On the Inf-Sup Constant of a Triangular Spectral Method for the Stokes Equations

Su, Yanhui; Chen, Lizhen; Li, Xianjuan; Xu, Chuanju

doi:10.1515/cmam-2016-0011

Cited by 8 publications

(4 citation statements)

References 20 publications

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“…Using proper degrees of freedom, this leads to a similar method on quadrilaterals as we use on triangles. Another approach, combining the tensor product structure on quadrilaterals and the advantage of approximating more complex geometries using triangles is analyzed in Su et al (2016). The key of this method is to use the Duffy transformation and a proper pair of approximation spaces which leads to β (k) = O(k − 1 2 ) with the drawback of using rational functions for the approximation.…”

Section: Introductionmentioning

confidence: 99%

Polynomial robust stability analysis for $H$(div)-conforming finite elements for the Stokes equations

Lederer

Schöberl

2017

IMA Journal of Numerical Analysis

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In this work we consider a discontinuous Galerkin method for the discretization of the Stokes problem. We use H(div)-conforming finite elements as they provide major benefits such as exact mass conservation and pressure-independent error estimates. The main aspect of this work lies in the analysis of high order approximations. We show that the considered method is uniformly stable with respect to the polynomial order k and provides optimal error estimates u − u h 1 h + Π Q h p − p h 0 c (h/k) s u s+1 . To derive those estimates, we prove a k-robust LBB condition. This proof is based on a polynomial H 2 -stable extension operator. This extension operator itself is of interest for the numerical analysis of C 0 -continuous discontinuous Galerkin methods for 4 th order problems.

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Section: Introductionmentioning

confidence: 99%

Polynomial robust stability analysis for $H$(div)-conforming finite elements for the Stokes equations

Lederer

Schöberl

2017

IMA Journal of Numerical Analysis

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“…Fractional derivatives, as quasi-differential operators, exhibit non-local characteristics and thus serve as a powerful tool for describing the properties of non-locality and long memory observed in various physical phenomena. With the rapid development of the fractional calculus in the last several decades, fractional partial differential equations (FPDEs) have been applied in various fields including, but not limited to, physics, fluid mechanics, biology, and finance, engineering [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29]. In the late 20th century, researchers discovered that financial markets exhibit fractal characteristics both domestically and internationally [9,30].…”

Section: Introductionmentioning

confidence: 99%

Review of the Fractional Black-Scholes Equations and Their Solution Techniques

Zhang,

Liu

et al. 2024

Fractal Fract

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The pioneering work in finance by Black, Scholes and Merton during the 1970s led to the emergence of the Black-Scholes (B-S) equation, which offers a concise and transparent formula for determining the theoretical price of an option. The establishment of the B-S equation, however, relies on a set of rigorous assumptions that give rise to several limitations. The non-local property of the fractional derivative (FD) and the identification of fractal characteristics in financial markets have paved the way for the introduction and rapid development of fractional calculus in finance. In comparison to the classical B-S equation, the fractional B-S equations (FBSEs) offer a more flexible representation of market behavior by incorporating long-range dependence, heavy-tailed and leptokurtic distributions, as well as multifractality. This enables better modeling of extreme events and complex market phenomena, The fractional B-S equations can more accurately depict the price fluctuations in actual financial markets, thereby providing a more reliable basis for derivative pricing and risk management. This paper aims to offer a comprehensive review of various FBSEs for pricing European options, including associated solution techniques. It contributes to a deeper understanding of financial model development and its practical implications, thereby assisting researchers in making informed decisions about the most suitable approach for their needs.

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“…Moreover, the RT instability also has been investigated under other physical factors, such as internal surface tension [12,17,41], magnetic fields [4, 5, 19-21, 23, 24, 39, 40], rotation [3,6,29], and so on. We also refer to [2,8,14,30,[33][34][35][36] for related progress in other mathematical problems of hydromechanics.…”

Section: Introductionmentioning

confidence: 99%

On classical solutions of Rayleigh–Taylor instability in inhomogeneous viscoelastic fluids

Tan

Wang

2019

Bound Value Probl

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We study the nonlinear Rayleigh-Taylor (RT) instability of an inhomogeneous incompressible viscoelastic fluid in a bounded domain. It is well known that there exist strong solutions of RT instability in H 2-norm in inhomogeneous incompressible viscoelastic fluids, when the elasticity coefficient κ is less than some threshold κ C. In this paper, we prove the existence of classical solutions of RT instability in L 1-norm in Lagrangian coordinates based on a bootstrap instability method with finer analysis, if κ < κ C. Moreover, we also get classical solutions of RT instability in L 1-norm in Eulerian coordinates by further applying an inverse transformation of Lagrangian coordinates.

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On the Inf-Sup Constant of a Triangular Spectral Method for the Stokes Equations

Cited by 8 publications

References 20 publications

Polynomial robust stability analysis for $H$(div)-conforming finite elements for the Stokes equations

Polynomial robust stability analysis for $H$(div)-conforming finite elements for the Stokes equations

Review of the Fractional Black-Scholes Equations and Their Solution Techniques

On classical solutions of Rayleigh–Taylor instability in inhomogeneous viscoelastic fluids

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