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Preface

Arif Masud
2004 Computer Methods in Applied Mechanics and Engineering  
In the GLS method a least-squares form of the residuals that is based on the corresponding Euler-Lagrange equations is added to the Galerkin finite element formulation.  ...  In order to correct this deficiency in the standard Galerkin approach they introduced the Streamline-Upwind-Petrov-Galerkin (SUPG) method.  ...  The paper then presents a procedure, based on Rothe method, to determine specific values for the potential onset of spatial oscillations.  ... 
doi:10.1016/j.cma.2004.01.003 fatcat:33jowznkmzgjtl5hmnh4iw4zai

Page 3334 of Mathematical Reviews Vol. , Issue 2001E [page]

2001 Mathematical Reviews  
The value of the unknown function at the initial instant is given. For this integral equation the Petrov- Galerkin finite element methods are considered.  ...  The numerical solution of PDEs in R” by a wavelet-based Galerkin  ... 

The Adjoint Petrov-Galerkin Method for Non-Linear Model Reduction [article]

Eric J. Parish, Christopher Wentland, Karthik Duraisamy
2019 arXiv   pre-print
We formulate a new projection-based reduced-ordered modeling technique for non-linear dynamical systems.  ...  The formulation is shown to be equivalent to a Petrov-Galerkin method with a non-linear, time-varying test basis, thus sharing some similarities with the least-squares Petrov-Galerkin method.  ...  Implicit Time Integration Schemes This section evaluates the computational cost of the Galerkin, Adjoint Petrov-Galerkin, and Least-Squares Petrov-Galerkin methods for implicit time integration schemes  ... 
arXiv:1810.03455v3 fatcat:ydwdteo5j5g2vcggcupyqyqmsi

Removing the Cell Resonance Error in the Multiscale Finite Element Method via a Petrov-Galerkin Formulation

Thomas Y. Hou, Xiao-Hui Wu, Yu Zhang
2004 Communications in Mathematical Sciences  
Here, we introduce a Petrov-Galerkin MsFEM formulation with nonconforming multiscale trial functions and linear test functions.  ...  We continue the study of the nonconforming multiscale finite element method (Ms-FEM) introduced in [17, 14] for second order elliptic equations with highly oscillatory coefficients.  ...  Yalchin Efendiev for helpful discussions on this work and Dr. Andrew Westhead for providing the report on his special MsFEM method.  ... 
doi:10.4310/cms.2004.v2.n2.a3 fatcat:ict4awxeajd4xktkjapqficceu

Page 2169 of Mathematical Reviews Vol. , Issue 2000c [page]

2000 Mathematical Reviews  
A detailed analysis has been carried out on the stability and convergence of the wavelet Petrov- Galerkin method.” Tian-Xiao He (1-ILCC-CS; Chicago, IL) 2000¢:65120 65N99 Eremin, Yu. A.  ...  Summary: “A macro-hybrid formulation based on overlapping domain decomposition is introduced and studied for a model elliptic partial differential equation.  ... 

Non-intrusive implementation of Multiscale Finite Element Methods: an illustrative example [article]

Rutger A. Biezemans, Claude Le Bris, Frederic Legoll, Alexei Lozinski
2022 arXiv   pre-print
They first compute local, oscillatory, problem-dependent basis functions which generate a specific discretization space, and next perform a Galerkin approximation of the problem on that space.  ...  Multiscale Finite Element Methods (MsFEM) are finite element type approaches dedicated to multiscale problems.  ...  Acknowledgments The first author acknowledges the support of DIM Math INNOV.  ... 
arXiv:2204.06852v1 fatcat:qolhgg46wreshgrugxfeakdpte

A stabilized finite element formulation for advection-diffusion using the generalized finite element framework [article]

D. Z. Turner and K. B. Nakshatrala and K. D. Hjelmstad
2008 arXiv   pre-print
To contextualize this method among other stabilized methods, we show by decomposition of the solution (in a multiscale manner) an equivalence to both Galerkin/least-squares type methods and those that  ...  The following work presents a generalized (extended) finite element formulation for the advection-diffusion equation.  ...  As such, one can construct an enrichment function based on the solution to the problem at low P e h , for which the classical Galerkin formulation is stable.  ... 
arXiv:0806.3963v1 fatcat:5d3vhugburhlnotlsfnpdqgp5y

On multiscale methods in Petrov–Galerkin formulation

Daniel Elfverson, Victor Ginting, Patrick Henning
2015 Numerische Mathematik  
In Section 3, we present the multiscale methods based on the LOD framework, starting from the usual Galerkin variational equation and concentrate further on the Petrov-Galerkin variational equation that  ...  As another application of the framework, we show how the Petrov-Galerkin framework can be used to construct a locally mass conservative solver for the Buckley-Leverett equation.  ...  Example 2: Discontinuous Galerkin Finite Element Method In this subsection, we apply the results of Section 3.2 to a LOD Method that is based on a Discontinuous Galerkin approach.  ... 
doi:10.1007/s00211-015-0703-z fatcat:rtqzjazzorakrl2qhoo47veqzm

Towards multiscale functions: enriching finite element spaces with local but not bubble-like functions

Leopoldo P. Franca, Alexandre L. Madureira, Frederic Valentin
2005 Computer Methods in Applied Mechanics and Engineering  
Bubbles are the choice for the test functions allowing static condensation, thus our method is of Petrov-Galerkin type.  ...  Such multiscale functions have an analytic expression, for triangles and rectangles.  ...  Franca was visiting the Applied Mathematics Department of LNCC. This author expresses his gratitude towards the colleagues of this Department for their hospitality.  ... 
doi:10.1016/j.cma.2004.07.029 fatcat:vv5pfzill5ch7dr4zsilzqo4xi

On the stability of projection-based model order reduction for convection-dominated laminar and turbulent flows [article]

Sebastian Grimberg, Charbel Farhat, Noah Youkilis
2020 arXiv   pre-print
The paper also shows that alternatively, a Petrov-Galerkin framework can be used to construct numerically stable PROMs for convection-dominated laminar as well as turbulent flow problems that are numerically  ...  real culprit behind most if not all reported numerical instabilities of PROMs for turbulence and convection-dominated turbulent flow problems is the Galerkin framework that has been used for constructing  ...  For problems characterized by a large Kolmogorov n-width of the HDM-based solution manifold, popular methods for constructing the right ROB V, such as the proper orthogonal decomposition (POD) method of  ... 
arXiv:2001.10110v1 fatcat:nkjvusozirdpvpj5uc335gb7aq

Wavelet Methods for Second-Order Elliptic Problems, Preconditioning, and Adaptivity

A. Cohen, R. Masson
1999 SIAM Journal on Scientific Computing  
For such domains, we discuss the construction of high order multiresolution approximation and wavelet bases, and in particular the choice of the wavelets near the boundary in order to optimize the e ciency  ...  A n d'am eliorer les performances du pr econditionnement diagonal en base d'ondelettes, nous introduisons un pr econditionnement presque diagonal obtenu en r esolvant des probl emes de Petrov-Galerkin  ...  We use both Galerkin and Petrov-Galerkin methods and compare isotropic and anisotropic bases in terms of preconditioning properties.  ... 
doi:10.1137/s1064827597330613 fatcat:yjm4orrjc5cxlaccdk5x3wt3ze

Multiscale discontinuous Petrov--Galerkin method for the multiscale elliptic problems [article]

Song Fei, Deng Weibing
2017 arXiv   pre-print
In this paper we present a new multiscale discontinuous Petrov--Galerkin method (MsDPGM) for multiscale elliptic problems.  ...  This method utilizes the classical oversampling multiscale basis in the framework of Petrov--Galerkin version of discontinuous Galerkin finite element method, allowing us to better cope with multiscale  ...  In this paper, we have proposed a new Petrov-Galerkin method based on the discontinuous multiscale approximation space for the multiscale elliptic in the L 2 , L ∞ and energy norm for the L-shaped problem  ... 
arXiv:1702.02317v1 fatcat:2dij4o5eozh45gvjb3augdoq7e

Petrov-Galerkin flux upwinding for mixed mimetic spectral elements, and its application to geophysical flow problems [article]

David Lee
2020 arXiv   pre-print
This involves a Petrov-Galerkin formulation by which the mass flux test functions are evaluated at downstream locations along velocity characteristics.  ...  As for the original advection operator, a material form advection operator may be constructed by similarly downwinding the trial functions of the tracer gradients.  ...  Numerous methods have been devised to address this issue in the context of finite element methods, including the streamwise-upwind Petrov-Galerkin method [1] and variational multiscale methods [2] .  ... 
arXiv:2004.13225v3 fatcat:ihzvrbdmq5bxvf4fnspr4rxfci

Revisiting stabilized finite element methods for the advective–diffusive equation

Leopoldo P. Franca, Guillermo Hauke, Arif Masud
2006 Computer Methods in Applied Mechanics and Engineering  
We give a brief overview of stabilized finite element methods and illustrate the developments applied to the advection-diffusion equation.  ...  This article presents a concise perspective of the developments emanated from the works started in the 1980s through today.  ...  Acknowledgements The authors acknowledge the following sources of support: L.P. Franca was supported by the National Science Foundation Grants No. 0325314 and 0339107. A.  ... 
doi:10.1016/j.cma.2005.05.028 fatcat:233etjme7zebhnvqribvfypnpm

Page 2725 of Mathematical Reviews Vol. , Issue 99d [page]

1991 Mathematical Reviews  
The generalized Stokes problem is studied in two dimensions for a MAC-like method, analyzed using a Petrov-Galerkin interpreta- tion.  ...  The momentum equation is in- tegrated over the dual grid and the incompressibility condition is integrated over the primal grid, by using test functions in the Petrov-Galerkin method that are defined over  ... 
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