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Journal articles on the topic 'Petrov-Galerkin'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 July 2025

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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 (1998): 406–34. http://dx.doi.org/10.1137/s0036142996297217.

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3

Donea, J. "Generalized Galerkin Methods for Convection Dominated Transport Phenomena." Applied Mechanics Reviews 44, no. 5 (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 approximation

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4

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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5

Galanin, M., and E. Savenkov. "FEDORENKO FINITE SUPERELEMENT METHOD AS SPECIAL GALERKIN APPROXIMATION." Mathematical Modelling and Analysis 7, no. 1 (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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6

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

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7

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

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8

Demkowicz, Leszek, and Jay Gopalakrishnan. "The discontinuous Petrov–Galerkin method." Acta Numerica 34 (July 2025): 293–384. https://doi.org/10.1017/s0962492924000102.

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The discontinuous Petrov–Galerkin (DPG) method is a Petrov–Galerkin finite element method with test functions designed for obtaining stability. These test functions are computable locally, element by element, and are motivated by optimal test functions which attain the supremum in an inf-sup condition. A profound consequence of the use of nearly optimal test functions is that the DPG method can inherit the stability of the (undiscretized) variational formulation, be it coercive or not. This paper combines a presentation of the fundamentals of the DPG ideas with a review of the ongoing research

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9

Führer, Thomas, Norbert Heuer, Michael Karkulik, and Rodolfo Rodríguez. "Combining the DPG Method with Finite Elements." Computational Methods in Applied Mathematics 18, no. 4 (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 t

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10

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 (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 deri

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11

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

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12

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

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13

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

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14

Bringmann, P., C. Carstensen, D. Gallistl, et al. "Towards adaptive discontinuous Petrov-Galerkin methods." PAMM 16, no. 1 (2016): 741–42. http://dx.doi.org/10.1002/pamm.201610359.

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15

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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16

Akin, J. E., T. Tezduyar, M. Ungor, and S. Mittal. "Stabilization Parameters and Smagorinsky Turbulence Model." Journal of Applied Mechanics 70, no. 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

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17

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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18

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 (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

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19

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 aske

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20

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 (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 c

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21

BACUTA, CONSTANTIN, CRISTINA BACUTA, and DANIEL HAYES. "COMPARISON OF VARIATIONAL DISCRETIZATIONS FOR A." Revue Roumaine Mathematiques Pures Appliquees LXIX, no. 3-4 (2024): 327–51. https://doi.org/10.59277/rrmpa.2024.327.351.

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For a model convection-diffusion problem, we obtain new error estimates for a general upwinding finite element discretization based on bubble modification of the test space. The key analysis tool is finding representations of the optimal norms on the trial spaces at the continuous and discrete levels. We analyze and compare three methods: the standard linear discretization, the saddle point least square and the upwinding Petrov–Galerkin methods. We conclude that the bubble upwinding the Petrov–Galerkin method is the most performant discretization for the one-dimensional model. Our results for

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22

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 p

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23

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 (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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24

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

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25

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

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26

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 (2006): 331–43. http://dx.doi.org/10.1080/00207160600747938.

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27

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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28

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

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29

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

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30

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

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31

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

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32

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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33

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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34

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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35

Meidner, Dominik, and Thomas Richter. "Goal-Oriented Error Estimation for the Fractional Step Theta Scheme." Computational Methods in Applied Mathematics 14, no. 2 (2014): 203–30. http://dx.doi.org/10.1515/cmam-2014-0002.

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Abstract. In this work, we derive a goal-oriented a posteriori error estimator for the error due to time-discretization of nonlinear parabolic partial differential equations by the fractional step theta method. This time-stepping scheme is assembled by three steps of the general theta method, that also unifies simple schemes like forward and backward Euler as well as the Crank–Nicolson method. Further, by combining three substeps of the theta time-stepping scheme, the fractional step theta time-stepping scheme is derived. It possesses highly desired stability and numerical dissipation properti

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36

Shi, Liwei. "A Petrov-Galerkin Finite Element Method for Solving the Time-fractional Diffusion Equation with Interface." Journal of Mathematics Research 10, no. 4 (2018): 136. http://dx.doi.org/10.5539/jmr.v10n4p136.

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Time-fractional partial differential equation is widely applied in a variety of disciplines, its numerical solution has attracted much attention from researchers in recent years. Time-fractional differential equations with interfaces is a more challenging problem because the governing equation has discontinuous coefficients at interfaces and sometimes singular source term exists. In this paper, we propose a Petrov-Galerkin finite element method for solving the two-dimensional time-fractional diffusion equation with interfaces. In this method, a finite difference scheme is employed in time and

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37

Carey, Graham F., A. L. Pardhanani, and S. W. Bova. "Advanced Numerical Methods and Software Approaches for Semiconductor Device Simulation." VLSI Design 10, no. 4 (2000): 391–414. http://dx.doi.org/10.1155/2000/43903.

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In this article we concisely present several modern strategies that are applicable to driftdominated carrier transport in higher-order deterministic models such as the driftdiffusion, hydrodynamic, and quantum hydrodynamic systems. The approaches include extensions of “upwind” and artificial dissipation schemes, generalization of the traditional Scharfetter – Gummel approach, Petrov – Galerkin and streamline-upwind Petrov Galerkin (SUPG), “entropy” variables, transformations, least-squares mixed methods and other stabilized Galerkin schemes such as Galerkin least squares and discontinuous Gale

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38

Zhao, Ting Gang, Yan Tang Liang, and He Ping Ma. "Chebyshev-Legendre Pseudo-Spectral Method for the Modified Kawahara Equation." Advanced Materials Research 143-144 (October 2010): 191–95. http://dx.doi.org/10.4028/www.scientific.net/amr.143-144.191.

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Chebyshev-Legendre pseudo-spectral method was applied to resolve the modified Kawahara equation. The proposed approach is based on Legendre-Petrov Galerkin formulation but only the Chebyshev-Gauss-Lobatto nodes are used in the computation. The Petrov method is considered for the unsymmetry of the fifth-order derivative term appeared in the modified Kawahara equation and Chebyshev-Legendre method is used for reducing the computational complexity. Numerical tests are presented to verify the efficiency of the proposed approach.

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39

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

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40

Loula, Abimael F. D., and Joa˜o Nisan C. Guerreiro. "New Mixed Finite Element Formulations for Transient and Steady State Creep Problems." Applied Mechanics Reviews 42, no. 11S (1989): S150—S156. http://dx.doi.org/10.1115/1.3152385.

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We apply the mixed Petrov–Galerkin formulation to construct finite element approximations for transient and steady-state creep problems. With the new approach we recover stability, convergence, and accuracy of some Galerkin unstable approximations. We also present the main results on the numerical analysis and error estimates of the proposed finite element approximation for the steady problem, and discuss the asymptotic behavior of the continuum and discrete transient problems.

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41

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

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42

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

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43

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 (2015): 1–4. http://dx.doi.org/10.1109/tmag.2014.2359213.

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44

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

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45

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

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46

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 (1993): 253–87. http://dx.doi.org/10.1016/0377-0257(93)80034-9.

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47

Aimi, A., and M. Diligenti. "Numerical integration schemes for Petrov–Galerkin infinite BEM." Applied Numerical Mathematics 58, no. 8 (2008): 1084–102. http://dx.doi.org/10.1016/j.apnum.2007.04.014.

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48

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 (2002): 3391–409. http://dx.doi.org/10.1016/s0045-7825(02)00254-2.

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49

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

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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 (2004): 210–18. http://dx.doi.org/10.1007/bf02437322.

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