Least squares finite element simulation of transonic flows

Chen, T. F.; Fix, George J.

doi:10.1016/0168-9274(86)90042-5

Cited by 24 publications

(17 citation statements)

References 19 publications

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“…where ρ 0 and H 0 denote the stagnation density and enthalpy, respectively; see, e.g., [64] or [83]. A boundary condition for (1.14) is given by…”

Section: Convection-diffusion and Potential Flowmentioning

confidence: 99%

“…• replace stronger (and impractical) norms by weighted L 2 norms, see [3], [4], [11], [12], [17], [19], [64], [75], [86], and [115]; • use nonconforming discretizations; see [3].…”

Section: Discretization Levelmentioning

confidence: 99%

“…The decomposition step consists of transforming the problem into a first-order system. For many problems, decomposition can be accomplished through the introduction of physically meaningful variables such as vorticity, stresses, or fluxes, and has been often exploited in both least-squares and Galerkin methods; see, e.g., [11]- [19], [33]- [38], [42], [43]- [45], [47]- [57], [59], [61], [62], [64], [66], [67], [72], [73], [86], [87], [90]- [96], [99], [103], [104], [107], [108], [111]- [113], and [117].…”

Section: A Recipe For Practicalitymentioning

confidence: 99%

“…As a result, least-squares finite element methods in these settings are among the best understood, studied, and tested from both the theoretical and computational viewpoints. Our discussion will also include least-squares methods for convection-diffusion and other second-order elliptic problems (see [6], [24], [26], [33], [35], [38], [42]- [45], [48], [52], [62], [71], [72], [75], [86], [95], [106], [107], and [104]), linear elasticity (see [34], [36], and [37]), inviscid, compressible flows (see [61], [64], [67], [99], and [118]), and electromagnetics (see [46], [58], [60], [97], and [116]. )…”

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confidence: 99%

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Finite Element Methods of Least-Squares Type

Bochev¹,

Gunzburger²

1998

SIAM Rev.

285

237

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Abstract. We consider the application of least-squares variational principles to the numerical solution of partial differential equations. Our main focus is on the development of least-squares finite element methods for elliptic boundary value problems arising in fields such as fluid flows, linear elasticity, and convection-diffusion. For many of these problems, least-squares principles offer numerous theoretical and computational advantages in the algorithmic design and implementation of corresponding finite element methods, that are not present in standard Galerkin discretizations. Most notably, the use of least-squares principles leads to symmetric and positive definite algebraic problems and allows one to circumvent stability conditions such as the inf-sup condition arising in mixed methods for the Stokes and Navier-Stokes equations. As a result, application of least-squares principles has led to the development of robust and efficient finite element methods for a large class of problems of practical importance.

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“…where ρ 0 and H 0 denote the stagnation density and enthalpy, respectively; see, e.g., [64] or [83]. A boundary condition for (1.14) is given by…”

Section: Convection-diffusion and Potential Flowmentioning

confidence: 99%

“…• replace stronger (and impractical) norms by weighted L 2 norms, see [3], [4], [11], [12], [17], [19], [64], [75], [86], and [115]; • use nonconforming discretizations; see [3].…”

Section: Discretization Levelmentioning

confidence: 99%

Section: A Recipe For Practicalitymentioning

confidence: 99%

mentioning

confidence: 99%

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Finite Element Methods of Least-Squares Type

Bochev¹,

Gunzburger²

1998

SIAM Rev.

285

237

View full text Add to dashboard Cite

show abstract

“…Introduction. Least-squares finite element methods have always held out the attraction of yielding discrete linear systems that are symmetric and positive definite even for problems for which other methods, e.g., mixed finite element methods, fail to do so; see, e.g., [2]- [48], [50]- [56], [58], and [60]- [84]. In many settings such as the primitive variable formulation of the Stokes equations, these methods suffer from two serious problems.…”

mentioning

confidence: 99%

Issues Related to Least-Squares Finite Element Methods for the Stokes Equations

Deang¹,

Gunzburger²

1998

SIAM J. Sci. Comput.

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Abstract. Least-squares finite element methods have become increasingly popular for the approximate solution of first-order systems of partial differential equations. Here, after a brief review of some existing theories, a number of issues connected with the use of such methods for the velocityvorticity-pressure formulation of the Stokes equations in two dimensions in realistic settings are studied through a series of computational experiments. Finite element spaces that are not covered by existing theories are considered; included in these are piecewise linear approximations for the velocity. Mixed boundary conditions, which are also not covered by existing theories, are also considered, as is enhancing mass conservation. Next, problems in nonconvex polygonal regions and the resulting nonsmooth solutions are considered with a view toward seeing how accuracy can be improved. A conclusion that can be drawn from this series of computational experiments is that the use of appropriate mesh-dependent weights in the least-squares functional almost always improves the accuracy of the approximations. Concluding remarks concerning three-dimensional problems, the nonlinear Navier-Stokes equations, and the conditioning of the discrete systems are provided. Key words. least squares, finite element methods, Stokes equations AMS subject classification. 65N30PII. S10648275952945261. Introduction. Least-squares finite element methods have always held out the attraction of yielding discrete linear systems that are symmetric and positive definite even for problems for which other methods, e.g., mixed finite element methods, fail to do so; see, e.g. , [2]-[48], [50]-[56], [58], and [60]-[84]. In many settings such as the primitive variable formulation of the Stokes equations, these methods suffer from two serious problems. The first is that conforming discretizations require the use of continuously differentiable finite element functions and the second is that the condition number of the discrete equations is often proportional to h −4 , where h denotes some measure of the grid size. However, least-squares finite element methods have recently been receiving increasing attention in both the engineering and mathematics communities; see, e.g., [3] [84]. The focus of this attention has been on the application of least-squares finite element methodologies to first-order systems of partial differential equations for which one can, in principle, use merely continuous finite element functions and for which one may often prove that the condition numbers of the discrete systems are proportional to h −2 . The mathematical references cited above consider least-squares finite element methods in idealized situations, i.e., for problems having simple boundary conditions and smooth solutions and in the asymptotic limit of the grid size measure h → 0. Unfortunately, these are usually far from true in most applications of the methods to practical problems; indeed, there are many such settings for which mathematical

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A three-dimensional least-squares finite element technique for deformation analysis

Siu

Lee

1997

Int. J. Numer. Meth. Engng.

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The development of a three-dimensional least-squares ÿnite element technique suitable for deformation analysis was presented. By adopting a spatial viewpoint, a consistent rate formulation that treats deformation as a process was established. The technique utilized the least-squares variational principle that minimizes the squares of errors encountered in any attempt to meet the ÿeld equations exactly. Both velocity and Cauchy stress rate ÿelds were discretized by the same linear interpolation function. The discretization always yields a sparse, symmetric, and positive-deÿnite coe cient matrix. A conjugate gradient iterative solver with incomplete-Choleski preconditioner was used to solve the resulting linear system of equations. Issues such as ÿnite element formulation, mesh design, code e ciency, and time integration were addressed. A set of linear elastic problems was used for patch-test; both homogeneous and non-homogeneous deformations were considered. Additionally, two ÿnite elastic deformation problems were analysed to gauge the overall performance of the technique. The results demonstrated the computational feasibility of a three-dimensional least-squares ÿnite element technique for deformation analysis.

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Least squares finite element simulation of transonic flows

Cited by 24 publications

References 19 publications

Finite Element Methods of Least-Squares Type

Finite Element Methods of Least-Squares Type

Issues Related to Least-Squares Finite Element Methods for the Stokes Equations

A three-dimensional least-squares finite element technique for deformation analysis

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