The Experts below are selected from a list of 360 Experts worldwide ranked by ideXlab platform
Bernardo Cockburn - One of the best experts on this subject based on the ideXlab platform.
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a hybridizable Discontinuous Galerkin formulation for non linear elasticity
Computer Methods in Applied Mechanics and Engineering, 2015Co-Authors: Hardik Kabaria, Bernardo CockburnAbstract:Abstract We revisit the hybridizable Discontinuous Galerkin method for non-linear elasticity introduced by S.-C. Soon (2008). We show that it can be recast as a minimization problem of a non-linear functional over a space of Discontinuous approximations to the displacement. The functional can be written as the sum over the elements of the classic potential energy plus a new energy associated to the inter-element jumps of the displacement. We then show that if this new energy is not properly weighted, the minimizers might not converge to the exact solution. We construct an example illustrating this phenomenon and show how to overcome it by suitably increasing the weight of the energy of the inter-element jumps. Finally, we explore the performance of the method for the case of piecewise-linear approximations in rather demanding situations in both two-dimensional and, for the first time, three-dimensional situations. They include almost incompressible materials, large deformations with large-shear layers, and cavitation. We also compare the method with the continuous Galerkin method and a previously explored Discontinuous Galerkin method, and show that, when using piecewise-linear approximations and a moderate number of degrees of freedom, the current method turns out to be more efficient for the computation of the gradient.
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Discontinuous Galerkin methods theory computation and applications
2011Co-Authors: Bernardo Cockburn, George Em Karniadakis, Chi-wang ShuAbstract:This volume contains current progress of a new class of finite element method, the Discontinuous Galerkin Method (DGM), which has been under rapid developments recently and has found its use very quickly in such diverse applications as aeroacoustics, semi-conductor device simulation, turbomachinery, turbulent flows, materials processing, Magneto-hydro-dynamics, plasma simulations and image processing. While there has been a lot of interest in DGM from mathematicians, physicists and engineers, only scattered information is available and there has been no prior effort in organizing and publishing the existing volume of knowledge on this subject. The current volume organizes this knowledge and it covers both theoretical as well as practical issues of the Discontinuous Galerkin method.
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an embedded Discontinuous Galerkin method for the compressible euler and navier stokes equations
20th AIAA Computational Fluid Dynamics Conference 2011, 2011Co-Authors: Jaime Peraire, Ngoc Cuong Nguyen, Bernardo CockburnAbstract:We present an Embedded Discontinuous Galerkin (EDG) method for the solution of the compressible Euler and Navier-Stokes equations. The method is devised by using the Discontinuous Galerkin approximation with a special choice of the numerical uxes and weakly imposing the continuity of the normal component of the numerical uxes across the element interfaces. This allows the approximate conserved variables dening the Discontinuous Galerkin solution to be locally condensed, thereby resulting in a reduced system which involves only the degrees of freedom of the approximate traces of the solution. The EDG method can be seen as a particular form of a Hybridizable Discontinuous Galerkin (HDG) method in which the hybrid uxes are required to belong to a smaller space than in standard HDG methods. In our EDG method, the hybrid unknown is taken to be continuous at the vertices, thus resulting in an even smaller number of coupled degrees of freedom than in the HDG method. In fact, the resulting stiness matrix has the same structure as that of the statically condensed continuous Galerkin method. In exchange for the reduced number of degrees of freedom, the EDG method looses the optimal converge property of the ux which characterizes other HDG methods. Thus, for convection-diusi on problems, the EDG solution converges optimally for the primal unknown but suboptimally for the ux.
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a hybridizable Discontinuous Galerkin method for stokes flow
Computer Methods in Applied Mechanics and Engineering, 2010Co-Authors: Ngoc Cuong Nguyen, J Peraire, Bernardo CockburnAbstract:Abstract In this paper, we introduce a hybridizable Discontinuous Galerkin method for Stokes flow. The method is devised by using the Discontinuous Galerkin methodology to discretize a velocity–pressure–gradient formulation of the Stokes system with appropriate choices of the numerical fluxes and by applying a hybridization technique to the resulting discretization. One of the main features of this approach is that it reduces the globally coupled unknowns to the numerical trace of the velocity and the mean of the pressure on the element boundaries, thereby leading to a significant reduction in the size of the resulting matrix. Moreover, by using an augmented lagrangian method, the globally coupled unknowns are further reduced to the numerical trace of the velocity only. Another important feature is that the approximations of the velocity, pressure, and gradient converge with the optimal order of k + 1 in the L 2 -norm, when polynomials of degree k ⩾ 0 are used to represent the approximate variables. Based on the optimal convergence of the HDG method, we apply an element-by-element postprocessing scheme to obtain a new approximate velocity, which converges with order k + 2 in the L 2 -norm for k ⩾ 1 . The postprocessing performed at the element level is less expensive than the solution procedure. Numerical results are provided to assess the performance of the method.
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a hybridizable Discontinuous Galerkin method for the compressible euler and navier stokes equations
48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, 2010Co-Authors: Jaime Peraire, Ngoc Cuong Nguyen, Bernardo CockburnAbstract:In this paper, we present a Hybridizable Discontinuous Galerkin (HDG) method for the solution of the compressible Euler and Navier-Stokes equations. The method is devised by using the Discontinuous Galerkin approximation with a special choice of the numerical fluxes and weakly imposing the continuity of the normal component of the numerical fluxes across the element interfaces. This allows the approximate conserved variables defining the Discontinuous Galerkin solution to be locally condensed, thereby resulting in a reduced system which involves only the degrees of freedom of the approximate traces of the solution. The HDG method inherits the geometric flexibility and arbitrary high order accuracy of Discontinuous Galerkin methods, but offers a significant reduction in the computational cost as well as improved accuracy and convergence properties. In particular, we show that HDG produces optimal converges rates for both the conserved quantities as well as the viscous stresses and the heat fluxes. We present some numerical results to demonstrate the accuracy and convergence properties of the method.
Chi-wang Shu - One of the best experts on this subject based on the ideXlab platform.
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Discontinuous Galerkin methods theory computation and applications
2011Co-Authors: Bernardo Cockburn, George Em Karniadakis, Chi-wang ShuAbstract:This volume contains current progress of a new class of finite element method, the Discontinuous Galerkin Method (DGM), which has been under rapid developments recently and has found its use very quickly in such diverse applications as aeroacoustics, semi-conductor device simulation, turbomachinery, turbulent flows, materials processing, Magneto-hydro-dynamics, plasma simulations and image processing. While there has been a lot of interest in DGM from mathematicians, physicists and engineers, only scattered information is available and there has been no prior effort in organizing and publishing the existing volume of knowledge on this subject. The current volume organizes this knowledge and it covers both theoretical as well as practical issues of the Discontinuous Galerkin method.
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superconvergence of Discontinuous Galerkin and local Discontinuous Galerkin schemes for linear hyperbolic and convection diffusion equations in one space dimension
SIAM Journal on Numerical Analysis, 2010Co-Authors: Yingda Cheng, Chi-wang ShuAbstract:In this paper, we study the superconvergence property for the Discontinuous Galerkin (DG) and the local Discontinuous Galerkin (LDG) methods for solving one-dimensional time dependent linear conservation laws and convection-diffusion equations. We prove superconvergence towards a particular projection of the exact solution when the upwind flux is used for conservation laws and when the alternating flux is used for convection-diffusion equations. The order of superconvergence for both cases is proved to be $k+\frac{3}{2}$ when piecewise $P^k$ polynomials with $k\geq1$ are used. The proof is valid for arbitrary nonuniform regular meshes and for piecewise $P^k$ polynomials with arbitrary $k\geq1$, improving upon the results in [Y. Cheng and C.-W. Shu, J. Comput. Phys., 227 (2008), pp. 9612-9627], [Y. Cheng and C.-W. Shu, Computers and Structures, 87 (2009), pp. 630-641] in which the proof based on Fourier analysis was given only for uniform meshes with periodic boundary condition and piecewise $P^1$ polynomials.
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superconvergence of local Discontinuous Galerkin methods for one dimensional convection diffusion equations
Computers & Structures, 2009Co-Authors: Yingda Cheng, Chi-wang ShuAbstract:In this paper, we study the convergence behavior of the local Discontinuous Galerkin (LDG) methods when applied to one-dimensional time dependent convection-diffusion equations. We show that the LDG solution will be superconvergent towards a particular projection of the exact solution, if this projection is carefully chosen based on the convection and diffusion fluxes. The order is observed to be at least k+2 when piecewise P^k polynomials are used. Moreover, the numerical traces for the solution are also superconvergent, sometimes, of higher-order. This is a continuation of our previous work [Cheng Y, Shu C-W. Superconvergence and time evolution of Discontinuous Galerkin finite element solutions. J Comput Phys 2008;227:9612-27], in which superconvergence of DG schemes for convection equations is discussed.
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runge kutta Discontinuous Galerkin method using weno limiters ii
Journal of Computational Physics, 2008Co-Authors: Jun Zhu, Chi-wang Shu, Jianxian Qiu, Michael DumbserAbstract:In J. Qiu, C.-W. Shu, Runge-Kutta Discontinuous Galerkin method using WENO limiters, SIAM Journal on Scientific Computing 26 (2005) 907-929], Qiu and Shu investigated using weighted essentially non-oscillatory (WENO) finite volume methodology as limiters for the Runge-Kutta Discontinuous Galerkin (RKDG) methods for solving nonlinear hyperbolic conservation law systems on structured meshes. In this continuation paper, we extend the method to solve two-dimensional problems on unstructured meshes, with the goal of obtaining a robust and high order limiting procedure to simultaneously obtain uniform high order accuracy and sharp, nonoscillatory shock transition for RKDG methods. Numerical results are provided to illustrate the behavior of this procedure.
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local Discontinuous Galerkin methods for moment models in device simulations formulation and one dimensional results
International Workshop on Computational Electronics, 2004Co-Authors: Yunxian Liu, Chi-wang ShuAbstract:We report, our preliminary work in applying the local Discontinuous Galerkin (LDG) finite element method to solve various time dependent and steady state moment models for semiconductor device simulations, in which both the first derivative convection terms and second derivative diffusion (heat conduction) terms exist and are discretized by the Discontinuous Galerkin method and the local Discontinuous Galerkin method (Cockburn and Shu, 2001) respectively. The potential equation for the electrical field is also discretized by the local Discontinuous Galerkin method. This is an ongoing project, with the objective of developing a numerical tool based on the Discontinuous Galerkin and local Discontinuous Galerkin methodology, capable of solving various models for semiconductor device simulations (hydrodynamic models, energy transport models, quantum drift-diffusion or quantum hydrodynamic models, kinetic models, etc.) in a unified treatment of first and higher spatial derivatives, including those for the potential equations, which would allow easy k-p adaptivity and efficient parallel implementation.
Ilaria Perugia - One of the best experts on this subject based on the ideXlab platform.
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plane wave Discontinuous Galerkin methods analysis of the h version
Mathematical Modelling and Numerical Analysis, 2009Co-Authors: Claude Jeffrey Gittelson, Ralf Hiptmair, Ilaria PerugiaAbstract:We are concerned with a finite element approximation for time-harmonic wave propagation governed by the Helmholtz equation. The usually oscillatory behavior of solutions, along with numerical dispersion, render standard finite element methods grossly inefficient already in medium-frequency regimes. As an alternative, methods that incorporate information about the solution in the form of plane waves have been proposed. We focus on a class of Trefftz-type Discontinuous Galerkin methods that employs trial and test spaces spanned by local plane waves. In this paper we give ap riori convergence estimates for the h-version of these plane wave Discontinuous Galerkin methods in two dimensions. To that end, we develop new inverse and approximation estimates for plane waves and use these in the context of duality techniques. Asymptotic optimality of the method in a mesh dependent norm can be established. However, the estimates require a minimal resolution of the mesh beyond what it takes to resolve the wavelength. We give numerical evidence that this requirement cannot be dispensed with. It reflects the presence of numerical dispersion.
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Discontinuous Galerkin approximation of the maxwell eigenproblem
SIAM Journal on Numerical Analysis, 2006Co-Authors: Annalisa Buffa, Ilaria PerugiaAbstract:A theoretical framework for the analysis of Discontinuous Galerkin approximations of the Maxwell eigenproblem with Discontinuous coefficients is presented. Necessary and sufficient conditions for a spurious-free approximation are established, and it is shown that, at least on conformal meshes, basically all the Discontinuous Galerkin methods in the literature actually fit into this framework. Relations with the classical theory for conforming approximations are also discussed.
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the hp local Discontinuous Galerkin method for low frequency time harmonic maxwell equations
Mathematics of Computation, 2003Co-Authors: Ilaria Perugia, Dominik SchotzauAbstract:The local Discontinuous Galerkin method for the numerical approximation of the time-harmonic Maxwell equations in a low-frequency regime is introduced and analyzed. Topologically nontrivial domains and heterogeneous media are considered, containing both conducting and insulating materials. The presented method involves Discontinuous Galerkin discretizations of the curl-curl and grad-div operators, derived by introducing suitable auxiliary variables and so-called numerical fluxes. An hp-analysis is carried out and error estimates that are optimal in the meshsize h and slightly suboptimal in the approximation degree p are obtained.
Michael Dumbser - One of the best experts on this subject based on the ideXlab platform.
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runge kutta Discontinuous Galerkin method using weno limiters ii
Journal of Computational Physics, 2008Co-Authors: Jun Zhu, Chi-wang Shu, Jianxian Qiu, Michael DumbserAbstract:In J. Qiu, C.-W. Shu, Runge-Kutta Discontinuous Galerkin method using WENO limiters, SIAM Journal on Scientific Computing 26 (2005) 907-929], Qiu and Shu investigated using weighted essentially non-oscillatory (WENO) finite volume methodology as limiters for the Runge-Kutta Discontinuous Galerkin (RKDG) methods for solving nonlinear hyperbolic conservation law systems on structured meshes. In this continuation paper, we extend the method to solve two-dimensional problems on unstructured meshes, with the goal of obtaining a robust and high order limiting procedure to simultaneously obtain uniform high order accuracy and sharp, nonoscillatory shock transition for RKDG methods. Numerical results are provided to illustrate the behavior of this procedure.
Jianxian Qiu - One of the best experts on this subject based on the ideXlab platform.
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runge kutta Discontinuous Galerkin method using weno limiters ii
Journal of Computational Physics, 2008Co-Authors: Jun Zhu, Chi-wang Shu, Jianxian Qiu, Michael DumbserAbstract:In J. Qiu, C.-W. Shu, Runge-Kutta Discontinuous Galerkin method using WENO limiters, SIAM Journal on Scientific Computing 26 (2005) 907-929], Qiu and Shu investigated using weighted essentially non-oscillatory (WENO) finite volume methodology as limiters for the Runge-Kutta Discontinuous Galerkin (RKDG) methods for solving nonlinear hyperbolic conservation law systems on structured meshes. In this continuation paper, we extend the method to solve two-dimensional problems on unstructured meshes, with the goal of obtaining a robust and high order limiting procedure to simultaneously obtain uniform high order accuracy and sharp, nonoscillatory shock transition for RKDG methods. Numerical results are provided to illustrate the behavior of this procedure.
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runge kutta Discontinuous Galerkin method using weno limiters
SIAM Journal on Scientific Computing, 2005Co-Authors: Jianxian QiuAbstract:The Runge--Kutta Discontinuous Galerkin (RKDG) method is a high order finite element method for solving hyperbolic conservation laws. It uses ideas from high resolution finite volume schemes, such ...