## A least-squares approach based on a discrete minus one inner product for first order systems (1997)

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Venue: | MATH. COMP |

Citations: | 64 - 12 self |

### BibTeX

@ARTICLE{Bramble97aleast-squares,

author = {James H. Bramble and Raytcho D. Lazarov and Joseph E. Pasciak},

title = {A least-squares approach based on a discrete minus one inner product for first order systems},

journal = {MATH. COMP},

year = {1997},

volume = {66},

number = {219},

pages = {935--955}

}

### Years of Citing Articles

### OpenURL

### Abstract

The purpose of this paper is to develop and analyze a least-squares approximation to a first order system. The first order system represents a reformulation of a second order elliptic boundary value problem which may be indefinite and/or nonsymmetric. The approach taken here is novel in that the least-squares functional employed involves a discrete inner product which is related to the inner product in H −1 (Ω) (the Sobolev space of order minus one on Ω). The use of this inner product results in a method of approximation which is optimal with respect to the required regularity as well as the order of approximation even when applied to problems with low regularity solutions. In addition, the discrete system of equations which needs to be solved in order to compute the resulting approximation is easily preconditioned, thus providing an efficient method for solving the algebraic equations. The preconditioner for this discrete system only requires the construction of preconditioners for standard second order problems, a task which is well understood.

### Citations

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Citation Context ...bspaces. It is well known that these properties hold for typical finite element spaces consisting of piecewise polynomials with respect to quasi-uniform triangulations of the domain Ω (cf., [1], [7], =-=[11]-=-, [13], [33]). Let r be an integer greater than or equal to one. (H.1) The subspace Vh has the following approximation property: For any η ∈ H r (Ω) d ∩ H 0 div (Ω), � � r (3.7) inf �η − δ� + h �∇ · (... |

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Citation Context ...), for i =1,... ,d. To describe and analyze the least-squares method, we shall use Sobolev spaces. For non-negative integers s, letH s (Ω) denote the Sobolev space of order s defined on Ω (see, e.g., =-=[19]-=-, [26], [29]). The norm in H s (Ω) will be denoted by �·� s . For s =0,H s (Ω) coincides with L 2 (Ω). In this case, the norm and inner product will be denoted by �·� and (·, ·) respectively. The spac... |

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Citation Context ... i =1,... ,d. To describe and analyze the least-squares method, we shall use Sobolev spaces. For non-negative integers s, letH s (Ω) denote the Sobolev space of order s defined on Ω (see, e.g., [19], =-=[26]-=-, [29]). The norm in H s (Ω) will be denoted by �·� s . For s =0,H s (Ω) coincides with L 2 (Ω). In this case, the norm and inner product will be denoted by �·� and (·, ·) respectively. The space W is... |

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Citation Context ...venting the inf-sup condition. Examples of application of the least-squares to potential flows, convection-diffusion problems, Stokes and Navier-Stokes equations can be found in [8], [9], [16], [17], =-=[20]-=-, [21], [23], [28]. In general, the corresponding problem is written as a system of partial differential equations of first order with possibly additional compatibility conditions. For example, −∇ · u... |

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Citation Context ...posed in a weak sense and approximated by finite element methods. In many cases (for example, Stokes equations), this procedure leads to a saddle point problem. Due largely to Babuˇska [3] and Brezzi =-=[10]-=-, it is now well understood that the finite element spaces approximating different physical quantities (pressure and velocity, or temperature and flux, or displacement and stresses, etc.) cannot be ch... |

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Citation Context ...is well known that these properties hold for typical finite element spaces consisting of piecewise polynomials with respect to quasi-uniform triangulations of the domain Ω (cf., [1], [7], [11], [13], =-=[33]-=-). Let r be an integer greater than or equal to one. (H.1) The subspace Vh has the following approximation property: For any η ∈ H r (Ω) d ∩ H 0 div (Ω), � � r (3.7) inf �η − δ� + h �∇ · (η − δ)� ≤ Ch... |

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Citation Context ...find u ∈ W satisfying (2.3) A(u, θ) =(f,θ) for all θ ∈ W. We assume that the solution of (2.3) is unique. This means that if v ∈ W and satisfies A(v, θ) = 0 for all θ ∈ W , then v = 0. As usual (cf., =-=[18]-=-, [26]), the uniqueness assumption implies the existence of solutions as well. The particular space H−1 (Ω) chosen above is related to the boundary conditions used in our boundary value problem (2.1).... |

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Citation Context ...imation subspaces. It is well known that these properties hold for typical finite element spaces consisting of piecewise polynomials with respect to quasi-uniform triangulations of the domain Ω (cf., =-=[1]-=-, [7], [11], [13], [33]). Let r be an integer greater than or equal to one. (H.1) The subspace Vh has the following approximation property: For any η ∈ H r (Ω) d ∩ H 0 div (Ω), � � r (3.7) inf �η − δ�... |

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Citation Context ...gives rise to full elliptic regularity, then it is known that the W-cycle multigrid algorithm with sufficiently many smoothings on each level gives rise to an operator Bh which satisfies (3.24) (cf., =-=[4]-=-). Another example of an operator Bh which satisfies (3.24) is the variable V-cycle introduced in [5]. It seems that in this case also sufficiently many smoothings are required on the finest level. Th... |

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Citation Context ... and Fix in [17] and by Carey and Shen in [14] that were a basis for the theoretical analysis of Pehlivanov, Carey and Lazarov in [30] for selfadjoint and of Cai, Lazarov, Manteuffel and McCormick in =-=[12]-=- for non-selfadjoint second order elliptic equations. The main result in [12], [30] is that the least-squares functional generates a bilinear form that is continuous and coercive in a properly defined... |

48 |
Some estimates for a weighted L projection
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Citation Context ...ubspaces. It is well known that these properties hold for typical finite element spaces consisting of piecewise polynomials with respect to quasi-uniform triangulations of the domain\Omega (cf., [1], =-=[7], [11-=-], [13], [33]). Let r be an integer greater than or equal to one. (H.1) The subspace V h has the following approximation property: For any j 2 H r (\Omega\Gamma d " H 0 div (3.7) inf ffi2V h \Phi... |

46 | Analysis of least squares finite element methods for the Stokes equations
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Citation Context ...ations and circumventing the inf-sup condition. Examples of application of the least-squares to potential flows, convection-diffusion problems, Stokes and Navier-Stokes equations can be found in [8], =-=[9]-=-, [16], [17], [20], [21], [23], [28]. In general, the corresponding problem is written as a system of partial differential equations of first order with possibly additional compatibility conditions. F... |

37 |
Some estimates for weighted L 2 projections
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Citation Context ...mation subspaces. It is well known that these properties hold for typical nite element spaces consisting of piecewise polynomials with respect to quasi-uniform triangulations of the domain (cf., [1], =-=[7]-=-, [11], [13], [33]). Let r be an integer greater than or equal to one. (H.1) The subspace Vh has the following approximation property: For any 2 Hr ( ) d \ H 0 div ( ), (3.7) inf 2Vh k ; k + h kr ( ; ... |

33 |
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Citation Context ...fficiently many smoothings on each level gives rise to an operator Bh which satisfies (3.24) (cf., [4]). Another example of an operator Bh which satisfies (3.24) is the variable V-cycle introduced in =-=[5]-=-. It seems that in this case also sufficiently many smoothings are required on the finest level. Though results of numerical calculations indicate that (3.24) holds also for the usual V-cycle, as far ... |

31 |
Least-squares mixed finite elements for second-order elliptic problems
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Citation Context ...sting computational experiments in this setting have been done by Chen and Fix in [17] and by Carey and Shen in [14] that were a basis for the theoretical analysis of Pehlivanov, Carey and Lazarov in =-=[30]-=- for selfadjoint and of Cai, Lazarov, Manteuffel and McCormick in [12] for non-selfadjoint second order elliptic equations. The main result in [12], [30] is that the least-squares functional generates... |

30 |
Least-squares methods for elliptic systems
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Citation Context ...in [16], [17]) for fluid flow computations. There are two main approaches for studying least-squares methods for systems of first order. The first approach introduced by Aziz, Kellogg and Stephens in =-=[2]-=- uses the general theory of elliptic boundary value problems of Agmon-Douglis-Nirenberg (ADN) and reduces the system to a minimization of a least-squares functional that consists of a weighted sum of ... |

20 |
Least-squares finite elements for the Stokes problem, Comput. Methods
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Citation Context ...inf-sup condition. Examples of application of the least-squares to potential flows, convection-diffusion problems, Stokes and Navier-Stokes equations can be found in [8], [9], [16], [17], [20], [21], =-=[23]-=-, [28]. In general, the corresponding problem is written as a system of partial differential equations of first order with possibly additional compatibility conditions. For example, −∇ · u = f, u = ∇ ... |

17 |
Finite element approximation for grad-div type of systems in the plane
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Citation Context ... boundary conditions. The weights occurring in the least-squares functional are determined by the indices that enter into the definition of the ADN boundary value problem. See also the paper of Chang =-=[15]-=-. This approach generalizes both the least-squares method of Jespersen [22], which is for the Poisson equation written as a grad − div system, and the method of Wendland [34], which is for elliptic sy... |

16 |
Accuracy of least-squares methods for the Navier{Stokes equations, Comput. Fluids 22
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Citation Context ...ormulations and circumventing the inf-sup condition. Examples of application of the least-squares to potential flows, convection-diffusion problems, Stokes and Navier-Stokes equations can be found in =-=[8]-=-, [9], [16], [17], [20], [21], [23], [28]. In general, the corresponding problem is written as a system of partial differential equations of first order with possibly additional compatibility conditio... |

15 |
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Citation Context ...To describe and analyze the least-squares method, we shall use Sobolev spaces. For non-negative integers s, let H s(\Omega\Gamma denote the Sobolev space of order s defined on\Omega (see, e.g., [19], =-=[26]-=-, [29]). The norm in H s(\Omega\Gamma will be denoted by k\Deltak s . For s = 0, H s (\Omega\Gamma coincides with L 2(\Omega\Gamma3 In this case, the norm and inner product will be denoted by k\Deltak... |

14 |
Least-squares mixed finite element methods for second order elliptic problems
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Citation Context ...Ω) and, therefore, any finite element approximation of Hdiv(Ω) can be used since the approximating space need not to satisfy the inf-sup condition. A recent paper by Pehlivanov, Carey and Vassilevski =-=[32]-=- considers a least-squares method for non-selfadjoint problems. One problem with the above mentioned least-squares methods is that the error estimates require relatively smooth solutions. The known es... |

12 |
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Citation Context ...that Wh is such that Qh, theL2 (Ω) orthogonal projection operator onto Wh, is a bounded operator with respect to the norm in W , i.e., (3.9) �Qhu�1 ≤ C �u�1 for all u ∈ W. Remark 3.1. It follows from =-=[6]-=- that if (H.1) and (H.2) hold for r =˜r, then they hold for r =1,2,... ,˜r. The property (H.3) is studied in [7]. We note some properties implied by the above assumptions. It follows from (3.8) that (... |

12 |
Optimal least-squares finite element methods for elliptic problems
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Citation Context ...ditional compatibility conditions. For example, −∇ · u = f, u = ∇ p provides a first order system for the Poisson equation −∆p = f which can be augmented by the compatibility equation curl u = 0 (see =-=[24]-=-, [28]). Alternatively, the system curl u = 0 and ∇· u+f = 0 has been used (cf. Chen and Fix in [16], [17]) for fluid flow computations. There are two main approaches for studying least-squares method... |

12 |
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Citation Context ...se the point Gauss Seidel smoothing iteration on all spaces except the first (with mesh size 1/2) on which we solve directly. The resulting multigrid iterative procedure is described in, for example, =-=[27]-=-. The multigrid preconditioner results from applying one step of the iterative procedure with zero starting iterate, [5]. The V-cycle uses one pre and post Gauss Seidel iteration sweep where the direc... |

12 |
Elliptic Systems in the Plane
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Citation Context ...e also the paper of Chang [15]. This approach generalizes both the least-squares method of Jespersen [22], which is for the Poisson equation written as a grad − div system, and the method of Wendland =-=[34]-=-, which is for elliptic systems of Cauchy-Riemann type. Recently, Bochev and Gunzburger [8], [9], have extended the ADN approach to velocity-vorticity-pressure formulation of Stokes and Navier-Stokes ... |

12 |
A mixed nite element method for 2nd order elliptic problems
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(Show Context)
Citation Context ... It is well known that these properties hold for typical nite element spaces consisting of piecewise polynomials with respect to quasi-uniform triangulations of the domain (cf., [1], [7], [11], [13], =-=[33]-=-). Let r be an integer greater than or equal to one. (H.1) The subspace Vh has the following approximation property: For any 2 Hr ( ) d \ H 0 div ( ), (3.7) inf 2Vh k ; k + h kr ( ; )k Ch r k k r : (H... |

10 |
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Citation Context ...s. It is well known that these properties hold for typical finite element spaces consisting of piecewise polynomials with respect to quasi-uniform triangulations of the domain Ω (cf., [1], [7], [11], =-=[13]-=-, [33]). Let r be an integer greater than or equal to one. (H.1) The subspace Vh has the following approximation property: For any η ∈ H r (Ω) d ∩ H 0 div (Ω), � � r (3.7) inf �η − δ� + h �∇ · (η − δ)... |

10 |
A least-square decomposition method for solving elliptic systems
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(Show Context)
Citation Context ... are determined by the indices that enter into the definition of the ADN boundary value problem. See also the paper of Chang [15]. This approach generalizes both the least-squares method of Jespersen =-=[22]-=-, which is for the Poisson equation written as a grad − div system, and the method of Wendland [34], which is for elliptic systems of Cauchy-Riemann type. Recently, Bochev and Gunzburger [8], [9], hav... |

10 |
A new nite element formulation for computational uid dynamics. V. Circumventing the Babuska{Brezzi condition: a stable Petrov{ Galerkin formulation of the Stokes problem accomodating equal{order interpolations
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(Show Context)
Citation Context ...cumventing the inf-sup condition. Examples of application of the least-squares to potential ows, convection-di usion problems, Stokes and Navier-Stokes equations can be found in [8], [9], [16], [17], =-=[20]-=-, [21], [23], [28]. In general, the corresponding problem is written as a system of partial di erential equations of rst order with possibly additional compatibility conditions. For example, ;r u = f,... |

9 |
Some estimates for weighted L
- Bramble, Xu
(Show Context)
Citation Context ...on subspaces. It is well known that these properties hold for typical finite element spaces consisting of piecewise polynomials with respect to quasi-uniform triangulations of the domain Ω (cf., [1], =-=[7]-=-, [11], [13], [33]). Let r be an integer greater than or equal to one. (H.1) The subspace Vh has the following approximation property: For any η ∈ H r (Ω) d ∩ H 0 div (Ω), � � r (3.7) inf �η − δ� + h ... |

9 |
On nite element approximation of the gradient for the solution to Poisson equation
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- 1981
(Show Context)
Citation Context ...p condition. Examples of application of the least-squares to potential flows, convection-diffusion problems, Stokes and Navier-Stokes equations can be found in [8], [9], [16], [17], [20], [21], [23], =-=[28]-=-. In general, the corresponding problem is written as a system of partial differential equations of first order with possibly additional compatibility conditions. For example, −∇ · u = f, u = ∇ p prov... |

8 |
On least-squares approximations to compressible flow problems
- Chen
- 1986
(Show Context)
Citation Context ...s and circumventing the inf-sup condition. Examples of application of the least-squares to potential flows, convection-diffusion problems, Stokes and Navier-Stokes equations can be found in [8], [9], =-=[16]-=-, [17], [20], [21], [23], [28]. In general, the corresponding problem is written as a system of partial differential equations of first order with possibly additional compatibility conditions. For exa... |

7 |
Least Squares Finite Element Simulation of Transonic Flows
- Chen, Fix
- 1986
(Show Context)
Citation Context ...circumventing the inf-sup condition. Examples of application of the least-squares to potential flows, convection-diffusion problems, Stokes and Navier-Stokes equations can be found in [8], [9], [16], =-=[17]-=-, [20], [21], [23], [28]. In general, the corresponding problem is written as a system of partial differential equations of first order with possibly additional compatibility conditions. For example, ... |

6 |
An optimal order process for solving nite element equations
- Bank, Dupont
- 1981
(Show Context)
Citation Context ... T gives rise to full elliptic regularity, then it is known that the W-cycle multigrid algorithm with su ciently many smoothings on each level gives rise to an operator Bh which satis es (3.24) (cf., =-=[4]-=-). Improved L2 ( ) estimates depend upon elliptic regularity. We consider the adjoint boundary value problem in weak form: Given g 2 L2 ( ) nd v 2 W such that (3.25) A( � v) =( � g) for all 2 W:LEAST... |

5 |
Convergence studies of least-squares finite elements for first order systems
- Carey, Shen
- 1989
(Show Context)
Citation Context ...ss and the coercivity of the corresponding bilinear form in an appropriate space. Interesting computational experiments in this setting have been done by Chen and Fix in [17] and by Carey and Shen in =-=[14]-=- that were a basis for the theoretical analysis of Pehlivanov, Carey and Lazarov in [30] for selfadjoint and of Cai, Lazarov, Manteuffel and McCormick in [12] for non-selfadjoint second order elliptic... |

4 |
On the Schwarz algorithm in the theory of differential equations of mathematical physics
- Babuska
- 1958
(Show Context)
Citation Context ...problems can be posed in a weak sense and approximated by finite element methods. In many cases (for example, Stokes equations), this procedure leads to a saddle point problem. Due largely to Babuska =-=[3]-=- and Brezzi [10], it is now well understood that the finite element spaces approximating different physical quantities (pressure and velocity, or temperature and flux, or displacement and stresses, et... |

4 |
Least squares mixed nite elements for second order elliptic problems
- Pehlivanov, Carey, et al.
(Show Context)
Citation Context ...resting computational experiments in this setting have beendoneby Chen and Fix in [17] and by Carey and Shen in [14] that were a basis for the theoretical analysis of Pehlivanov, Carey and Lazarov in =-=[30]-=- for selfadjoint and of Cai, Lazarov, Manteu el and McCormick in [12] for non-selfadjoint second order elliptic equations. The main result in [12], [30] is that the least-squares functional generates ... |

3 |
Convergence analysis of least-squares mixed finite elements
- Pehlivanov, Carey, et al.
- 1993
(Show Context)
Citation Context ...r some positive numbers C0, C1, (2.8) C0(�δ� 2 + �v�2 Hdiv 1 ) ≤�∇·δ+Xv�2 + ≤ C1(�δ� 2 + �v�2 Hdiv 1 ) � � �A −1/2 � � (δ+A∇v) for all v ∈ W and δ ∈ H0 div (Ω). The one dimensional case was proved in =-=[31]-=- and the case of higher dimensions was proved in [12]. Numerical approximations are defined by introducing spaces of approximating functions Vh ⊆ H0 div (Ω) and Wh ⊆ W . The discrete approximations ar... |

3 |
Analysis of least-squares nite element methods for the Stokes equations
- Bochev, Gunzburger
(Show Context)
Citation Context ...mulations and circumventing the inf-sup condition. Examples of application of the least-squares to potential ows, convection-di usion problems, Stokes and Navier-Stokes equations can be found in [8], =-=[9]-=-, [16], [17], [20], [21], [23], [28]. In general, the corresponding problem is written as a system of partial di erential equations of rst order with possibly additional compatibility conditions. For ... |

2 |
On the Schwarz algorithm in the theory of differential equations of mathematical physics
- Babuˇska
- 1958
(Show Context)
Citation Context ...roblems can be posed in a weak sense and approximated by finite element methods. In many cases (for example, Stokes equations), this procedure leads to a saddle point problem. Due largely to Babuˇska =-=[3]-=- and Brezzi [10], it is now well understood that the finite element spaces approximating different physical quantities (pressure and velocity, or temperature and flux, or displacement and stresses, et... |

2 |
On the Schwarz algorithm in the theory of dierential equations of mathematical physics
- Babuska
- 1958
(Show Context)
Citation Context ...e problems can be posed in a weak sense and approximated by nite element methods. In many cases (for example, Stokes equations), this procedure leads to a saddle point problem. Due largely to Babuska =-=[3]-=- and Brezzi [10], it is now well understood that the nite element spaces approximating di erent physical quantities (pressure and velocity, or temperature and ux, or displacement and stresses, etc.) c... |

2 |
Least-squares nite elements for the Stokes problem
- Jiang, Chang
- 1990
(Show Context)
Citation Context ...he inf-sup condition. Examples of application of the least-squares to potential ows, convection-di usion problems, Stokes and Navier-Stokes equations can be found in [8], [9], [16], [17], [20], [21], =-=[23]-=-, [28]. In general, the corresponding problem is written as a system of partial di erential equations of rst order with possibly additional compatibility conditions. For example, ;r u = f, u = r p pro... |

2 |
Optimal least-squares nite element method for elliptic problems
- Jiang, Povinelli
- 1993
(Show Context)
Citation Context ...y additional compatibility conditions. For example, ;r u = f, u = r p provides a rst order system for the Poisson equation ; p = f which can be augmented by the compatibility equation curl u = 0 (see =-=[24]-=-, [28]). Alternatively, the system curl u = 0 and r u + f = 0 has been used (cf. Chen and Fix in [16], [17]) for uid ow computations. There are two main approaches for studying least-squares methods f... |

1 |
Convergence studies of least-squares nite elements for rst order systems
- Carey, Shen
- 1989
(Show Context)
Citation Context ...ness and the coercivity of the corresponding bilinear form in an appropriate space. Interesting computational experiments in this setting have beendoneby Chen and Fix in [17] and by Carey and Shen in =-=[14]-=- that were a basis for the theoretical analysis of Pehlivanov, Carey and Lazarov in [30] for selfadjoint and of Cai, Lazarov, Manteu el and McCormick in [12] for non-selfadjoint second order elliptic ... |

1 |
On the least-squares approximations to compressible ow problems
- Chen
- 1986
(Show Context)
Citation Context ...ions and circumventing the inf-sup condition. Examples of application of the least-squares to potential ows, convection-di usion problems, Stokes and Navier-Stokes equations can be found in [8], [9], =-=[16]-=-, [17], [20], [21], [23], [28]. In general, the corresponding problem is written as a system of partial di erential equations of rst order with possibly additional compatibility conditions. For exampl... |

1 |
Least-squares nite element simulation of transonic ows
- Chen, Fix
- 1986
(Show Context)
Citation Context ...nd circumventing the inf-sup condition. Examples of application of the least-squares to potential ows, convection-di usion problems, Stokes and Navier-Stokes equations can be found in [8], [9], [16], =-=[17]-=-, [20], [21], [23], [28]. In general, the corresponding problem is written as a system of partial di erential equations of rst order with possibly additional compatibility conditions. For example, ;r ... |

1 |
Convergence analysis of leastsquares mixed nite elements, Computing 51
- Pehlivanov, Carey, et al.
- 1993
(Show Context)
Citation Context ... positive numbers C0, (2.8) C0(k k 2 + kvk2 Hdiv 1 ) kr + X vk2 + A ;1=2 ( + Arv) 2 C1(k k 2 + kvk2 Hdiv 1 )6 BRAMBLE, ET. AL. for all v 2 W and 2 H 0 div ( ). The one dimensional case was proved in =-=[31]-=- and the case of higher dimensions was proved in [12]. Numerical approximations are de ned by introducing spaces of approximating functions Vh H 0 div ( ) and Wh W . The discrete approximations are de... |

1 |
Carey and P.S.Vassilevski, Least-squares mixed nite element methods for non-selfadjoint elliptic problems: I. Error estimates, UniversityofTexas at Austin CNA271
- Pehlivanov, F
- 1994
(Show Context)
Citation Context ... ( ) and, therefore, any nite element approximation of Hdiv( ) can be used since the approximating space need not to satisfy the inf-sup condition. A recent paper by Pehlivanov, Carey and Vassilevski =-=[32]-=- considers a least-squares method for non-selfadjoint problems. One problem with the above mentioned least-squares methods is that the error estimates require relatively smooth solutions. The known es... |