Finite Element Approximation for Grad-Div Type Systems in the Plane

Chang, Ching L.

doi:10.1137/0729027

Cited by 42 publications

(36 citation statements)

References 12 publications

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“…In fact, (24) and (25) are equivalent in that whenver φ h is a solution of (25), then φ h and u h = ∇φ h are a solution pair for (24) and conversely. In this way we see that for (24), i.e., using LSFEMs.…”

Section: Mfems For the Poisson Problemmentioning

confidence: 99%

“…the Dirichlet principle, the required inf-sup condition is completely benign in the sense that it can be avoided by eliminating the flux approximation u h from (24) and solving (25) instead. The required inf-sup condition is implicitly satisfied by the pair of spaces S 0 m and S 1 m = ∇(S 0 m ).…”

Section: Mfems For the Poisson Problemmentioning

confidence: 99%

“…Formally, a least-squares principle for (1) can be easily extended to handle (47) by modifying (4) and (3) to min u∈S J G (u; f, g), where J G (u; f, g) = Lu + G(u) − f 2 HΩ + Ru − g 2 HΓ (48) and then define a LSFEM by restricting (48) to a family S h ⊂ S. While the extension of LSFEMs to (47) is trivial, its analysis is not and remains one of the open problems in LSFEMs. Compared with the well-developed mathematical theory for linear elliptic problems [2,13,18,21,23,24,26,32,38], analyses of LSFEMs for nonlinear problems are mostly confined to the Navier-Stokes equations [7][8][9]. It can be shown that the Euler-Lagrange equation associated with the leastsquares principle (48) for the Navier-Stokes equations has the abstract form…”

Section: Lsfems For Nonlinear Problems Consider the Nonlinear Versiomentioning

confidence: 99%

“…If one insists on solving (24), then one needs to explicitly produce a basis for S 1 m ; this is easily accomplished. From either (24) or (25) …”

Section: Mfems For the Poisson Problemmentioning

confidence: 99%

“…In addition, excepting in one special case, such methods produce suboptimally accurate (with respect to L 2 (Ω) norms) flux approximations. 14 Already in [38], optimal L 2 error estimates for LSFEMs were established for the scalar variable; however, there and in all subsequent analyses, optimal L 2 error estimates for the flux could not be obtained 15 without the addition of a "redundant" curl equation; see, e.g., [23,24,26,39,44]. Moreover, computational studies in [32] strongly suggested that optimal L 2 convergence for flux approximations may in fact be nearly impossible to obtain if one uses pairs of standard, nodalbased, continuous finite element spaces.…”

Section: Compatible Lsfemsmentioning

confidence: 99%

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Least Squares Finite Element Methods

Bochev¹,

Gunzburger²

2015

Encyclopedia of Applied and Computational Mathematics

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Abstract. Least-squares finite element methods are an attractive class of methods for the numerical solution of partial differential equations. They are motivated by the desire to recover, in general settings, the advantageous features of Rayleigh-Ritz methods such as the avoidance of discrete compatibility conditions and the production of symmetric and positive definite discrete systems. The methods are based on the minimization of convex functionals that are constructed from equation residuals. This paper focuses on theoretical and practical aspects of least-square finite element methods and includes discussions of what issues enter into their construction, analysis, and performance. It also includes a discussion of some open problems.

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Section: Mfems For the Poisson Problemmentioning

confidence: 99%

Section: Mfems For the Poisson Problemmentioning

confidence: 99%

Section: Lsfems For Nonlinear Problems Consider the Nonlinear Versiomentioning

confidence: 99%

“…If one insists on solving (24), then one needs to explicitly produce a basis for S 1 m ; this is easily accomplished. From either (24) or (25) …”

Section: Mfems For the Poisson Problemmentioning

confidence: 99%

Section: Compatible Lsfemsmentioning

confidence: 99%

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Least Squares Finite Element Methods

Bochev¹,

Gunzburger²

2015

Encyclopedia of Applied and Computational Mathematics

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A locally conservative least‐squares method for Darcy flows

Bochev

Gunzburger

2006

Commun. Numer. Meth. Engng.

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SUMMARYLeast-squares finite-element methods for Darcy flow offer several advantages relative to the mixedGalerkin method: the avoidance of stability conditions between finite-element spaces, the efficiency of solving symmetric and positive definite systems, and the convenience of using standard, continuous nodal elements for all variables. However, conventional C 0 implementations conserve mass only approximately and for this reason they have found limited acceptance in applications where locally conservative velocity fields are of primary interest. In this paper, we show that a properly formulated compatible least-squares method offers the same level of local conservation as a mixed method. The price paid for gaining favourable conservation properties is that one has to give up what is arguably the least important advantage attributed to least-squares finite-element methods: one can no longer use continuous nodal elements for all variables. As an added benefit, compatible least-squares methods inherit the best computational properties of both Galerkin and mixed-Galerkin methods and, in some cases, yield identical results, while offering the advantages of not having to deal with stability conditions and yielding positive definite discrete problems. Numerical results that illustrate our findings are provided.

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On the subdomain‐Galerkin/least squares method for 2‐ and 3‐D mixed elliptic problems with reaction terms

Yang

2002

Numerical Methods Partial

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In this article we apply the subdomain-Galerkin/least squares method, which is first proposed by Chang and Gunzburger for first-order elliptic systems without reaction terms in the plane, to solve second-order non-selfadjoint elliptic problems in two-and three-dimensional bounded domains with triangular or tetrahedral regular triangulations. This method can be viewed as a combination of a direct cell vertex finite volume discretization step and an algebraic least-squares minimization step in which the pressure is approximated by piecewise linear elements and the flux by the lowest order Raviart-Thomas space. This combined approach has the advantages of both finite volume and least-squares methods. Among other things, the combined method is not subject to the Ladyzhenskaya-Babuska-Brezzi condition, and the resulting linear system is symmetric and positive definite. An optimal error estimate in the H 1 (⍀) ϫ H(div; ⍀) norm is derived. An equivalent residual-type a posteriori error estimator is also given.

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Finite Element Approximation for Grad-Div Type Systems in the Plane

Cited by 42 publications

References 12 publications

Least Squares Finite Element Methods

Least Squares Finite Element Methods

A locally conservative least‐squares method for Darcy flows

On the subdomain‐Galerkin/least squares method for 2‐ and 3‐D mixed elliptic problems with reaction terms

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