Least-squares methods for the velocity-pressure-stress formulation of the Stokes equations

Bochev, Pavel B.; Gunzburger, Max

doi:10.1016/0045-7825(95)00826-m

Cited by 78 publications

(63 citation statements)

References 25 publications

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“…The analysis and implementation of such methods have drawn most of the attention of researchers interested in modern least-squares finite element methods and there exists an abundant mathematical and engineering literature devoted to this subject; see, e.g., [3], [9], [11]- [19], [25], [34], [36], [47], [54], [57], [50], [51], [53], [59], [68], [87], [90]- [94], [96], [99], [105], [112], and [117] among others. 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.…”

mentioning

confidence: 99%

Finite Element Methods of Least-Squares Type

Bochev¹,

Gunzburger²

1998

SIAM Rev.

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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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mentioning

confidence: 99%

Finite Element Methods of Least-Squares Type

Bochev¹,

Gunzburger²

1998

SIAM Rev.

Self Cite

285

237

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“…Similar weighted functionals were studied in [6] and [12] for the incompressible Stokes problem. However, for (4.21) one can only show that i.e., (4.21) is only quasi norm-equivalent.…”

Section: Theorem 42 Let the Assumption (24) Hold And Assume Furthementioning

confidence: 99%

“…However, for (4.21) one can only show that i.e., (4.21) is only quasi norm-equivalent. Because (1.1) is an incompletely elliptic system, analysis of [6] and [12] cannot be extended to the present case. Nevertheless, simplicity of the weighted functional makes it very attractive computationally.…”

Section: Theorem 42 Let the Assumption (24) Hold And Assume Furthementioning

confidence: 99%

Analysis and computation of least-squares methods for a compressible Stokes problem

Bochev

Kim

Shin

2006

Numer. Methods Partial Differential Eq.

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We develop and analyze a negative norm least-squares method for the compressible Stokes equations with an inflow boundary condition. Least-squares principles are derived for a first-order form of the equations obtained by using ω = ∇ × u and φ = ∇ · u as new dependent variables. The resulting problem is incompletely elliptic, i.e., it combines features of elliptic and hyperbolic equations. As a result, wellposedness of least-squares functionals cannot be established using the ADN elliptic theory and so we use direct approaches to prove their norm-equivalence. The article concludes with numerical examples that illustrate the theoretical convergence rates.

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“…[2]. Using quadratic L 2 -norms, a related three-field least-squares functional can be constructed for both systems as with some physical weights w 1 = 1/ √ ρ and w 2 = 1/ √ ρν for the first formulation and w 3 = √ 2ν for the second formulation, respectively.…”

Section: Least-squares Methodsmentioning

confidence: 99%

Influence of higher interpolation orders in mixed LSFEM for the incompressible Navier‐Stokes equations

Serdas

Schwarz

Schröder

et al. 2013

Proc Appl Math and Mech

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The focus of this contribution is the examination of the influence of different interpolation orders (especially for the pressure field) by solving the stationary incompressible Navier‐Stokes equations with least‐squares finite elements. We consider two different div‐grad first‐order systems which result in a three‐field formulation with stresses, velocitites and pressure, see e.g. SCHWARZ & SCHRÖDER [1] and BOCHEV & GUNZBURGER [2]. Additionally, a numerical example is presented to show the performance of the considered formulations. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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Least-squares methods for the velocity-pressure-stress formulation of the Stokes equations

Cited by 78 publications

References 25 publications

Finite Element Methods of Least-Squares Type

Finite Element Methods of Least-Squares Type

Analysis and computation of least-squares methods for a compressible Stokes problem

Influence of higher interpolation orders in mixed LSFEM for the incompressible Navier‐Stokes equations

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