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Galerkin Methods (galerkin + methods)

Distribution by Scientific Domains
Engineering56%
Mathematics and Statistics40%

Kinds of Galerkin Methods

  • discontinuous galerkin methods
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    Selected Abstracts

    Analysis and performance of a predictor-multicorrector Time Discontinuous Galerkin method in non-linear elastodynamics

    EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, Issue 10 2002
    Oreste S. Bursi
    Abstract A predictor-multicorrector implementation of a Time Discontinuous Galerkin method for non-linear dynamic analysis is described. This implementation is intended to limit the high computational expense typically required by implicit Time Discontinuous Galerkin methods, without degrading their accuracy and stability properties. The algorithm is analysed with reference to conservative Duffing oscillators for which closed-form solutions are available. Therefore, insight into the accuracy and stability properties of the predictor-multicorrector algorithm for different approximations of non-linear internal forces is gained, showing that the properties of the underlying scheme can be substantially retained. Finally, the results of representative numerical simulations relevant to Duffing oscillators and to a stiff spring pendulum discretized with finite elements illustrate the performance of the numerical scheme and confirm the analytical estimates. Copyright © 2002 John Wiley & Sons, Ltd. [source]

    Piecewise divergence-free discontinuous Galerkin methods for Stokes flow

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 5 2008
    Peter Hansbo
    Abstract In this paper, we consider different possibilities of using divergence-free discontinuous Galerkin methods for the Stokes problem in order to eliminate the pressure from the discrete problem. We focus on three different approaches: one based on a C0 approximation of the stream function in two dimensions (the vector potential in three dimensions), one based on the non-conforming Morley element (which corresponds to a divergence-free non-conforming Crouzeix,Raviart approximation of the velocities), and one fully discontinuous Galerkin method with a stabilization of the pressure that allows the edgewise elimination of the pressure variable before solving the discrete system. We limit the analysis in the stream function case to two spatial dimensions, while the analysis of the fully discontinuous approach is valid also in three dimensions. Copyright © 2006 John Wiley & Sons, Ltd. [source]

    Imposition of essential boundary conditions by displacement constraint equations in meshless methods

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 3 2001
    Xiong Zhang
    Abstract One of major difficulties in the implementation of meshless methods is the imposition of essential boundary conditions as the approximations do not pass through the nodal parameter values. As a consequence, the imposition of essential boundary conditions in meshless methods is quite awkward. In this paper, a displacement constraint equations method (DCEM) is proposed for the imposition of the essential boundary conditions, in which the essential boundary conditions is treated as a constraint to the discrete equations obtained from the Galerkin methods. Instead of using the methods of Lagrange multipliers and the penalty method, a procedure is proposed in which unknowns are partitioned into two subvectors, one consisting of unknowns on boundary ,u, and one consisting of the remaining unknowns. A simplified displacement constraint equations method (SDCEM) is also proposed, which results in a efficient scheme with sufficient accuracy for the imposition of the essential boundary conditions in meshless methods. The present method results in a symmetric, positive and banded stiffness matrix. Numerical results show that the accuracy of the present method is higher than that of the modified variational principles. The present method is a exact method for imposing essential boundary conditions in meshless methods, and can be used in Galerkin-based meshless method, such as element-free Galerkin methods, reproducing kernel particle method, meshless local Petrov,Galerkin method. Copyright © 2001 John Wiley & Sons, Ltd. [source]

    An adaptive stabilization strategy for enhanced strain methods in non-linear elasticity

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 11 2010
    Alex Ten Eyck
    Abstract This paper proposes and analyzes an adaptive stabilization strategy for enhanced strain (ES) methods applied to quasistatic non-linear elasticity problems. The approach is formulated for any type of enhancements or material models, and it is distinguished by the fact that the stabilization term is solution dependent. The stabilization strategy is first constructed for general linearized elasticity problems, and then extended to the non-linear elastic regime via an incremental variational principle. A heuristic choice of the stabilization parameters is proposed, which in the numerical examples proved to provide stable approximations for a large range of deformations, different problems and material models. We also provide explicit lower bounds for the stabilization parameters that guarantee that the method will be stable. These are not advocated, since they are generally larger than the ones based on heuristics, and hence prone to deteriorate the locking-free behavior of ES methods. Numerical examples with two different non-linear elastic models in thin geometries and incompressible situations show that the method remains stable and locking free over a large range of deformations. Finally, the method is strongly based on earlier developments for discontinuous Galerkin methods, and hence throughout the paper we offer a perspective about the similarities between the two. Copyright © 2009 John Wiley & Sons, Ltd. [source]

    Local maximum-entropy approximation schemes: a seamless bridge between finite elements and meshfree methods

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 13 2006
    M. Arroyo
    Abstract We present a one-parameter family of approximation schemes, which we refer to as local maximum-entropy approximation schemes, that bridges continuously two important limits: Delaunay triangulation and maximum-entropy (max-ent) statistical inference. Local max-ent approximation schemes represent a compromise,in the sense of Pareto optimality,between the competing objectives of unbiased statistical inference from the nodal data and the definition of local shape functions of least width. Local max-ent approximation schemes are entirely defined by the node set and the domain of analysis, and the shape functions are positive, interpolate affine functions exactly, and have a weak Kronecker-delta property at the boundary. Local max-ent approximation may be regarded as a regularization, or thermalization, of Delaunay triangulation which effectively resolves the degenerate cases resulting from the lack or uniqueness of the triangulation. Local max-ent approximation schemes can be taken as a convenient basis for the numerical solution of PDEs in the style of meshfree Galerkin methods. In test cases characterized by smooth solutions we find that the accuracy of local max-ent approximation schemes is vastly superior to that of finite elements. Copyright © 2005 John Wiley & Sons, Ltd. [source]

    Anisotropic adaptive simulation of transient flows using discontinuous Galerkin methods

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 7 2005
    Jean-François Remacle
    Abstract An anisotropic adaptive analysis procedure based on a discontinuous Galerkin finite element discretization and local mesh modification of simplex elements is presented. The procedure is applied to transient two- and three-dimensional problems governed by Euler's equation. A smoothness indicator is used to isolate jump features where an aligned mesh metric field in specified. The mesh metric field in smooth portions of the domain is controlled by a Hessian matrix constructed using a variational procedure to calculate the second derivatives. The transient examples included demonstrate the ability of the mesh modification procedures to effectively track evolving interacting features of general shape as they move through a domain. Copyright © 2004 John Wiley & Sons, Ltd. [source]

    On discontinuous Galerkin methods

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 8 2003
    O. C. Zienkiewicz
    Abstract Discontinuous Galerkin methods have received considerable attention in recent years for problems in which advection and diffusion terms are present. Several alternatives for treating the diffusion and advective fluxes have been introduced. This report summarizes some of the methods that have been proposed. Several numerical examples are included in the paper. These present discontinuous Galerkin solutions of one-dimensional problems with a scalar variable. Results are presented for diffusion,reaction problems and advection,diffusion problems. We discuss the performance of various formulations with respect to accuracy as well as stability of the method. Copyright © 2003 John Wiley & Sons, Ltd. [source]

    On the stability of bubble functions and a stabilized mixed finite element formulation for the Stokes problem

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2009
    D. Z. Turner
    Abstract In this paper we investigate the relationship between stabilized and enriched finite element formulations for the Stokes problem. We also present a new stabilized mixed formulation for which the stability parameter is derived purely by the method of weighted residuals. This new formulation allows equal-order interpolation for the velocity and pressure fields. Finally, we show by counterexample that a direct equivalence between subgrid-based stabilized finite element methods and Galerkin methods enriched by bubble functions cannot be constructed for quadrilateral and hexahedral elements using standard bubble functions. Copyright © 2008 John Wiley & Sons, Ltd. [source]

    Well-balanced finite volume evolution Galerkin methods for the shallow water equations with source terms

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 10-11 2005
    M. Luká, ová-Medvid'ová
    Abstract The goal of this paper is to present a new well-balanced genuinely multi-dimensional high-resolution finite volume evolution Galerkin method for systems of balance laws. The derivation of the method will be illustrated for the shallow water equation with geometrical source term modelling the bottom topography. The results can be generalized to more complex systems of balance laws. Copyright © 2005 John Wiley & Sons, Ltd. [source]

    Superconvergence and H(div) projection for discontinuous Galerkin methods

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 10 2003
    Peter Bastian
    Abstract We introduce and analyse a projection of the discontinuous Galerkin (DG) velocity approximations that preserve the local mass conservation property. The projected velocities have the additional property of continuous normal component. Both theoretical and numerical convergence rates are obtained which show that the accuracy of the DG velocity field is maintained. Superconvergence properties of the DG methods are shown. Finally, numerical simulations of complicated flow and transport problem illustrate the benefits of the projection. Copyright © 2003 John Wiley & Sons, Ltd. [source]

    The use of classical Lax,Friedrichs Riemann solvers with discontinuous Galerkin methods

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3-4 2002
    W. J. Rider
    Abstract While conducting a von Neumann stability analysis of discontinuous Galerkin methods we discovered that the classic Lax,Friedrichs Riemann solver is unstable for all time-step sizes. We describe a simple modification of the Riemann solver's dissipation returns the method to stability. Furthermore, the method has a smaller truncation error than the corresponding method with an upwind flux for the RK2-DG(1) method. These results are verified upon testing. Copyright © 2002 John Wiley & Sons, Ltd. [source]

    Numerical evaluation of two discontinuous Galerkin methods for the compressible Navier,Stokes equations

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 1-2 2002
    F. Bassi
    This paper presents a critical comparison between two recently proposed discontinuous Galerkin methods for the space discretization of the viscous terms of the compressible Navier,Stokes equations. The robustness and accuracy of the two methods has been numerically evaluated by considering simple but well documented classical two-dimensional test cases, including the flow around the NACA0012 airfoil, the flow along a flat plate and the flow through a turbine nozzle. Copyright © 2002 John Wiley & Sons, Ltd. [source]

    A Parallelised High Performance Monte Carlo Simulation Approach for Complex Polymerisation Kinetics

    MACROMOLECULAR THEORY AND SIMULATIONS, Issue 6 2007
    Hugh Chaffey-Millar
    Abstract A novel, parallelised approach to Monte Carlo simulations for the computation of full molecular weight distributions (MWDs) arising from complex polymerisation reactions is presented. The parallel Monte Carlo method constitutes perhaps the most comprehensive route to the simulation of full MWDs of multiple chain length polymer entities and can also provide detailed microstructural information. New fundamental insights have been developed with regard to the Monte Carlo process in at least three key areas: (i) an insufficient system size is demonstrated to create inaccuracies via poor representation of the most improbable events and least numerous species; (ii) advanced algorithmic principles and compiler technology known to computer science have been used to provide speed improvements and (iii) the parallelisability of the algorithm has been explored and excellent scalability demonstrated. At present, the parallel Monte Carlo method presented herein compares very favourably in speed with the latest developments in the h-p Galerkin method-based PREDICI software package while providing significantly more detailed microstructural information. It seems viable to fuse parallel Monte Carlo methods with those based on the h-p Galerkin methods to achieve an optimum of information depths for the modelling of complex macromolecular kinetics and the resulting microstructural information. [source]

    Stability and convergence of optimum spectral non-linear Galerkin methods

    MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 5 2001
    He Yinnian
    Abstract Our objective in this article is to present some numerical schemes for the approximation of the 2-D Navier,Stokes equations with periodic boundary conditions, and to study the stability and convergence of the schemes. Spatial discretization can be performed by either the spectral Galerkin method or the optimum spectral non-linear Galerkin method; time discretization is done by the Euler scheme and a two-step scheme. Our results show that under the same convergence rate the optimum spectral non-linear Galerkin method is superior to the usual Galerkin methods. Finally, numerical example is provided and supports our results. Copyright © 2001 John Wiley & Sons, Ltd. [source]

    A CFL-free explicit characteristic interior penalty scheme for linear advection-reaction equations

    NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 3 2010
    Kaixin Wang
    Abstract We develop a CFL-free, explicit characteristic interior penalty scheme (CHIPS) for one-dimensional first-order advection-reaction equations by combining a Eulerian-Lagrangian approach with a discontinuous Galerkin framework. The CHIPS method retains the numerical advantages of the discontinuous Galerkin methods as well as characteristic methods. An optimal-order error estimate in the L2 norm for the CHIPS method is derived and numerical experiments are presented to confirm the theoretical estimates. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010 [source]

    Equilibrated error estimators for discontinuous Galerkin methods

    NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 5 2008
    Sarah Cochez-Dhondt
    Abstract We consider some diffusion problems in domains of ,d, d = 2 or 3 approximated by a discontinuous Galerkin method with polynomials of any degree. We propose a new a posteriori error estimator based on H(div)-conforming elements. It is shown that this estimator gives rise to an upper bound where the constant is one up to higher order terms. The lower bound is also established with a constant depending on the aspect ratio of the mesh, the dependence with respect to the coefficients being also traced. The reliability and efficiency of the proposed estimator is confirmed by some numerical tests. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008 [source]

    An Eulerian-Lagrangian discontinuous Galerkin method for transient advection-diffusion equations

    NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2007
    Kaixin Wang
    Abstract We develop an Eulerian-Lagrangian discontinuous Galerkin method for time-dependent advection-diffusion equations. The derived scheme has combined advantages of Eulerian-Lagrangian methods and discontinuous Galerkin methods. The scheme does not contain any undetermined problem-dependent parameter. An optimal-order error estimate and superconvergence estimate is derived. Numerical experiments are presented, which verify the theoretical estimates.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007 [source]

    Iteratively coupled mixed and Galerkin finite element methods for poro-elasticity

    NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 4 2007
    Mary F. Wheeler
    Abstract We present a finite element formulation for coupled flow and geomechanics. We use mixed finite element spaces to approximate pressure and continuous Galerkin methods for displacements. In solving the coupled system, pressure and displacements can be solved either simultaneously in a fully coupled scheme or sequentially in a loosely coupled scheme. In this paper we formulate an iterative method where pressure and displacement solutions are staggered during a time step until a convergence tolerance is satisfied. A priori convergence results for the iterative coupling are also presented, along with a summary of the convergence results for the fully coupled scheme. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 785,797, 2007 [source]

    Discontinuous Galerkin methods for periodic boundary value problems

    NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 3 2007
    Kumar Vemaganti
    Abstract This article considers the extension of well-known discontinuous Galerkin (DG) finite element formulations to elliptic problems with periodic boundary conditions. Such problems routinely appear in a number of applications, particularly in homogenization of composite materials. We propose an approach in which the periodicity constraint is incorporated weakly in the variational formulation of the problem. Both H1 and L2 error estimates are presented. A numerical example confirming theoretical estimates is shown. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007 [source]

    Convergence analysis of the streamline diffusion and discontinuous Galerkin methods for the Vlasov-Fokker-Planck system

    NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 3 2005
    M. Asadzadeh
    Abstract We prove stability estimates and derive optimal convergence rates for the streamline diffusion and discontinuous Galerkin finite element methods for discretization of the multi-dimensional Vlasov-Fokker-Planck system. The focus is on the theoretical aspects, where we deal with construction and convergence analysis of the discretization schemes. Some related special cases are implemented in M. Asadzadeh [Appl Comput Meth 1(2) (2002), 158,175] and M. Asadzadeh and A. Sopasakis [Comput Meth Appl Mech Eng 191(41,42) (2002), 4641,4661]. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005 [source]

    Uniform stability of spectral nonlinear Galerkin methods

    NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 5 2004
    Yinnian He
    Abstract This article provides a stability analysis for the backward Euler schemes of time discretization applied to the spatially discrete spectral standard and nonlinear Galerkin approximations of the nonstationary Navier-Stokes equations with some appropriate assumption of the data (,, u0, f). If the backward Euler scheme with the semi-implicit nonlinear terms is used, the spectral standard and nonlinear Galerkin methods are uniform stable under the time step constraint ,t , (2/,,1). Moreover, if the backward Euler scheme with the explicit nonlinear terms is used, the spectral standard and nonlinear Galerkin methods are uniform stable under the time step constraints ,t = O(,) and ,t = O(,), respectively, where , , ,, which shows that the restriction on the time step of the spectral nonlinear Galerkin method is less than that of the spectral standard Galerkin method. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004 [source]

    Nonconforming Galerkin methods for the Helmholtz equation

    NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 5 2001
    Jim Douglas Jr.
    Abstract Nonconforming Galerkin methods for a Helmholtz-like problem arising in seismology are discussed both for standard simplicial linear elements and for several new rectangular elements related to bilinear or trilinear elements. Optimal order error estimates in a broken energy norm are derived for all elements and in L2 for some of the elements when proper quadrature rules are applied to the absorbing boundary condition. Domain decomposition iterative procedures are introduced for the nonconforming methods, and their convergence at a predictable rate is established. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 475,494, 2001 [source]