditions and for almost and incompressible elastic media. The use of such inner products (applied to second order problems) was proposed in an earlier paper by Bramble, Lazarov and Pasciak [6]. 1. Introduction There are many papers written on the subject of approximation schemes for Stokes equations and the equations of linear elasticity (see, 13] 15] 16] 17] 24] 36] and the included references) Mixed nite element methods involving a pair of approximation spaces are ....

J.H. Bramble, R.D. Lazarov and J.E. Pasciak, A least-squares approach based on a discrete minus one inner product for rst order systems, Math. Comp., 66 (1997), 935-955.

....element method which is unconditionally stable for problems with traction type of boundary conditions and for almost incompressible and incompressible elastic media. The use of such inner products (applied to second order problems) was proposed in an earlier paper by Bramble, Lazarov and Pasciak [8]. 1. Introduction There is a great deal of literature on the subject of approximation schemes for Stokes equations and the equations of linear elasticity (see, 16] 18] 19] 20] 31] 43] and the included references) Mixed finite element methods involving a pair of approximation spaces ....

....on part of the boundary Dirichlet data are specified, while on the remaining part a traction boundary condition is given. The approach of this paper is based on a discrete negative norm for one of the terms defining the least squares functional and is an extension of some of the work done in [8] and [9] The use of such norms gives rise to two important advantages. It results in approximation methods which are optimal both in order of approximation as well as required regularity. In addition, the corresponding algebraic systems can be easily preconditioned. The preconditioner for the ....

[Article contains additional citation context not shown here]

J.H. Bramble, R.D. Lazarov and J.E. Pasciak, A least-squares approach based on a discrete minus one inner product for first order systems, Math. Comp., 66 (1997), 935-955.

No context found.

J.H. Bramble, R.D. Lazarov, and J.E. Pasciak. A least-squares approach based on a discrete minus one inner product for first order systems. Math. Comp., 66:935--955, 1997.

No context found.

J.H. Bramble, R.D. Lazarov, and J.E. Pasciak. A least-squares approach based on a discrete minus one inner product for rst order systems. Math. Comp., 66:935955, 1997. 63

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