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Journal articles on the topic 'Poisson's equation'

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Published: 4 June 2021

Last updated: 29 July 2024

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1

Khairi, Fathul, and Malahayati. "Penerapan Fungsi Green dari Persamaan Poisson pada Elektrostatika." Quadratic: Journal of Innovation and Technology in Mathematics and Mathematics Education 1, no. 1 (April 30, 2021): 56–79. http://dx.doi.org/10.14421/quadratic.2021.011-08.

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The Dirac delta function is a function that mathematically does not meet the criteria as a function, this is because the function has an infinite value at a point. However, in physics the Dirac Delta function is an important construction, one of which is in constructing the Green function. This research constructs the Green function by utilizing the Dirac Delta function and Green identity. Furthermore, the construction is directed at the Green function of the Poisson's equation which is equipped with the Dirichlet boundary condition. After the form of the Green function solution from the Poisson's equation is obtained, the Green function is determined by means of the expansion of the eigen functions in the Poisson's equation. These results are used to analyze the application of the Poisson equation in electrostatic.

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2

Laugesen, Richard S., Prashant G. Mehta, Sean P. Meyn, and Maxim Raginsky. "Poisson's Equation in Nonlinear Filtering." SIAM Journal on Control and Optimization 53, no. 1 (January 2015): 501–25. http://dx.doi.org/10.1137/13094743x.

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3

Fox, Bennett L., and Paul Glasserman. "Estimating Derivatives Via Poisson's Equation." Probability in the Engineering and Informational Sciences 5, no. 4 (October 1991): 415–28. http://dx.doi.org/10.1017/s0269964800002205.

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Let x(j) be the expected reward accumulated up to hitting an absorbing set in a Markov chain, starting from state j. Suppose the transition probabilities and the one-step reward function depend on a parameter, and denote by y(j) the derivative of x(j) with respect to that parameter. We estimate y(0) starting from the respective Poisson equations that x = [x(0),x(l),…] and y = [y(0),y(l),…] satisfy. Relative to a likelihood-ratio-method (LRM) estimator, our estimator generally has (much) smaller variance; in a certain sense, it is a conditional expectation of that estimator given x. Unlike LRM, however, we have to estimate certain components of x. Our method has broader scope than LRM: we can estimate sensitivity to opening arcs.

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4

Chicone, C., and B. Mashhoon. "Nonlocal gravity: Modified Poisson's equation." Journal of Mathematical Physics 53, no. 4 (April 2012): 042501. http://dx.doi.org/10.1063/1.3702449.

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5

Zhou, Yecheng, and Angus Gray-Weale. "A numerical model for charge transport and energy conversion of perovskite solar cells." Physical Chemistry Chemical Physics 18, no. 6 (2016): 4476–86. http://dx.doi.org/10.1039/c5cp05371d.

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6

Taufer, Jiří, and Emil Vitásek. "Transfer of boundary conditions for Poisson's equation on a circle." Applications of Mathematics 39, no. 1 (1994): 15–23. http://dx.doi.org/10.21136/am.1994.134240.

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7

Hanley, Mary. "Existence of Solutions to Poisson's Equation." Canadian Mathematical Bulletin 51, no. 2 (June 1, 2008): 229–35. http://dx.doi.org/10.4153/cmb-2008-024-8.

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AbstractLet Ω be a domain in ℝn (n ≥ 2). We find a necessary and sufficient topological condition on Ω such that, for anymeasure μ on ℝn, there is a function u with specified boundary conditions that satisfies the Poisson equation Δu = μ on Δ in the sense of distributions.

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8

Berg, M. van den, and D. Bucur. "Sign changing solutions of Poisson's equation." Proceedings of the London Mathematical Society 121, no. 3 (April 29, 2020): 513–36. http://dx.doi.org/10.1112/plms.12334.

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9

Chalupka, Slavko, Jaroslava Sisákova, and Eva Vargová. "Boundary perturbation formalism for Poisson's equation." Studia Geophysica et Geodaetica 36, no. 4 (December 1992): 325–28. http://dx.doi.org/10.1007/bf01625485.

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10

Nicholson, D. M. C., and W. A. Shelton. "Removed sphere method for Poisson's equation." Journal of Physics: Condensed Matter 14, no. 22 (May 24, 2002): 5601–8. http://dx.doi.org/10.1088/0953-8984/14/22/312.

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11

Hemmati, Mostafa. "Electron shock waves moving into an ionized medium." Laser and Particle Beams 13, no. 3 (September 1995): 377–82. http://dx.doi.org/10.1017/s0263034600009502.

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The propagation of electron driven shock waves has been investigated by employing a one-dimensional, three-component fluid model. In the fluid model, the basic set of equations consists of equations of conservation of mass, momentum, and energy, plus Poisson's equation. The wave is assumed to be a shock front followed by a dynamical transition region. Following Fowler's (1976) categorization of breakdown waves, the waves propagating into a preionized medium will be referred to as Class II Waves. To describe the breakdown waves, Shelton and Fowler (1968) used the terms proforce and antiforce waves, depending on whether the applied electric field force on electrons was with or against the direction of wave propagation. Breakdown waves, i.e., return strokes of lightning flashes, therefore, will be referred to as Antiforce Class II waves. The shock boundary conditions and Poisson's equation for Antiforce Class II waves are different from those for Antiforce Waves. The use of a newly derived set of boundary conditions and Poisson's equation for Antiforce Class II waves allows for a successful integration of the set of fluid equations through the dynamical transition region. The wave structure, i.e., electric field, electron concentration, electron temperature, and electron velocity, are very sensitive to the ion concentration ahead of the wave.

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12

Kitterød, Nils-Otto, and Étienne Leblois. "Estimation of sediment thickness by solving Poisson's equation with bedrock outcrops as boundary conditions." Hydrology Research 52, no. 3 (March 11, 2021): 597–619. http://dx.doi.org/10.2166/nh.2021.102.

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Abstract Sediment thickness and bedrock topography are vital for the terrestrial hydrosphere. In this study, we estimated sediment thickness by using information from digital elevation models, geological maps, and public databases. We discuss two different approaches: First, the horizontal distances to the nearest bedrock outcrop were used as a secondary function in kriging and cokriging. Second, we applied Poisson's equation to estimate the local trend of the sediment thickness where bedrock outcrops were used as boundary conditions. Differences between point observations and the parabolic surface from Poisson's equation were minimized by inverse modelling. Ordinary kriging was applied to the residuals. These two approaches were evaluated with data from the Øvre Eiker, Norway. Estimates derived from Poisson's equation gave the smallest mean absolute error, and larger soil depths were reproduced better if the local trend was included in the estimation procedure. An independent cross-validation was undertaken. The results showed the best accuracy and precision for kriging on the residuals from Poisson's equation. Solutions of Poisson's equation are sensitive to the boundary conditions, which in this case were locations of the bedrock outcrops. Bedrock outcrops are available for direct observations; hence, the quality of the estimates can be improved by updating input from high-resolution mapping.

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13

CARRILLO, JOSÉ A., and SIMON LABRUNIE. "GLOBAL SOLUTIONS FOR THE ONE-DIMENSIONAL VLASOV–MAXWELL SYSTEM FOR LASER-PLASMA INTERACTION." Mathematical Models and Methods in Applied Sciences 16, no. 01 (January 2006): 19–57. http://dx.doi.org/10.1142/s0218202506001042.

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We analyze a reduced 1D Vlasov–Maxwell system introduced recently in the physical literature for studying laser-plasma interaction. This system can be seen as a standard Vlasov equation in which the field is split into two terms: an electrostatic field obtained from Poisson's equation and a vector potential term satisfying a nonlinear wave equation. Both nonlinearities in the Poisson and wave equations are due to the coupling with the Vlasov equation through the charge density. We show global existence of weak solutions in the nonrelativistic case, and global existence of characteristic solutions in the quasi-relativistic case. Moreover, these solutions are uniquely characterized as fixed points of a certain operator. We also find a global energy functional for the system allowing us to obtain Lp-nonlinear stability of some particular equilibria in the periodic setting.

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14

Aranda, Carlos Cesar. "ON THE POISSON'S EQUATION -△u = ∞." Cubo (Temuco) 15, no. 1 (March 2013): 151–58. http://dx.doi.org/10.4067/s0719-06462013000100010.

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15

Lussardi, Luca. "On a Poisson's equation arising from magnetism." Discrete & Continuous Dynamical Systems - S 8, no. 4 (2015): 769–72. http://dx.doi.org/10.3934/dcdss.2015.8.769.

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16

Delaurentis, J. M., and L. A. Romero. "A Monte Carlo method for poisson's equation." Journal of Computational Physics 90, no. 1 (September 1990): 123–40. http://dx.doi.org/10.1016/0021-9991(90)90199-b.

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17

Weatherford *, Charles, Eddie Red, and Philip Hoggan. "Solution of Poisson's equation using spectral forms." Molecular Physics 103, no. 15-16 (August 10, 2005): 2169–72. http://dx.doi.org/10.1080/00268970500137261.

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18

Earn, David J. D. "Potential-Density Basis Sets for Three-Dimensional Disks." Symposium - International Astronomical Union 169 (1996): 509–10. http://dx.doi.org/10.1017/s0074180900230192.

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Poisson's equation, ∇2ψ = 4πGρ, can be solved approximately using basis sets of potential-density pairs. A given density is approximated by a truncated expansion; the computed expansion coefficients immediately yield the corresponding potential since each basis density function is paired with a basis potential function, and Poisson's equation is linear.

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19

NAITO, Takahiro, Toru TAKAHASHI, and Toshiro MATSUMOTO. "1706 BOUNDARY ELEMENT METHOD FOR POISSON EQUATION BASED ON THE FUNDAMENTAL SOLUTION OF SIMULTANEOUS POISSON'S EQUATIONS." Proceedings of The Computational Mechanics Conference 2007.20 (2007): 305–6. http://dx.doi.org/10.1299/jsmecmd.2007.20.305.

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20

Roulstone, I., B. Banos, J. D. Gibbon, and V. N. Roubtsov. "A geometric interpretation of coherent structures in Navier–Stokes flows." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, no. 2107 (April 2009): 2015–21. http://dx.doi.org/10.1098/rspa.2008.0483.

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The pressure in the incompressible three-dimensional Navier–Stokes and Euler equations is governed by Poisson's equation: this equation is studied using the geometry of three-forms in six dimensions. By studying the linear algebra of the vector space of three-forms Λ 3 W * where W is a six-dimensional real vector space, we relate the characterization of non-degenerate elements of Λ 3 W * to the sign of the Laplacian of the pressure—and hence to the balance between the vorticity and the rate of strain. When the Laplacian of the pressure, Δ p , satisfies Δ p >0, the three-form associated with Poisson's equation is the real part of a decomposable complex form and an almost-complex structure can be identified. When Δ p <0, a real decomposable structure is identified. These results are discussed in the context of coherent structures in turbulence.

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21

Ma, Zu-Hui, Weng Cho Chew, and Li Jun Jiang. "A Novel Efficient Numerical Solution of Poisson's Equation for Arbitrary Shapes in Two Dimensions." Communications in Computational Physics 20, no. 5 (November 2016): 1381–404. http://dx.doi.org/10.4208/cicp.230813.291113a.

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AbstractEven though there are various fast methods and preconditioning techniques available for the simulation of Poisson problems, little work has been done for solving Poisson's equation by using the Helmholtz decomposition scheme. To bridge this issue, we propose a novel efficient algorithm to solve Poisson's equation in irregular two dimensional domains for electrostatics through a quasi-Helmholtz decomposition technique—the loop-tree basis decomposition. It can handle Dirichlet, Neumann or mixed boundary problems in which the filling media can be homogeneous or inhomogeneous. A novel point of this method is to first find the electric flux efficiently by applying the loop-tree basis functions. Subsequently, the potential is obtained by finding the inverse of the gradient operator. Furthermore, treatments for both Dirichlet and Neumann boundary conditions are addressed. Finally, the validation and efficiency are illustrated by several numerical examples. Through these simulations, it is observed that the computational complexity of our proposed method almost scales as , where N is the triangle patch number of meshes. Consequently, this new algorithm is a feasible fast Poisson solver.

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22

Ábrahámová, Andrea, and Margita Vajsáblová. "A Comparison of Variational Projection and Cartographic Projection by Ritz’s Method." Slovak Journal of Civil Engineering 30, no. 2 (June 1, 2022): 22–29. http://dx.doi.org/10.2478/sjce-2022-0011.

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Abstract The application of alternative mathematical methods in creating cartographic projections is an interesting factor, which affects the optimization of distortions and their distribution in the projected territory. This article presents the methodology for the creation and comparison of conformal cartographic projections formed by alternative mathematical methods of minimizing the integral criterion for scale distortion in Slovakia. The creation of the variational projection is based on the Airy-Kavraiskii criterion of evaluating the projection on the displayed area by solving Laplace's equation. The second projection is created by solving Poisson's equation using Ritz's method. Our analysis showed that the variational projection of Slovakia achieves more satisfactory distortion values than the cartographic projection created using Ritz's method. The advantage of Ritz's method is that it is possible to choose a boundary condition for a predefined undistorted convex closed curve. In this paper, we have also derived specific members of the map equations for cartographic projection based on solving Poisson's equation by Ritz's method.

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23

Anderson, Michael L., Andrew P. Bassom, and Neville Fowkes. "Boundary tracing and boundary value problems: II. Applications." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 463, no. 2084 (June 5, 2007): 1925–38. http://dx.doi.org/10.1098/rspa.2007.1859.

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This is the second of a pair of papers describing the use of boundary tracing for boundary value problems. In the preceding article, the theory of the technique was explained and it was shown how it enables one to use known exact solutions of partial differential equations to generate new solutions. Here, we illustrate the use of the technique by applying it to three equations of practical significance: Helmholtz's equation, Poisson's equation and the nonlinear constant mean curvature equation. A variety of new solutions are obtained and the potential of the technique for further application outlined.

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24

Khokhlov, A. V. "NON-MONOTONICITY, SIGN CHANGES AND OTHER FEATURES OF POISSON'S RATIO EVOLUTION FOR ISOTROPIC LINEAR VISCOELASTIC MATERIALS UNDER TENSION AT CONSTANT STRESS RATES." Problems of strenght and plasticity 81, no. 3 (2019): 271–91. http://dx.doi.org/10.32326/1814-9146-2019-81-3-271-291.

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We study analytically the Boltzmann - Volterra linear constitutive equation for isotropic non-aging viscoelastic media in order to elucidate its capabilities to provide a qualitative simulation of rheological phenomena related to different types of evolution of triaxial strain state and of the lateral contraction ratio (the Poisson ratio) observed in uni-axial tests of viscoelastic materials under tension or compression at constant stress rate. In particular, we consider such effects as increasing, decreasing or non-monotone dependences of lateral strain and Poisson's ratio on time, sign changes and negativity of Poisson's ratio (auxeticity effect) and its stabilization at large times. The viscoelasticity equation implies that the hydrostatic and deviatoric parts of stress and strain tensors don't depend on each other. It is governed by two material functions of a positive real argument (that is shear and bulk creep compliances). Assuming both creep compliances are arbitrary positive, differentiable, increasing and convex up functions on time semi-axis, we analyze general expressions for the Poisson ratio and strain triaxiality ratio (which is equal to volumetric strain divided by deviatoric strain) generated by the viscoelasticity relation under uni-axial tension or compression. We investigate qualitative properties and peculiarities of their evolution in time and their dependences on material functions characteristics. We obtain the universal accurate two-sided bound for the Poisson ratio range and criteria for the Poisson ratio increase or decrease and for extrema existence. We derive necessary and sufficient restrictions on shear and bulk creep compliances providing sign changes of the Poisson ratio and negative values of Poisson's ratio on some interval of time. The properties of the Poisson ratio under tension at constant stress rates found in the study we compare to properties the Poisson ratio evolution under constant stress (in virtual creep tests) and illustrate them using popular classical and fractal models with shear and bulk creep functions each one controlled by three parameters. The analysis carried out let us to conclude that the linear viscoelasticity theory (supplied with common creep functions which are non-exotic from any point of view) is able to simulate qualitatively the main effects associated with different types of the Poisson ratio evolution under tension or compression at constant stress rate except for dependence of Poisson's ratio on stress rate. It is proved that the linear theory can reproduce increasing, decreasing or non-monotone and convex up or down dependences of lateral strain and Poisson's ratio on time and it can provide existence of minimum, maximum or inflection points and sign changes from minus to plus and vice versa and asymptotic stabilization at large times.

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25

Darmaev M.V., Sangadiev S.Sh., Mashanov A.A., and Kim T.B. "On the derivation of the Belomestnykh--Tesleva's formula." Physics of the Solid State 64, no. 8 (2022): 1079. http://dx.doi.org/10.21883/pss.2022.08.54629.327.

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The Belomestnykh--Tesleva's formula is interesting in that it establishes an unambiguous relationship between the Poisson's ratio and the Gruneisen's parameter. The derivation of this formula from the generally accepted Gruneisen's equation is discussed. The Belomestnykh--Tesleva's formula, obtained earlier from other assumptions, is derived using the theory of elasticity and the Leontiev's equation. For a number of silicate glasses and glassy metaphosphates of alkaline earth metals, the proposed approach finds a fairly satisfactory agreement with the experimental data. Keywords: Gruneisen's equation, Gruneisen's parameter, Poisson's ratio, shear modulus, isothermal bulk modulus, acoustic wave velocities.

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26

OKITA, Shunsuke, Hiroshi OKUDA, and Genki YAGAWA. "Asynchronous Analysis of Poisson's Equation using Neural Network." Proceedings of the JSME annual meeting 2002.1 (2002): 33–34. http://dx.doi.org/10.1299/jsmemecjo.2002.1.0_33.

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27

Sadiku, M. N. O., S. O. Ajose, and Zhibao Fu. "Applying the Exodus method to solve Poisson's equation." IEEE Transactions on Microwave Theory and Techniques 42, no. 4 (April 1994): 661–66. http://dx.doi.org/10.1109/22.285073.

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28

Abushama, Abeer Ali, and Bernard Bialecki. "Modified Nodal Cubic Spline Collocation For Poisson's Equation." SIAM Journal on Numerical Analysis 46, no. 1 (January 2008): 397–418. http://dx.doi.org/10.1137/050629033.

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29

Alotto, Piergiorgio, Fabio Freschi, and Maurizio Repetto. "Algebraic Second Order Hodge Operator for Poisson's Equation." IEEE Transactions on Magnetics 49, no. 5 (May 2013): 1761–64. http://dx.doi.org/10.1109/tmag.2013.2241406.

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30

Nikoshkinen, K. I., and I. V. Lindell. "Image solution for Poisson's equation in wedge geometry." IEEE Transactions on Antennas and Propagation 43, no. 2 (1995): 179–87. http://dx.doi.org/10.1109/8.366380.

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31

Hoyles, Matthew, Serdar Kuyucak, and Shin-Ho Chung. "Solutions of Poisson's equation in channel-like geometries." Computer Physics Communications 115, no. 1 (December 1998): 45–68. http://dx.doi.org/10.1016/s0010-4655(98)00090-3.

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32

Kourkoutas, C. D., and G. J. Papaioannou. "A fast solution of poisson's equation in FETs." Solid-State Electronics 37, no. 2 (February 1994): 373–76. http://dx.doi.org/10.1016/0038-1101(94)90090-6.

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33

Lakhani, G. "Enhancing Poisson's Equation-Based Approach for DCT Prediction." IEEE Transactions on Image Processing 17, no. 3 (March 2008): 427–30. http://dx.doi.org/10.1109/tip.2007.915560.

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34

Alotto, Piergiorgio, and Fabio Freschi. "A Second-Order Cell Method for Poisson's Equation." IEEE Transactions on Magnetics 47, no. 5 (May 2011): 1430–33. http://dx.doi.org/10.1109/tmag.2010.2092419.

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35

Golberg, Michael A. "The method of fundamental solutions for Poisson's equation." Engineering Analysis with Boundary Elements 16, no. 3 (October 1995): 205–13. http://dx.doi.org/10.1016/0955-7997(95)00062-3.

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36

Garg, Rahul, and Daniel Spector. "On the regularity of solutions to Poisson's equation." Comptes Rendus Mathematique 353, no. 9 (September 2015): 819–23. http://dx.doi.org/10.1016/j.crma.2015.07.001.

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37

Plagne, Laurent, and Jean-Yves Berthou. "Tensorial Basis Spline Collocation Method for Poisson's Equation." Journal of Computational Physics 157, no. 2 (January 2000): 419–40. http://dx.doi.org/10.1006/jcph.1999.6338.

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38

Wang, Xiantao, and Jian-Feng Zhu. "Boundary Schwarz lemma for solutions to Poisson's equation." Journal of Mathematical Analysis and Applications 463, no. 2 (July 2018): 623–33. http://dx.doi.org/10.1016/j.jmaa.2018.03.043.

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39

Shahri, Mojtaba P., and Stefan Z. Miska. "In-Situ Poisson's Ratio Determination From Interference Transient Well Test." SPE Journal 20, no. 05 (October 20, 2015): 1041–52. http://dx.doi.org/10.2118/166074-pa.

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Summary Poisson's ratio is usually determined with well logging, fracturing data, and core samples. However, these methods provide us with a Poisson's ratio that is representative of only near-wellbore regions. In this paper, a technique is proposed by extending currently used pressure-transient-testing concepts to include reservoir stresses. More specifically, the interference well test is generalized to find not only conventional flow parameters such as reservoir transmissivity and storage capacity, but also the average in-situ Poisson's ratio. This is accomplished with the generalized diffusivity equation, which takes into account flow-induced stress changes. First, a generalized diffusivity equation is formulated by considering a deformable porous medium. The main goal of the generalized diffusivity equation is to extend current well-testing methods to include both fluid-flow and rock-mechanics aspects, and to present a way to determine the rock-mechanics-related property, Poisson's ratio, from the interference-well test. The line-source solution to the diffusivity equation is used to modify the current interference well-test technique. A synthetic example is presented to show the main steps of the proposed transient well-testing analysis technique. In addition, application of the proposed method is illustrated with interference-well-test field data. With a Monte Carlo simulation, effects of uncertainty in the input data on the prediction of Poisson's ratio are investigated, as well. In addition, a coupled fluid-flow/geomechanical simulation is performed to show the validity of the proposed formulation and corresponding improvement over the current analytical approach. One can put in practice an average in-situ value in different applications requiring accurate value of Poisson's ratio on the reservoir scale. Some examples of these include in-situ-stress-field determination, stress distribution and rock-mass deformation, and the next generation of coupled fluid-flow/geomechanical simulators. By use of Poisson's ratio that could capture flow-induced stress changes, we would be able to find the stress distribution caused by production/injection within the reservoir more precisely as well.

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40

Nakamura, Y., and I. Tsukabayashi. "Modified Korteweg—de Vries ion-acoustic solitons in a plasma." Journal of Plasma Physics 34, no. 3 (December 1985): 401–15. http://dx.doi.org/10.1017/s0022377800002968.

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Propagation of nonlinear ion-acoustic waves in a multi-component plasma with negative ions is investigated experimentally. At a critical concentration of negative ions, both compressive and rarefactive solitons are observed. The velocities and widths of the solitons are measured and compared with the soliton solutions of the modified Korteweg–de Vries equation and of the pseudopotential method. The modified Korteweg–de Vries equation is solved numerically to investigate overtaking collisions of a positive and a negative soliton. Fluid equations together with Poisson's equation are numerically integrated to simulate their head-on collisions.

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41

Knackstedt, Mark A., Christoph H. Arns, and W. Val Pinczewski. "Velocity-porosity relationships, 1: Accurate velocity model for clean consolidated sandstones, GEOPHYSICS, 68, 1822–1834." GEOPHYSICS 71, no. 2 (March 2006): Y3. http://dx.doi.org/10.1190/1.2191109.

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42

Edelson, Allan L. "Asymptotic Properties of Semilinear Equations." Canadian Mathematical Bulletin 32, no. 1 (March 1, 1989): 34–46. http://dx.doi.org/10.4153/cmb-1989-006-3.

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AbstractWe study the asymptotic properties of positive solutions to the semilinear equation — Δu = f(x, u). Existence and asymptotic estimates are obtained for solutions in exterior domains, as well as entire solutions, for n ≧ 2. The study uses integral operator equations in Rn, and convergence theorems for solutions of Poisson's equation in bounded domains. A consequence of the method is that more precise estimates can be obtained for the growth of solutions at infinity, than have been obtained by other methods. As a special case the results are applied to the generalized Emden-Fowler equation — Δu = p(x)uγ, for γ > 0

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43

Li, Hu, and Jin Huang. "A High-Accuracy Mechanical Quadrature Method for Solving the Axisymmetric Poisson's Equation." Advances in Applied Mathematics and Mechanics 9, no. 2 (January 9, 2017): 393–406. http://dx.doi.org/10.4208/aamm.2015.m1287.

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AbstractIn this article, we consider the numerical solution for Poisson's equation in axisymmetric geometry. When the boundary condition and source term are axisymmetric, the problem reduces to solving Poisson's equation in cylindrical coordinates in the two-dimensional (r,z) region of the original three-dimensional domain S. Hence, the original boundary value problem is reduced to a two-dimensional one. To make use of the Mechanical quadrature method (MQM), it is necessary to calculate a particular solution, which can be subtracted off, so that MQM can be used to solve the resulting Laplace problem, which possesses high accuracy order and low computing complexities. Moreover, the multivariate asymptotic error expansion of MQM accompanied with for all mesh widths hi is got. Hence, once discrete equations with coarse meshes are solved in parallel, the higher accuracy order of numerical approximations can be at least by the splitting extrapolation algorithm (SEA). Meanwhile, a posteriori asymptotic error estimate is derived, which can be used to construct self-adaptive algorithms. The numerical examples support our theoretical analysis.

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44

Fonf, Vladimir P., Michael Lin, and Przemysław Wojtaszczyk. "Poisson's equation and characterizations of reflexivity of Banach spaces." Colloquium Mathematicum 124, no. 2 (2011): 225–35. http://dx.doi.org/10.4064/cm124-2-7.

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45

Glynn, Peter W. "Poisson's equation for the recurrent M/G/1 queue." Advances in Applied Probability 26, no. 4 (December 1994): 1044–62. http://dx.doi.org/10.2307/1427904.

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This paper shows how to calculate solutions to Poisson's equation for the waiting time sequence of the recurrent M/G/l queue. The solutions are used to construct martingales that permit us to study additive functionals associated with the waiting time sequence. These martingales provide asymptotic expressions, for the mean of additive functionals, that reflect dependence on the initial state of the process. In addition, we show how to explicitly calculate the scaling constants that appear in the central limit theorems for additive functionals of the waiting time sequence.

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46

ATKINSON, K. E. "The Numerical Evaluation of Particular Solutions for Poisson's Equation." IMA Journal of Numerical Analysis 5, no. 3 (1985): 319–38. http://dx.doi.org/10.1093/imanum/5.3.319.

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Evans, D. J. "The solution of poisson's equation in a triangular region." International Journal of Computer Mathematics 39, no. 1-2 (January 1991): 81–98. http://dx.doi.org/10.1080/00207169108803981.

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Evans, D. J. "The numerical solution of poisson's equation in a rhombus." International Journal of Computer Mathematics 42, no. 3-4 (January 1992): 193–211. http://dx.doi.org/10.1080/00207169208804062.

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Ushakov, A. L. "FAST SOLUTION OF THE MODEL PROBLEM FOR POISSON'S EQUATION." Bulletin of the South Ural State University series "Mathematics. Mechanics. Physics" 9, no. 4 (2017): 36–42. http://dx.doi.org/10.14529/mmph170405.

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Glynn, Peter W. "Poisson's equation for the recurrent M/G/1 queue." Advances in Applied Probability 26, no. 04 (December 1994): 1044–62. http://dx.doi.org/10.1017/s0001867800026756.

Full text

Abstract:

This paper shows how to calculate solutions to Poisson's equation for the waiting time sequence of the recurrent M/G/l queue. The solutions are used to construct martingales that permit us to study additive functionals associated with the waiting time sequence. These martingales provide asymptotic expressions, for the mean of additive functionals, that reflect dependence on the initial state of the process. In addition, we show how to explicitly calculate the scaling constants that appear in the central limit theorems for additive functionals of the waiting time sequence.

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