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Zeitschriftenartikel zum Thema „Discrete duality finite volumes“

Autor: Grafiati

Veröffentlicht am 28. September 2024

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1

ANDREIANOV, B., M. BENDAHMANE, and K. H. KARLSEN. "DISCRETE DUALITY FINITE VOLUME SCHEMES FOR DOUBLY NONLINEAR DEGENERATE HYPERBOLIC-PARABOLIC EQUATIONS." Journal of Hyperbolic Differential Equations 07, no. 01 (2010): 1–67. http://dx.doi.org/10.1142/s0219891610002062.

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We consider a class of doubly nonlinear degenerate hyperbolic-parabolic equations with homogeneous Dirichlet boundary conditions, for which we first establish the existence and uniqueness of entropy solutions. We then turn to the construction and analysis of discrete duality finite volume schemes (in the spirit of Domelevo and Omnès [43]) for these problems in two and three spatial dimensions. We derive a series of discrete duality formulas and entropy dissipation inequalities for the schemes. We establish the existence of solutions to the discrete problems, and prove that sequences of approximate solutions generated by the discrete duality finite volume schemes converge strongly to the entropy solution of the continuous problem. The proof revolves around basic a priori estimates, the discrete duality features, Minty–Browder type arguments, and "hyperbolic" L∞weak-⋆ compactness arguments (i.e. propagation of compactness along the lines of Tartar, DiPerna, …). Our results cover the case of non-Lipschitz nonlinearities.

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2

Coudière, Yves, and Gianmarco Manzini. "The Discrete Duality Finite Volume Method for Convection-diffusion Problems." SIAM Journal on Numerical Analysis 47, no. 6 (2010): 4163–92. http://dx.doi.org/10.1137/080731219.

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3

Handlovicova, Angela. "Stability estimates for Discrete duality finite volume scheme of Heston model." Computer Methods in Material Science 17, no. 2 (2017): 101–10. http://dx.doi.org/10.7494/cmms.2017.2.0596.

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Tensor diffusion equation represents an important model in many fields of science. We focused our attention to the problem which arises in financial mathematics and is known as 2D Heston model. Stability estimates for discrete duality finite volume scheme for proposed model is presented. Numerical experiments using proposed method and comparing it with previous numerical scheme are included

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4

Tomek, Lukáš, and Karol Mikula. "Discrete duality finite volume method with tangential redistribution of points for surfaces evolving by mean curvature." ESAIM: Mathematical Modelling and Numerical Analysis 53, no. 6 (2019): 1797–840. http://dx.doi.org/10.1051/m2an/2019040.

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We propose a new discrete duality finite volume method for solving mean curvature flow of surfaces in ℝ3. In the cotangent scheme, which is widely used discretization of Laplace–Beltrami operator, a two-dimensional surface is usually approximated by a triangular mesh. In the cotangent scheme the unknowns are the vertices of the triangulation. A finite volume around each vertex is constructed as a surface patch bounded by a piecewise linear curve with nodes in the midpoints of the neighbouring edges and a representative point of each adjacent triangle. The basic idea of our new approach is to include the representative points into the numerical scheme as supplementary unknowns and generalize discrete duality finite volume method from ℝ2 to 2D surfaces embedded in ℝ3. To improve the quality of the mesh we use an area-oriented tangential redistribution of the grid points. We derive the numerical scheme for both closed surfaces and surfaces with boundary, and present numerical experiments. Surface evolution models are applied to construction of minimal surfaces with given set of boundary curves.

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5

Chainais-Hillairet, C., S. Krell, and A. Mouton. "Study of Discrete Duality Finite Volume Schemes for the Peaceman Model." SIAM Journal on Scientific Computing 35, no. 6 (2013): A2928—A2952. http://dx.doi.org/10.1137/130910555.

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6

Coudière, Yves, and Florence Hubert. "A 3D Discrete Duality Finite Volume Method for Nonlinear Elliptic Equations." SIAM Journal on Scientific Computing 33, no. 4 (2011): 1739–64. http://dx.doi.org/10.1137/100786046.

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7

Handlovičová, Angela, and Dana Kotorová. "Stability of the Semi-Implicit Discrete Duality Finite Volume Scheme for the Curvature Driven Level Set Equation in 2D." Tatra Mountains Mathematical Publications 61, no. 1 (2014): 117–29. http://dx.doi.org/10.2478/tmmp-2014-0031.

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Abstract Stability of the linear semi-implicit discrete duality finite volume (DDFV) numerical scheme for the solution of the regularized curvature driven level set equation is proved. Our scheme is linear, it is efficient regarding computational times. Numerical experiments confirm accuracy of the proposed scheme

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8

Andreianov, Boris, Mostafa Bendahmane, and Florence Hubert. "On 3D DDFV Discretization of Gradient and Divergence Operators: Discrete Functional Analysis Tools and Applications to Degenerate Parabolic Problems." Computational Methods in Applied Mathematics 13, no. 4 (2013): 369–410. http://dx.doi.org/10.1515/cmam-2013-0011.

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Abstract. We present a detailed survey of discrete functional analysis tools (consistency results, Poincaré and Sobolev embedding inequalities, discrete W1,p compactness, discrete compactness in space and in time) for the so-called Discrete Duality Finite Volume (DDFV) schemes in three space dimensions. We concentrate mainly on the 3D CeVe-DDFV scheme presented in [IMA J. Numer. Anal., 32 (2012), pp. 1574–1603]. Some of our results are new, such as a general time-compactness result based upon the idea of Kruzhkov (1969); others generalize the ideas known for the 2D DDFV schemes or for traditional two-point-flux finite volume schemes. We illustrate the use of these tools by studying convergence of discretizations of nonlinear elliptic-parabolic problems of Leray–Lions kind, and provide numerical results for this example.

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9

Njifenjou, Abdou, Abel Toudna Mansou, and Moussa Sali. "A New Second-order Maximum-principle-preserving Finite-volume Method for Flow Problems Involving Discontinuous Coefficients." American Journal of Applied Mathematics 12, no. 4 (2024): 91–110. http://dx.doi.org/10.11648/j.ajam.20241204.12.

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A new development of Finite Volumes (FV, for short) and its theoretical analysis are the purpose of this work. Recall that FV are known as powerful tools to address equations of conservation laws (mass, energy, momentum,...). Over the last two decades investigators have succeeded in putting in place a mathematical framework for the theoretical analysis of FV. A perfect illustration of this progress is the design and mathematical analysis of Discrete Duality Finite Volumes (DDFV, for short). We propose now a new class of DDFV for 2nd order elliptic equations involving discontinuous diffusion coefficients or nonlinearities. A one-dimensional linear elliptic equation is addressed here for illustrating the ideas behind our numerical strategy. The algebraic structure of the discrete system we have got is different from that of standard DDFV. The main novelty is that the so-called diamond mesh elements are confined in homogeneous zones for flow problems governed by piecewise constant coefficients. This is got from our judicious definition of the primal mesh. The gain is that there is no need to compute homogenized coefficients to be allocated to the so-called diamond cells as required to conventional DDFV. Notice that poor homogenized permeability allocated to diamond elements leads to poor approximations of fluxes across grid-block interfaces. Moreover for 1-D flow problems in a porous medium involving permeability discontinuities (piecewise constant permeability for instance) the proposed FV scheme leads to a symmetric positive-definite discrete system that meets the discrete maximum principle; we have shown its second order convergence under relevant assumptions.

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10

Andreianov, Boris, Mostafa Bendahmane, Kenneth H. Karlsen, and Charles Pierre. "Convergence of discrete duality finite volume schemes for the cardiac bidomain model." Networks & Heterogeneous Media 6, no. 2 (2011): 195–240. http://dx.doi.org/10.3934/nhm.2011.6.195.

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11

Rhoudaf, M., and N. Staïli. "A Discrete Duality Finite Volume Method for Coupling Darcy and Stokes Equations." Moroccan Journal of Pure and Applied Analysis 5, no. 1 (2019): 46–62. http://dx.doi.org/10.2478/mjpaa-2019-0005.

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AbstractWe present a general finite volume method to solve a coupled Stokes-Darcy problem, we propose two domains corresponding to fluid region and porous region with a physical intersection. At the contact interface between the fluid region and the porous media we impose two conditions; the first one is the normal continuity of the velocity and the second one is the continuity of the pressure. Furthermore, due to the lack of information about both the velocity and the pressure on the interface, we will use Schwarz domain decomposition. In Darcy equations, the tensor of permeability will be considered as variable, since it depends on both the properties of the porous medium and the viscosity of the fluid. Numerical examples are presented to demonstrate the efficiency of the proposed method.

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12

Kinfack Jeutsa, A., A. Njifenjou, and J. Nganhou. "Convergence Analysis on Unstructured Meshes of a DDFV Method for Flow Problems with Full Neumann Boundary Conditions." Journal of Applied Mathematics 2016 (2016): 1–22. http://dx.doi.org/10.1155/2016/5891064.

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A Discrete Duality Finite Volume (DDFV) method to solve on unstructured meshes the flow problems in anisotropic nonhomogeneous porous media with full Neumann boundary conditions is proposed in the present work. We start with the derivation of the discrete problem. A result of existence and uniqueness of a solution for that problem is given thanks to the properties of its associated matrix combined with adequate assumptions on data. Their theoretical properties, namely, stability and error estimates (in discrete energy norms andL2-norm), are investigated. Numerical test is provided.

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13

Dubois, Francois, Isabelle Greff, and Charles Pierre. "Raviart–Thomas finite elements of Petrov–Galerkin type." ESAIM: Mathematical Modelling and Numerical Analysis 53, no. 5 (2019): 1553–76. http://dx.doi.org/10.1051/m2an/2019020.

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Finite volume methods are widely used, in particular because they allow an explicit and local computation of a discrete gradient. This computation is only based on the values of a given scalar field. In this contribution, we wish to achieve the same goal in a mixed finite element context of Petrov–Galerkin type so as to ensure a local computation of the gradient at the interfaces of the elements. The shape functions are the Raviart–Thomas finite elements. Our purpose is to define test functions that are in duality with these shape functions: precisely, the shape and test functions will be asked to satisfy some orthogonality property. This paradigm is addressed for the discrete solution of the Poisson problem. The general theory of Babuška brings necessary and sufficient stability conditions for a Petrov–Galerkin mixed problem to be convergent. In order to ensure stability, we propose specific constraints for the dual test functions. With this choice, we prove that the mixed Petrov–Galerkin scheme is identical to the four point finite volume scheme of Herbin, and to the mass lumping approach developed by Baranger, Maitre and Oudin. Convergence is proven with the usual techniques of mixed finite elements.

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14

Krell, Stella, and Gianmarco Manzini. "The Discrete Duality Finite Volume Method for Stokes Equations on Three-Dimensional Polyhedral Meshes." SIAM Journal on Numerical Analysis 50, no. 2 (2012): 808–37. http://dx.doi.org/10.1137/110831593.

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15

Le, Anh Ha, and Pascal Omnes. "Ana posteriorierror estimation for the discrete duality finite volume discretization of the Stokes equations." ESAIM: Mathematical Modelling and Numerical Analysis 49, no. 3 (2015): 663–93. http://dx.doi.org/10.1051/m2an/2014057.

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16

Chainais-Hillairet, C. "Discrete duality finite volume schemes for two-dimensional drift-diffusion and energy-transport models." International Journal for Numerical Methods in Fluids 59, no. 3 (2009): 239–57. http://dx.doi.org/10.1002/fld.1393.

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17

He, Zhengkang, Rui Li, Jie Chen, and Zhangxin Chen. "The discrete duality finite volume method for a class of quasi‐Newtonian Stokes flows." Numerical Methods for Partial Differential Equations 35, no. 6 (2019): 2193–220. http://dx.doi.org/10.1002/num.22408.

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18

Gander, Martin J., Laurence Halpern, Florence Hubert, and Stella Krell. "Optimized Schwarz methods with general Ventcell transmission conditions for fully anisotropic diffusion with discrete duality finite volume discretizations." Moroccan Journal of Pure and Applied Analysis 7, no. 2 (2020): 182–213. http://dx.doi.org/10.2478/mjpaa-2021-0014.

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Abstract We introduce a new non-overlapping optimized Schwarz method for fully anisotropic diffusion problems. Optimized Schwarz methods take into account the underlying physical properties of the problem at hand in the transmission conditions, and are thus ideally suited for solving anisotropic diffusion problems. We first study the new method at the continuous level for two subdomains, prove its convergence for general transmission conditions of Ventcell type using energy estimates, and also derive convergence factors to determine the optimal choice of parameters in the transmission conditions. We then derive optimized Robin and Ventcell parameters at the continuous level for fully anisotropic diffusion, both for the case of unbounded and bounded domains. We next present a discretization of the algorithm using discrete duality finite volumes, which are ideally suited for fully anisotropic diffusion on very general meshes. We prove a new convergence result for the discretized optimized Schwarz method with two subdomains using energy estimates for general Ventcell transmission conditions. We finally study the convergence of the new optimized Schwarz method numerically using parameters obtained from the continuous analysis. We find that the predicted optimized parameters work very well in practice, and that for certain anisotropies which we characterize, our new bounded domain analysis is important.

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19

Cancès, Clément, Claire Chainais-Hillairet, and Stella Krell. "Numerical Analysis of a Nonlinear Free-Energy Diminishing Discrete Duality Finite Volume Scheme for Convection Diffusion Equations." Computational Methods in Applied Mathematics 18, no. 3 (2018): 407–32. http://dx.doi.org/10.1515/cmam-2017-0043.

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AbstractWe propose a nonlinear Discrete Duality Finite Volume scheme to approximate the solutions of drift diffusion equations. The scheme is built to preserve at the discrete level even on severely distorted meshes the energy/energy dissipation relation. This relation is of paramount importance to capture the long-time behavior of the problem in an accurate way. To enforce it, the linear convection diffusion equation is rewritten in a nonlinear form before being discretized. We establish the existence of positive solutions to the scheme. Based on compactness arguments, the convergence of the approximate solution towards a weak solution is established. Finally, we provide numerical evidences of the good behavior of the scheme when the discretization parameters tend to 0 and when time goes to infinity.

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20

Hermeline, Francois. "Discrete Duality Finite Volume Discretization of the Thermal-$P_N$ Radiative Transfer Equations on General Meshes." Communications in Computational Physics 31, no. 2 (2022): 398–448. http://dx.doi.org/10.4208/cicp.oa-2021-0084.

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21

Omnes, Pascal, Yohan Penel, and Yann Rosenbaum. "A Posteriori Error Estimation for the Discrete Duality Finite Volume Discretization of the Laplace Equation." SIAM Journal on Numerical Analysis 47, no. 4 (2009): 2782–807. http://dx.doi.org/10.1137/080735047.

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22

Boyer, Franck, Stella Krell, and Flore Nabet. "Inf-Sup stability of the discrete duality finite volume method for the 2D Stokes problem." Mathematics of Computation 84, no. 296 (2015): 2705–42. http://dx.doi.org/10.1090/mcom/2956.

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23

Njifenjou, A., H. Donfack, and I. Moukouop-Nguena. "Analysis on general meshes of a discrete duality finite volume method for subsurface flow problems." Computational Geosciences 17, no. 2 (2013): 391–415. http://dx.doi.org/10.1007/s10596-012-9339-6.

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24

Andreianov, Boris, Franck Boyer, and Florence Hubert. "Discrete duality finite volume schemes for Leray−Lions−type elliptic problems on general 2D meshes." Numerical Methods for Partial Differential Equations 23, no. 1 (2006): 145–95. http://dx.doi.org/10.1002/num.20170.

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25

KRELL, STELLA. "FINITE VOLUME METHOD FOR GENERAL MULTIFLUID FLOWS GOVERNED BY THE INTERFACE STOKES PROBLEM." Mathematical Models and Methods in Applied Sciences 22, no. 05 (2012): 1150025. http://dx.doi.org/10.1142/s0218202511500254.

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We study the approximation of solutions to the steady Stokes problem with a discontinuous viscosity coefficient (interface Stokes problem) in the 2D "Discrete Duality Finite Volume" (DDFV) framework. In order to take into account the discontinuities of the viscosity and to prevent consistency defect in the scheme, we propose to modify the definition of the numerical fluxes on the edges of the mesh where the discontinuity occurs. We first show how to design our modified scheme, called m-DDFV, and we analyze its well-posedness and its convergence properties. Finally, we provide numerical results which confirm that the m-DDFV scheme significantly improves the convergence rate of the usual DDFV method for Stokes problems.

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26

Boyer, Franck, and Flore Nabet. "A DDFV method for a Cahn−Hilliard/Stokes phase field model with dynamic boundary conditions." ESAIM: Mathematical Modelling and Numerical Analysis 51, no. 5 (2017): 1691–731. http://dx.doi.org/10.1051/m2an/2016073.

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In this paper we propose a “Discrete Duality Finite Volume” method (DDFV for short) for the diffuse interface modelling of incompressible two-phase flows. This numerical method is, conservative, robust and is able to handle general geometries and meshes. The model we study couples the Cahn−Hilliard equation and the unsteady Stokes equation and is endowed with particular nonlinear boundary conditions called dynamic boundary conditions. To implement the scheme for this model we have to derive new discrete consistent DDFV operators that allows an energy stable coupling between both discrete equations. We are thus able to obtain the existence of a family of solutions satisfying a suitable energy inequality, even in the case where a first order time-splitting method between the two subsystems is used. We illustrate various properties of such a model with some numerical results.

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27

Bazirha, Z., and L. Azrar. "DDFV scheme for nonlinear parabolic reaction-diffusion problems on general meshes." Mathematical Modeling and Computing 11, no. 1 (2024): 96–108. http://dx.doi.org/10.23939/mmc2024.01.096.

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This paper focuses on the nonlinear anisotropic parabolic model of the form ∂tC(u)−div(Λ∇u)+R(u)=f, where C, R, f, and Λ are respectively: two nonlinear functions, a source term and an anisotropic tensor diffusion. For space discretization, various types of the Discrete Duality Finite Volume (DDFV) scheme are elaborated leading to positive definite stiffness matrices for the diffusion term. A general mesh is used and hard anisotropic tensor with discontinuous effects is considered. An implicit time scheme is developed as well as the Newton–Raphson method to solve the resulting nonlinear system. An iterative incremental approach is elaborated handling the effects of anisotropy, discontinuity and non-linearity. The performance of the presented direct and indirect DDFV schemes for different meshes has been demonstrated by various numerical tests. A super-convergence in the discrete L2 and H1-norms is also demonstrated.

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28

Su, Shuai, Qiannan Dong, and Jiming Wu. "A decoupled and positivity-preserving discrete duality finite volume scheme for anisotropic diffusion problems on general polygonal meshes." Journal of Computational Physics 372 (November 2018): 773–98. http://dx.doi.org/10.1016/j.jcp.2018.06.052.

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29

Goudon, Thierry, Stella Krell, and Giulia Lissoni. "Non-overlapping Schwarz algorithms for the incompressible Navier–Stokes equations with DDFV discretizations." ESAIM: Mathematical Modelling and Numerical Analysis 55, no. 4 (2021): 1271–321. http://dx.doi.org/10.1051/m2an/2021024.

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We propose and analyze non-overlapping Schwarz algorithms for the domain decomposition of the unsteady incompressible Navier–Stokes problem with Discrete Duality Finite Volume (DDFV) discretization. The design of suitable transmission conditions for the velocity and the pressure is a crucial issue. We establish the well-posedness of the method and the convergence of the iterative process, pointing out how the numerical fluxes influence the asymptotic problem which is intended to be a discretization of the Navier–Stokes equations on the entire computational domain. Finally we numerically illustrate the behavior and performances of the algorithm. We discuss on numerical grounds the impact of the parameters for several mesh geometries and we perform simulations of the flow past an obstacle with several domains.

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30

Delcourte, Sarah, Komla Domelevo, and Pascal Omnes. "A Discrete Duality Finite Volume Approach to Hodge Decomposition and div‐curl Problems on Almost Arbitrary Two‐Dimensional Meshes." SIAM Journal on Numerical Analysis 45, no. 3 (2007): 1142–74. http://dx.doi.org/10.1137/060655031.

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31

Delcourte, Sarah, and Pascal Omnes. "A discrete duality finite volume discretization of the vorticity-velocity-pressure stokes problem on almost arbitrary two-dimensional grids." Numerical Methods for Partial Differential Equations 31, no. 1 (2014): 1–30. http://dx.doi.org/10.1002/num.21890.

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32

Kalanta, Stanislovas. "DUAL MATHEMATICAL MODELS OF LIMIT LOAD ANALYSIS PROBLEMS OF STRUCTURES BY MIXED FINITE ELEMENTS." JOURNAL OF CIVIL ENGINEERING AND MANAGEMENT 3, no. 10 (1997): 43–51. http://dx.doi.org/10.3846/13921525.1997.10531683.

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The general and discrete dual mathematical models of the limit load analysis and optimization problems of rigid-plastic body are created in the article. The discrete models are formulated by mixed finite elements and presented in terms of kinematic and static formulation. In these models the velocity of the energy dissipation is estimated not only within the volume of finite elements, but also at the plastic surfaces between elements, where the discontinuities of displacement velocities functions appear. The theory of plastic flow, the theory of duality and mathematical programming are applied. The mixed energy functional (1) and (3) of both problems are formulated using the general static formulations of these problems, presented in the article [10], and Lagrangian multipliers method. The mixed finite elements are used for their discretization. The discrete expressions (8), (9) and (13) of mixed functionals are given choosing the interpolation functions (7) for the stress, displacement velocities, plastic multipliers and external load. Stationary conditions are created by static variables (stress and load vectors) of theses functionals. The discrete expressions of the geometric compatibility equations and constraint of load power are received from them. Using them as preliminary conditions for the functionals (8) and (9), the mathematical models (14), (15) and (17) of kinematic formulation of limit load analysis and optimization problems are formulated. The model (20) with a smaller number of unknowns is formed by elimination the displacement velocities. Using Lagrangian multipliers method, the mathematical models (21)-(23) of static formulation for the limit load parameter analysis problem and the models (24)-(26) for the load optimization problem are derived. All of them are the problems of mathematical programming. The mathematical models of static formulation for engineering purposes are more important and fit better. They are easier solved (a smaller quantity of unknowns), besides, they allow to determine the optimum distribution of the load. The formulated mathematical models allow to determine upper values of limit load, stresses, displacement and plastic multipliers velocities. Together with equilibrium models of these problems, presented in the article [10], they allow to determine the lower and upper values of aforementioned parameters. So, a good possibility is created to check reliability and exactness of numerical calculation results and to establish, if the computing net density of finite elements is sufficient.

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33

Denicolai, Emilie, Stéphane Honoré, Florence Hubert, and Rémi Tesson. "Microtubules (MT) a key target in oncology: mathematical modeling of anti-MT agents on cell migration." Mathematical Modelling of Natural Phenomena 15 (2020): 63. http://dx.doi.org/10.1051/mmnp/2020004.

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Microtubules (MTs) are protein filaments found in all eukaryotic cells which are crucial for many cellular processes including cell movement, cell differentiation, and cell division, making them a key target for anti-cancer treatment. In particular, it has been shown that at low dose, MT targeted agents (MTAs) may induce an anti-migratory effect on cancer and endothelial cells, leading to new prospects in cancer therapy. In that context, we propose to better understand the role of MT dynamics and thus of MTAs on cell migration using a mathematical cell centered model of cell migration taking into account the action of microtubules in the process. The model use a fluid based approach that describes, through level-set techniques, the deformation of the membrane during cell migration. The fluid part of the model is mainly composed of Stokes equations and the biochemical state of the cell is described using Reaction-Diffusion equations. Microtubules act on the biochemical state by activating or inactivating proteins of the Rho-GTPases family. The numerical simulation of the model is performed using Discrete Duality Finite Volume techniques. We describe the different schemes used for the simulation, focusing on the adaptation of preexisting methods to our particular case. Numerical simulation are performed, showing a realistic behavior of the simulated cells in term of shape, speed and microtubules dynamics. Different strategies for a depolymerizing MTA (Vincristin) mechanisms are investigated and show the robutness of our model.

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34

Celani, Sergio, and Leonardo Cabrer. "Duality for finite Hilbert algebras." Discrete Mathematics 305, no. 1-3 (2005): 74–99. http://dx.doi.org/10.1016/j.disc.2005.09.002.

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35

Bartels, Sören, and Zhangxian Wang. "Orthogonality relations of Crouzeix–Raviart and Raviart–Thomas finite element spaces." Numerische Mathematik 148, no. 1 (2021): 127–39. http://dx.doi.org/10.1007/s00211-021-01199-3.

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AbstractIdentities that relate projections of Raviart–Thomas finite element vector fields to discrete gradients of Crouzeix–Raviart finite element functions are derived under general conditions. Various implications such as discrete convex duality results and a characterization of the image of the projection of the Crouzeix–Ravaiart space onto elementwise constant functions are deduced.

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36

Loten, Cynthia, and Claude Tardif. "Majority functions on structures with finite duality." European Journal of Combinatorics 29, no. 4 (2008): 979–86. http://dx.doi.org/10.1016/j.ejc.2007.11.007.

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37

Jiang, Lining, Maozheng Guo, and Min Qian. "The duality theory of a finite dimensional discrete quantum group." Proceedings of the American Mathematical Society 132, no. 12 (2004): 3537–47. http://dx.doi.org/10.1090/s0002-9939-04-07397-6.

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38

CHAJDA, I., R. HALAŠ, A. G. PINUS, and I. G. ROSENBERG. "DUALITY OF NORMALLY PRESENTED VARIETIES." International Journal of Algebra and Computation 10, no. 05 (2000): 651–64. http://dx.doi.org/10.1142/s0218196700000212.

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Let [Formula: see text] be a variety of the form [Formula: see text] where [Formula: see text] is a finite subdirectly irreducible algebra. We show that if [Formula: see text] is naturally dualizable (in the sense of D. M. Clark and B. A. Davey, i.e. with respect to the discrete topology) then the variety [Formula: see text][Formula: see text] determined by all normal identities of [Formula: see text] (the so called nilpotent shift of [Formula: see text]) is also naturally dualizable. We give a finite algebra [Formula: see text] and a relational system [Formula: see text], constructed explicitly from the system [Formula: see text] for [Formula: see text], such that [Formula: see text] and [Formula: see text] dualizes [Formula: see text] .

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39

Nešetřil, Jaroslav, and Yared Nigussie. "Finite duality for some minor closed classes." Electronic Notes in Discrete Mathematics 29 (August 2007): 579–85. http://dx.doi.org/10.1016/j.endm.2007.07.092.

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40

Aschieri, P., L. Castellani, and A. P. Isaev. "Discretized Yang–Mills and Born–Infeld Actions on Finite Group Geometries." International Journal of Modern Physics A 18, no. 20 (2003): 3555–85. http://dx.doi.org/10.1142/s0217751x03015209.

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Discretized non-Abelian gauge theories living on finite group spaces G are defined by means of a geometric action ∫ Tr F ∧ *F. This technique is extended to obtain discrete versions of the Born–Infeld action. The discretizations are in 1–1 correspondence with differential calculi on finite groups. A consistency condition for duality invariance of the discretized field equations is derived for discretized U(1) actions S[F] living on a four-dimensional Abelian G. Discretized electromagnetism satisfies this condition and therefore admits duality rotations. Yang–Mills and Born–Infeld theories are also considered on product spaces MD×G, and we find the corresponding field theories on MD after Kaluza–Klein reduction on the G discrete internal spaces. We examine in detail the case G=ZN, and discuss the limit N→∞. A self-contained review on the noncommutative differential geometry of finite groups is included.

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41

Bradley, David M. "Duality for finite multiple harmonic q-series." Discrete Mathematics 300, no. 1-3 (2005): 44–56. http://dx.doi.org/10.1016/j.disc.2005.06.008.

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42

Li, Shuxing, Alexander Pott, and Robert Schüler. "Formal duality in finite abelian groups." Journal of Combinatorial Theory, Series A 162 (February 2019): 354–405. http://dx.doi.org/10.1016/j.jcta.2018.11.005.

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43

Huang, Ming-Deh, and Wayne Raskind. "Global Duality, Signature Calculus and the Discrete Logarithm Problem." LMS Journal of Computation and Mathematics 12 (2009): 228–63. http://dx.doi.org/10.1112/s1461157000001509.

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AbstractWe develop a formalism for studying the discrete logarithm problem for the multiplicative group and for elliptic curves over finite fields by lifting the respective group to an algebraic number field and using global duality. One of our main tools is the signature of a Dirichlet character (in the multiplicative group case) or principal homogeneous space (in the elliptic curve case), which is a measure of its ramification at certain places. We then develop signature calculus, which generalizes and refines the index calculus method. Finally, using some heuristics, we show the random polynomial time equivalence for these two cases between the problem of computing signatures and the discrete logarithm problem. This relates the discrete logarithm problem to some very well-known problems in algebraic number theory and arithmetic geometry.

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Dubejko, T. "Discrete Solutions of Dirichlet Problems, Finite Volumes, and Circle Packings." Discrete & Computational Geometry 22, no. 1 (1999): 19–39. http://dx.doi.org/10.1007/pl00009447.

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45

Cao, Yuan, Yonglin Cao, and Fang-Wei Fu. "Hermitian duality of left dihedral codes over finite fields." Discrete Mathematics 346, no. 1 (2023): 113179. http://dx.doi.org/10.1016/j.disc.2022.113179.

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46

He, Yinnian, and Jun Zou. "A prioriestimates and optimal finite element approximation of the MHD flow in smooth domains." ESAIM: Mathematical Modelling and Numerical Analysis 52, no. 1 (2018): 181–206. http://dx.doi.org/10.1051/m2an/2018006.

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We study a finite element approximation of the initial-boundary value problem of the 3D incompressible magnetohydrodynamic (MHD) system under smooth domains and data. We first establish several important regularities anda prioriestimates for the velocity, pressure and magnetic field (u,p,B) of the MHD system under the assumption that ∇u∈L4(0,T;L2(Ω)3 × 3) and ∇ ×B∈L4(0,T;L2(Ω)3). Then we formulate a finite element approximation of the MHD flow. Finally, we derive the optimal error estimates of the discrete velocity and magnetic field in energy-norm and the discrete pressure inL2-norm, and the optimal error estimates of the discrete velocity and magnetic field inL2-norm by means of a novel negative-norm technique, without the help of the standard duality argument for the Navier-Stokes equations.

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47

Rump, Wolfgang. "The geometry of discrete L-algebras." Advances in Geometry 23, no. 4 (2023): 543–65. http://dx.doi.org/10.1515/advgeom-2023-0023.

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Abstract The relationship of discrete L-algebras to projective geometry is deepened and made explicit in several ways. Firstly, a geometric lattice is associated to any discrete L-algebra. Monoids of I-type are obtained as a special case where the perspectivity relation is trivial. Secondly, the structure group of a non-degenerate discrete L-algebra X is determined and shown to be a complete invariant. It is proved that X ∖ {1} is a projective space with an orthogonality relation. A new definition of non-symmetric quantum sets, extending the recursive definition of symmetric quantum sets, is provided and shown to be equivalent to the former one. Quantum sets are characterized as complete projective spaces with an anisotropic duality, and they are also characterized in terms of their complete lattice of closed subspaces, which is one-sided orthomodular and semimodular. For quantum sets of finite cardinality n > 3, a representation as a projective space with duality over a skew-field is given. Quantum sets of cardinality 2 are classified, and the structure group of their associated L-algebra is determined.

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48

Garanzha, Vladimir A., Liudmila N. Kudryavtseva, and Dmitry A. Makarov. "Discrete curvatures for planar curves based on Archimedes’ duality principle." Russian Journal of Numerical Analysis and Mathematical Modelling 37, no. 2 (2022): 85–98. http://dx.doi.org/10.1515/rnam-2022-0007.

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Abstract We introduce discrete curvatures for planar curves based on the construction of sequences of pairs of mutually dual polylines. For piecewise-regular curves consisting of a finite number of fragments of regular generalized spirals with definite (positive or negative) curvatures our discrete curvatures approximate the exact averaged curvature from below and from above. In order to derive these estimates one should provide a distance function allowing to compute the closest point on the curve for an arbitrary point on the plane.With refinement of the polylines, the averaged curvature over refined curve segments converges to the pointwise values of the curvature and, thus, we obtain a good and stable local approximation of the curvature. For the important engineering case when the curve is approximated only by the inscribed (primal) polyline and the exact distance function is not available, we provide a comparative analysis for several techniques allowing to build dual polylines and discrete curvatures and evaluate their ability to create lower and upper estimates for the averaged curvature.

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Zalmai, G. J. "Duality models for some nonclassical problems in the calculus of variations." International Journal of Mathematics and Mathematical Sciences 2003, no. 66 (2003): 4145–82. http://dx.doi.org/10.1155/s0161171203303370.

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Parametric and nonparametric necessary and sufficient optimality conditions are established for a class of nonconvex variational problems with generalized fractional objective functions and nonlinear inequality constraints containing arbitrary norms. Based on these optimality criteria, ten parametric and parameter-free dual problems are constructed and appropriate duality theorems are proved. These optimality and duality results contain, as special cases, similar results for minmax fractional variational problems involving square roots of positive semidefinite quadratic forms as well as for variational problems with fractional, discrete max, and conventional objective functions, which are particular cases of the main problem considered in this paper. The duality models presented here subsume various existing duality formulations for variational problems and include variational generalizations of a great variety of cognate dual problems investigated previously in the area of finite-dimensional nonlinear programming by an assortment of ad hoc methods.

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Lan, Shi-Yi, and Dao-Qing Dai. "Discrete solutions of Dirichlet problems by circle patterns and finite volumes." Applicable Analysis 95, no. 4 (2015): 902–18. http://dx.doi.org/10.1080/00036811.2015.1042460.

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