Discontinuous Galerkin method

In applied mathematics, discontinuous Galerkin methods (DG methods) form a class of numerical methods for solving differential equations. They combine features of the finite element and the finite volume framework and have been successfully applied to hyperbolic, elliptic, parabolic and mixed form problems arising from a wide range of applications. DG methods have in particular received considerable interest for problems with a dominant first-order part, e.g. in electrodynamics, fluid mechanics and plasma physics. Discontinuous Galerkin methods were first proposed and analyzed in the early 1970s as a technique to numerically solve partial differential equations. In 1973 Reed and Hill introduced a DG method to solve the hyperbolic neutron transport equation. The origin of the DG method for elliptic problems cannot be traced back to a single publication as features such as jump penalization in the modern sense were developed gradually. However, among the early influential contributors

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Convergence of a stabilized discontinuous Galerkin method for incompressible nonlinear elasticity

Davide Baroli, Alfio Quarteroni, Ricardo Esteban Ruiz Baier

In this paper we present a discontinuous Galerkin method applied to incompressible nonlinear elastostatics in a total Lagrangian deformation-pressure formulation, for which a suitable interior penalty stabilization is applied. We prove that the proposed discrete formulation for the linearized problem is well-posed, asymptotically consistent and that it converges to the corresponding weak solution. The derived convergence rates are optimal and further confirmed by a set of numerical examples in two and three spatial dimensions.

Springer Verlag2013

Model order reduction for compressible flows solved using the discontinuous Galerkin methods

Jan Sickmann Hesthaven

Projection-based reduced order models (ROM) based on the weak form and the strong form of the discontinuous Galerkin (DG) method are proposed and compared for shock-dominated problems. The incorporation of dissipation components of DG in a consistent manner, including the upwinding flux and the localized artificial viscosity model, is employed to enhance stability of the ROM. To ensure efficiency, the discrete empirical interpolation method (DEIM) is adopted to enable hyper-reduction, for which the upwinding flux is decomposed into the central part and the dissipation part. The maximum local wave speed in the upwinding dissipation part is compressed and approximated using the DEIM approach, and the same strategy is applied to the artificial viscosity. Energy stability is proved with the strong-form-based ROM prior to hyper-reduction for the one-dimensional scalar cases. Eigenvalue spectrum is analyzed to verify and compare the stability properties of the two proposed ROMs. Several benchmark cases are conducted to test the performance of the proposed models. Results show that stable computations with reasonable acceleration for shock-dominated cases can be achieved with the ROM built on the strong form. (C) 2022 Elsevier Inc. All rights reserved.

ACADEMIC PRESS INC ELSEVIER SCIENCE2022

High order discontinuous Galerkin methods on simplicial elements for the elastodynamics equation

Carlo Marcati, Alfio Quarteroni

In this work we apply the discontinuous Galekin (dG) spectral element method on meshes made of simplicial elements for the approximation of the elastodynamics equation. Our approach combines the high accuracy of spectral methods, the geometrical flexibility of simplicial elements and the computational efficiency of dG methods. We analyze the dissipation, dispersion and stability properties of the resulting scheme, with a focus on the choice of different sets of basis functions. Finally, we apply the method on benchmark as well as realistic test cases.

Springer Verlag2016

ME-372: Finite element method

L'étudiant acquiert une initiation théorique à la méthode des éléments finis qui constitue la technique la plus courante pour la résolution de problèmes elliptiques en mécanique. Il apprend à appliquer cette méthode à des cas simples et à l'exploiter pour résoudre les problèmes de la pratique.

MATH-451: Numerical approximation of PDEs

The course is about the derivation, theoretical analysis and implementation of the finite element method for the numerical approximation of partial differential equations in one and two space dimensions.

MATH-459: Numerical methods for conservation laws

Introduction to the development, analysis, and application of computational methods for solving conservation laws with an emphasis on finite volume, limiter based schemes, high-order essentially non-oscillatory schemes, and discontinuous Galerkin methods.