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Journal articles on the topic 'Petrov-Galerkin'

Author: Grafiati

Published: 4 June 2021

Last updated: 16 February 2022

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1

Wang, Kai, Shen Jie Zhou, and Zhi Feng Nie. "Development of the Smoothed Natural Neighbor Petrov-Galerkin Method." Applied Mechanics and Materials 201-202 (October 2012): 198–201. http://dx.doi.org/10.4028/www.scientific.net/amm.201-202.198.

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The strain smoothing technique is employed in the natural neighbor Petrov-Galerkin method (NNPG), and the so-called smoothed natural neighbor Petrov-Galerkin method is proposed and studied. This method inherits the advantages of the generalized MLPG method and possesses the easy imposition of essential boundary condition and the domain integration is completely avoided. In comparison with the traditional NNPG, the smoothed natural neighbor Petrov-Galerkin method can obtained more stable and accurate result without increasing the computational cost.

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2

Chen, Zhongying, and Yuesheng Xu. "The Petrov--Galerkin and Iterated Petrov--Galerkin Methods for Second-Kind Integral Equations." SIAM Journal on Numerical Analysis 35, no. 1 (February 1998): 406–34. http://dx.doi.org/10.1137/s0036142996297217.

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3

Heuer, Norbert, and Michael Karkulik. "Discontinuous Petrov–Galerkin boundary elements." Numerische Mathematik 135, no. 4 (July 29, 2016): 1011–43. http://dx.doi.org/10.1007/s00211-016-0824-z.

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4

Carstensen, C., P. Bringmann, F. Hellwig, and P. Wriggers. "Nonlinear discontinuous Petrov–Galerkin methods." Numerische Mathematik 139, no. 3 (March 6, 2018): 529–61. http://dx.doi.org/10.1007/s00211-018-0947-5.

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5

Donea, J. "Generalized Galerkin Methods for Convection Dominated Transport Phenomena." Applied Mechanics Reviews 44, no. 5 (May 1, 1991): 205–14. http://dx.doi.org/10.1115/1.3119502.

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A brief survey is made of recent advances in the development of finite element methods for convection dominated transport phenomena. Because of the nonsymmetric character of convection operators, the standard Galerkin formulation of the method of weighted residuals does not possess optimal approximation properties in application to problems in this class. As a result, numerical solutions are often corrupted by spurious node-to-node oscillations. For steady problems describing convection and diffusion, spurious oscillations can be precluded by the use of upwind-type finite element approximations that are constructed through a proper Petrov-Galerkin weighted residual formulation. Various upwind finite element formulations are reviewed in this paper, with a special emphasis on the major breakthroughs represented by the so-called streamline upwind Petrov-Galerkin and Galerkin least-squares methods. The second part of the paper is devoted to a review of time-accurate finite element methods recently developed for the solution of unsteady problems governed by first-order hyperbolic equations. This includes Petrov-Galerkin, Taylor-Galerkin, least-squares, and various characteristic Galerkin methods. The extension of these methods to deal with unsteady convection-diffusion problems is also considered.

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6

Liu, Xiaowei, and Jin Zhang. "Comparison of SUPG with Bubble Stabilization Parameters and the Standard SUPG." Abstract and Applied Analysis 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/364675.

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We study a streamline upwind Petrov-Galerkin method (SUPG) with bubble stabilization coefficients on quasiuniform triangular meshes. The new algorithm is a consistent Petrov-Galerkin method and shows similar numerical performances as the standard SUPG when the mesh Péclet number is greater than 1. Relationship between the new algorithm and the standard SUPG will be explored. Numerical experiments support these results.

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7

Galanin, M., and E. Savenkov. "FEDORENKO FINITE SUPERELEMENT METHOD AS SPECIAL GALERKIN APPROXIMATION." Mathematical Modelling and Analysis 7, no. 1 (June 30, 2002): 41–50. http://dx.doi.org/10.3846/13926292.2002.9637176.

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In this work we introduce variational equation which natural Petrov‐Galerkin approximation leads to Fedorenko Finite Superelement Method (FSEM). FSEM is considered as Petrov‐Galerkin approximation of the certain problem for traces of boundary‐value problem solution at the boundaries of some subdomains (superelements). Iterative methods of solution of the same problem are well known domain decomposition methods. Some numerical results are presented.

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8

Morton, K. W. "The convection-diffusion Petrov-Galerkin story." IMA Journal of Numerical Analysis 30, no. 1 (June 23, 2009): 231–40. http://dx.doi.org/10.1093/imanum/drp002.

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9

Grigorieff, R. D., and I. H. Solan. "Spline petrov-galerkin methods with quadrature." Numerical Functional Analysis and Optimization 17, no. 7-8 (January 1996): 755–84. http://dx.doi.org/10.1080/01630569608816723.

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10

Bringmann, P., C. Carstensen, D. Gallistl, F. Hellwig, D. Peterseim, S. Puttkammer, H. Rabus, and J. Storn. "Towards adaptive discontinuous Petrov-Galerkin methods." PAMM 16, no. 1 (October 2016): 741–42. http://dx.doi.org/10.1002/pamm.201610359.

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11

Ganjoo, D. K., and T. E. Tezduyar. "Petrov-Galerkin formulations for electrochemical processes." Computer Methods in Applied Mechanics and Engineering 65, no. 1 (November 1987): 61–83. http://dx.doi.org/10.1016/0045-7825(87)90183-6.

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12

Bassett, Brody, and Brian Kiedrowski. "Meshless local Petrov–Galerkin solution of the neutron transport equation with streamline-upwind Petrov–Galerkin stabilization." Journal of Computational Physics 377 (January 2019): 1–59. http://dx.doi.org/10.1016/j.jcp.2018.10.028.

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13

Führer, Thomas, Norbert Heuer, Michael Karkulik, and Rodolfo Rodríguez. "Combining the DPG Method with Finite Elements." Computational Methods in Applied Mathematics 18, no. 4 (October 1, 2018): 639–52. http://dx.doi.org/10.1515/cmam-2017-0041.

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AbstractWe propose and analyze a discretization scheme that combines the discontinuous Petrov–Galerkin and finite element methods. The underlying model problem is of general diffusion-advection-reaction type on bounded domains, with decomposition into two sub-domains. We propose a heterogeneous variational formulation that is of the ultra-weak (Petrov–Galerkin) form with broken test space in one part, and of Bubnov–Galerkin form in the other. A standard discretization with conforming approximation spaces and appropriate test spaces (optimal test functions for the ultra-weak part and standard test functions for the Bubnov–Galerkin part) gives rise to a coupled DPG-FEM scheme. We prove its well-posedness and quasi-optimal convergence. Numerical results confirm expected convergence orders.

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14

Wang, Li, W. Kyle Anderson, J. Taylor Erwin, and Sagar Kapadia. "Discontinuous Galerkin and Petrov Galerkin methods for compressible viscous flows." Computers & Fluids 100 (September 2014): 13–29. http://dx.doi.org/10.1016/j.compfluid.2014.04.035.

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15

Yan, B., H. Zhou, and D. Li. "Numerical simulation of the filling stage for plastic injection moulding based on the Petrov-Galerkin methods." Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture 221, no. 10 (October 1, 2007): 1573–77. http://dx.doi.org/10.1243/09544054jem823sc.

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Simulation of plastic injection moulding is carried out essentially to solve non-isothermal, viscous, incompressible, and non-Newtonian convection-diffusion flow equations. However, finite element methods present spurious numerical oscillations, which give a failure result associated with the classical Galerkin formulations of viscous incompressible Navier-Stokes equations. The streamline-upwind/Petrov-Galerkin (SUPG) and pressure-stabilizing/Petrov-Galerkin (PSPG) formulations were employed to prevent these potential numerical instabilities by adding to the weighting functions with their derivatives, thus resulting in stabilized finite element formulations using equal-order interpolation functions for velocity and pressure. Numerical experiments showed that the numerical algorithms developed perform in a stable way and give accurate results compared with the well-known commercial software Moldflow.

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16

Barry, W. "A Wachspress Meshless Local Petrov–Galerkin method." Engineering Analysis with Boundary Elements 28, no. 5 (May 2004): 509–23. http://dx.doi.org/10.1016/s0955-7997(03)00104-8.

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17

Simoncini, Valeria, and Daniel B. Szyld. "Interpreting IDR as a Petrov–Galerkin Method." SIAM Journal on Scientific Computing 32, no. 4 (January 2010): 1898–912. http://dx.doi.org/10.1137/090774756.

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18

Gallistl, Dietmar, Daniel Peterseim, and Carsten Carstensen. "Multiscale Petrov-Galerkin FEM for Acoustic Scattering." PAMM 16, no. 1 (October 2016): 745–46. http://dx.doi.org/10.1002/pamm.201610361.

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19

Mashat, D. S., L. A. Wazzan, and M. S. Ismail. "Petrov–Galerkin method andK(2, 2) equation." International Journal of Computer Mathematics 83, no. 3 (March 2006): 331–43. http://dx.doi.org/10.1080/00207160600747938.

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20

Ghorbany, Mehrzad, and Ali Reza Soheili. "Moving element free petrov‐galerkin viscous method." Journal of the Chinese Institute of Engineers 27, no. 4 (June 2004): 473–79. http://dx.doi.org/10.1080/02533839.2004.9670897.

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21

Gamba, Irene M., and Sergej Rjasanow. "Galerkin–Petrov approach for the Boltzmann equation." Journal of Computational Physics 366 (August 2018): 341–65. http://dx.doi.org/10.1016/j.jcp.2018.04.017.

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22

Stern, Ari. "Banach space projections and Petrov–Galerkin estimates." Numerische Mathematik 130, no. 1 (July 17, 2014): 125–33. http://dx.doi.org/10.1007/s00211-014-0658-5.

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23

Elfverson, Daniel, Victor Ginting, and Patrick Henning. "On multiscale methods in Petrov–Galerkin formulation." Numerische Mathematik 131, no. 4 (January 11, 2015): 643–82. http://dx.doi.org/10.1007/s00211-015-0703-z.

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24

Dubois, Francois, Isabelle Greff, and Charles Pierre. "Raviart–Thomas finite elements of Petrov–Galerkin type." ESAIM: Mathematical Modelling and Numerical Analysis 53, no. 5 (August 6, 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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25

KAMITANI, Atsushi, Teruou TAKAYAMA, Taku ITOH, and Hiroaki NAKAMURA. "Extension of Meshless Galerkin/Petrov-Galerkin Approach without Using Lagrange Multipliers." Plasma and Fusion Research 6 (2011): 2401074. http://dx.doi.org/10.1585/pfr.6.2401074.

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26

Carlberg, Kevin, Matthew Barone, and Harbir Antil. "Galerkin v. least-squares Petrov–Galerkin projection in nonlinear model reduction." Journal of Computational Physics 330 (February 2017): 693–734. http://dx.doi.org/10.1016/j.jcp.2016.10.033.

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27

Akin, J. E., T. Tezduyar, M. Ungor, and S. Mittal. "Stabilization Parameters and Smagorinsky Turbulence Model." Journal of Applied Mechanics 70, no. 1 (January 1, 2003): 2–9. http://dx.doi.org/10.1115/1.1526569.

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For the streamline-upwind/Petrov-Galerkin and pressure-stabilizing/Petrov-Galerkin formulations for flow problems, we present in this paper a comparative study of the stabilization parameters defined in different ways. The stabilization parameters are closely related to the local length scales (“element length”), and our comparisons include parameters defined based on the element-level matrices and vectors, some earlier definitions of element lengths, and extensions of these to higher-order elements. We also compare the numerical viscosities generated by these stabilized formulations with the eddy viscosity associated with a Smagorinsky turbulence model that is based on element length scales.

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28

Hou, Dianming, Mohammad Tanzil Hasan, and Chuanju Xu. "Müntz Spectral Methods for the Time-Fractional Diffusion Equation." Computational Methods in Applied Mathematics 18, no. 1 (January 1, 2018): 43–62. http://dx.doi.org/10.1515/cmam-2017-0027.

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AbstractIn this paper, we propose and analyze a fractional spectral method for the time-fractional diffusion equation (TFDE). The main novelty of the method is approximating the solution by using a new class of generalized fractional Jacobi polynomials (GFJPs), also known as Müntz polynomials. We construct two efficient schemes using GFJPs for TFDE: one is based on the Galerkin formulation and the other on the Petrov–Galerkin formulation. Our theoretical or numerical investigation shows that both schemes are exponentially convergent for general right-hand side functions, even though the exact solution has very limited regularity (less than {H^{1}}). More precisely, an error estimate for the Galerkin-based approach is derived to demonstrate its spectral accuracy, which is then confirmed by numerical experiments. The spectral accuracy of the Petrov–Galerkin-based approach is only verified by numerical tests without theoretical justification. Implementation details are provided for both schemes, together with a series of numerical examples to show the efficiency of the proposed methods.

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29

Sheikhi, Nasrin, Mohammad Najafi, and Vali Enjilela. "Extending the Meshless Local Petrov–Galerkin Method to Solve Stabilized Turbulent Fluid Flow Problems." International Journal of Computational Methods 16, no. 01 (November 21, 2018): 1850086. http://dx.doi.org/10.1142/s021987621850086x.

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The aim of this paper is to extend the meshless local Petrov–Galerkin method to solve stabilized turbulent fluid flow problems. For the unsteady incompressible turbulent fluid flow problems, the Spalart–Allmaras model is used to stabilize the governing equations, and the meshless local Petrov–Galerkin method is extended based on the vorticity-stream function to solve the turbulent flow problems. In this study, the moving least squares scheme interpolates the field variables. The proposed method solves three standard test cases of the turbulent flow over a flat plate, turbulent flow through a channel, and turbulent flow over a backward-facing step for evaluation of the method’s capability, accuracy, and validity purposes. Based on the comparison of the three test cases results with those of the experimental and conventional numerical works available in the literature, the proposed method shows to be accurate and quite implemental. The new extended method in this study together with the previously published works of the authors (on extending the meshless local Petrov–Galerkin method to solve laminar flow problems) now, for the first time, empower the meshless method to solve both laminar and turbulent flow problems.

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30

Garijo, D., ÓF Valencia, FJ Gómez-Escalonilla, and J. López Díez. "Bernstein–Galerkin approach in elastostatics." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 228, no. 3 (April 24, 2013): 391–404. http://dx.doi.org/10.1177/0954406213486733.

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Since 1994, two main meshless methods have been developed and widely used: these are the element free Galerkin method and the meshless local Petrov-Galerkin method. Both methods solve partial differential equations by posing a numerical approximation to the solution using the moving least squares technique. Using Bernstein polynomials as the shape functions of Galerkin weak form-based methods improves the numerical approximation achieved at boundaries without losing accuracy inside the domain.

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31

Duarte, Julio, and Francisco Reyes. "Soluciones Numéricas para la Ecuación KdV Usando el MétodoWavelet-Petrov-Galerkin." Selecciones Matemáticas 6, no. 2 (December 30, 2019): 148–55. http://dx.doi.org/10.17268/sel.mat.2019.02.02.

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32

Dalík, Josef. "A Petrov-Galerkin approximation of convection-diffusion and reaction-diffusion problems." Applications of Mathematics 36, no. 5 (1991): 329–54. http://dx.doi.org/10.21136/am.1991.104471.

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33

Palitta, Davide, and Valeria Simoncini. "Optimality Properties of Galerkin and Petrov–Galerkin Methods for Linear Matrix Equations." Vietnam Journal of Mathematics 48, no. 4 (March 5, 2020): 791–807. http://dx.doi.org/10.1007/s10013-020-00390-7.

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34

Xia, Ping, and Ke Xiang Wei. "Application of the Meshless Local Petrov-Galerkin Method in the Elasto-Plastic Fracture Problem of Moderately Thick Plate." Applied Mechanics and Materials 155-156 (February 2012): 485–90. http://dx.doi.org/10.4028/www.scientific.net/amm.155-156.485.

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A meshless local Petrov-Galerkin method for the analysis of the elasto-plastic fracture problem of the moderately thick plate is presented. The discretized system equations of the moderately thick plate are obtained using a locally weighted residual method. It uses a radial basis function coupled with a polynomial basis function as a trial function, and uses the quartic spline function as a test function of the weighted residual method. An incremental Newton-Raphson iterative algorithm is employed to solve incremental nonlinear local Petrov-Galerkin equations. Numerical results show that the present method possesses not only feasibility, but also rapid convergence for the elasto-plastic fracture problem of the moderately thick plate.

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35

Chuong, Nguyen Minh, and Bui Kien Cuong. "The convergence estimates for Galerkin-wavelet solution of periodic pseudodifferential initial value problems." International Journal of Mathematics and Mathematical Sciences 2003, no. 14 (2003): 857–67. http://dx.doi.org/10.1155/s0161171203203100.

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36

Roshan, Thoudam. "A Petrov–Galerkin method for equal width equation." Applied Mathematics and Computation 218, no. 6 (November 2011): 2730–39. http://dx.doi.org/10.1016/j.amc.2011.08.013.

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37

Hossain, M. Akram, and Michael E. Barber. "Optimized Petrov–Galerkin model for advective–dispersive transport." Applied Mathematics and Computation 115, no. 1 (October 2000): 1–10. http://dx.doi.org/10.1016/s0096-3003(99)00127-7.

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38

Bottasso, Carlo L., Stefano Micheletti, and Riccardo Sacco. "The discontinuous Petrov–Galerkin method for elliptic problems." Computer Methods in Applied Mechanics and Engineering 191, no. 31 (May 2002): 3391–409. http://dx.doi.org/10.1016/s0045-7825(02)00254-2.

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39

Wang, Yuan-ming, and Ben-yu Guo. "Petrov–Galerkin methods for nonlinear systems without monotonicity." Applied Numerical Mathematics 36, no. 1 (January 2001): 57–78. http://dx.doi.org/10.1016/s0168-9274(99)00054-9.

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40

Correa, Bruno C., Renato C. Mesquita, and Lucas P. Amorim. "CUDA Approach for Meshless Local Petrov–Galerkin Method." IEEE Transactions on Magnetics 51, no. 3 (March 2015): 1–4. http://dx.doi.org/10.1109/tmag.2014.2359213.

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41

Böttcher, Albrecht, and Hartmut Wolf. "Galerkin–Petrov Methods for Bergman Space Toeplitz Operators." SIAM Journal on Numerical Analysis 30, no. 3 (June 1993): 846–63. http://dx.doi.org/10.1137/0730043.

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42

Freund, J., and E. M. Salonen. "A logic for simple Petrov-Galerkin weighting functions." International Journal for Numerical Methods in Engineering 34, no. 3 (May 15, 1992): 805–22. http://dx.doi.org/10.1002/nme.1620340308.

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43

Soares, D., V. Sladek, and J. Sladek. "Modified meshless local Petrov-Galerkin formulations for elastodynamics." International Journal for Numerical Methods in Engineering 90, no. 12 (April 17, 2012): 1508–828. http://dx.doi.org/10.1002/nme.3373.

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44

Brueckner, Frank P., and Juan C. Heinrich. "Petrov-Galerkin finite element model for compressible flows." International Journal for Numerical Methods in Engineering 32, no. 2 (August 5, 1991): 255–74. http://dx.doi.org/10.1002/nme.1620320203.

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45

Mirzaei, Davoud. "A Petrov--Galerkin Kernel Approximation on the Sphere." SIAM Journal on Numerical Analysis 56, no. 1 (January 2018): 274–95. http://dx.doi.org/10.1137/16m1106626.

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46

Yuan-Ming, Wang. "Petrov-galerkin methods for nonlinear reaction-diffusion equations." International Journal of Computer Mathematics 69, no. 1-2 (January 1998): 123–45. http://dx.doi.org/10.1080/00207169808804713.

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47

Deeks, A. J., and C. E. Augarde. "A meshless local Petrov-Galerkin scaled boundary method." Computational Mechanics 36, no. 3 (February 28, 2005): 159–70. http://dx.doi.org/10.1007/s00466-004-0649-y.

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48

Li, M., F. F. Dou, T. Korakianitis, C. Shi, and P. H. Wen. "Boundary node Petrov–Galerkin method in solid structures." Computational and Applied Mathematics 37, no. 1 (April 12, 2016): 135–59. http://dx.doi.org/10.1007/s40314-016-0335-7.

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49

Carew, E. O. A., P. Townsend, and M. F. Webster. "A taylor-petrov-galerkin algorithm for viscoelastic flow." Journal of Non-Newtonian Fluid Mechanics 50, no. 2-3 (December 1993): 253–87. http://dx.doi.org/10.1016/0377-0257(93)80034-9.

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50

Yuan-bo, Xiong, and Long Shu-yao. "Local Petrov-Galerkin method for a thin plate." Applied Mathematics and Mechanics 25, no. 2 (February 2004): 210–18. http://dx.doi.org/10.1007/bf02437322.

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