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AuthorGuermond, Jean-Luc (6)Popov, Bojan (4)Guermond, J.-L. (2)Pasquetti, Richard (2)Bangerth, Wolfgang (1)View MoreJournalComputer Methods in Applied Mechanics and Engineering (2)Journal of Computational Physics (2)SIAM Journal on Numerical Analysis (2)ESAIM: Mathematical Modelling and Numerical Analysis (1)Journal of Mathematical Analysis and Applications (1)View MoreKAUST Grant Number

KUS-C1-016-04 (11)

PublisherElsevier BV (6)Society for Industrial & Applied Mathematics (SIAM) (3)American Mathematical Society (AMS) (1)EDP Sciences (1)Subject
Finite elements (11)

Conservation laws (2)Discontinuous Galerkin (2)Entropy viscosity (2)Euler equations (2)View MoreTypeArticle (11)Year (Issue Date)2015 (1)2013 (3)2011 (3)2010 (3)2009 (1)Item AvailabilityMetadata Only (11)

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Optimal bounds for a Lagrange interpolation inequality for piecewise linear continuous finite elements in two space dimensions

Muhamadiev, Èrgash; Nazarov, Murtazo (Journal of Mathematical Analysis and Applications, Elsevier BV, 2015-03) [Article]

© 2014 Elsevier Inc. In this paper the interpolation inequality of Szepessy [12, Lemma 4.2] is revisited. The lower bound in the above reference is proven to be proportional to p<sup>-2</sup>, where p is a polynomial degree, that goes fast to zero as p increases. We prove that the lower bound is proportional to ln<sup>2</sup> p which is an increasing function. Moreover, we prove that this estimate is sharp.

Stability analysis of explicit entropy viscosity methods for non-linear scalar conservation equations

Bonito, Andrea; Guermond, Jean-Luc; Popov, Bojan (Mathematics of Computation, American Mathematical Society (AMS), 2013-10-03) [Article]

We establish the L2-stability of an entropy viscosity technique applied to nonlinear scalar conservation equations. First-and second-order explicit time-stepping techniques using continuous finite elements in space are considered. The method is shown to be stable independently of the polynomial degree of the space approximation under the standard CFL condition. © 2013 American Mathematical Society.

A correction technique for the dispersive effects of mass lumping for transport problems

Guermond, Jean-Luc; Pasquetti, Richard (Computer Methods in Applied Mechanics and Engineering, Elsevier BV, 2013-01) [Article]

This paper addresses the well-known dispersion effect that mass lumping induces when solving transport-like equations. A simple anti-dispersion technique based on the lumped mass matrix is proposed. The method does not require any non-trivial matrix inversion and has the same anti-dispersive effects as the consistent mass matrix. A novel quasi-lumping technique for P2 finite elements is introduced. Higher-order extensions of the method are also discussed. © 2012 Elsevier B.V.

Implementation of the entropy viscosity method with the discontinuous Galerkin method

Zingan, Valentin; Guermond, Jean-Luc; Morel, Jim; Popov, Bojan (Computer Methods in Applied Mechanics and Engineering, Elsevier BV, 2013-01) [Article]

The notion of entropy viscosity method introduced in Guermond and Pasquetti [21] is extended to the discontinuous Galerkin framework for scalar conservation laws and the compressible Euler equations. © 2012 Elsevier B.V.

Effects of discontinuous magnetic permeability on magnetodynamic problems

Guermond, J.-L.; Léorat, J.; Luddens, F.; Nore, C.; Ribeiro, A. (Journal of Computational Physics, Elsevier BV, 2011-07) [Article]

A novel approximation technique using Lagrange finite elements is proposed to solve magneto-dynamics problems involving discontinuous magnetic permeability and non-smooth interfaces. The algorithm is validated on benchmark problems and is used for kinematic studies of the Cadarache von Kármán Sodium 2 (VKS2) experimental fluid dynamo. © 2011 Elsevier Inc.

Entropy viscosity method for nonlinear conservation laws

Guermond, Jean-Luc; Pasquetti, Richard; Popov, Bojan (Journal of Computational Physics, Elsevier BV, 2011-05) [Article]

A new class of high-order numerical methods for approximating nonlinear conservation laws is described (entropy viscosity method). The novelty is that a nonlinear viscosity based on the local size of an entropy production is added to the numerical discretization at hand. This new approach does not use any flux or slope limiters, applies to equations or systems supplemented with one or more entropy inequalities and does not depend on the mesh type and polynomial approximation. Various benchmark problems are solved with finite elements, spectral elements and Fourier series to illustrate the capability of the proposed method. © 2010 Elsevier Inc.

Error Analysis of a Fractional Time-Stepping Technique for Incompressible Flows with Variable Density

Guermond, J.-L.; Salgado, Abner J. (SIAM Journal on Numerical Analysis, Society for Industrial & Applied Mathematics (SIAM), 2011-01) [Article]

In this paper we analyze the convergence properties of a new fractional time-stepping technique for the solution of the variable density incompressible Navier-Stokes equations. The main feature of this method is that, contrary to other existing algorithms, the pressure is determined by just solving one Poisson equation per time step. First-order error estimates are proved, and stability of a formally second-order variant of the method is established. © 2011 Society for Industrial and Applied Mathematics.

Finite element discretization of Darcy's equations with pressure dependent porosity

Girault, Vivette; Murat, François; Salgado, Abner (ESAIM: Mathematical Modelling and Numerical Analysis, EDP Sciences, 2010-02-23) [Article]

We consider the flow of a viscous incompressible fluid through a rigid homogeneous porous medium. The permeability of the medium depends on the pressure, so that the model is nonlinear. We propose a finite element discretization of this problem and, in the case where the dependence on the pressure is bounded from above and below, we prove its convergence to the solution and propose an algorithm to solve the discrete system. In the case where the dependence on the pressure is exponential, we propose a splitting scheme which involves solving two linear systems, but parts of the analysis of this method are still heuristic. Numerical tests are presented, which illustrate the introduced methods. © 2010 EDP Sciences, SMAI.

Asymptotic Analysis of Upwind Discontinuous Galerkin Approximation of the Radiative Transport Equation in the Diffusive Limit

Guermond, Jean-Luc; Kanschat, Guido (SIAM Journal on Numerical Analysis, Society for Industrial & Applied Mathematics (SIAM), 2010-01) [Article]

We revisit some results from M. L. Adams [Nu cl. Sci. Engrg., 137 (2001), pp. 298- 333]. Using functional analytic tools we prove that a necessary and sufficient condition for the standard upwind discontinuous Galerkin approximation to converge to the correct limit solution in the diffusive regime is that the approximation space contains a linear space of continuous functions, and the restrictions of the functions of this space to each mesh cell contain the linear polynomials. Furthermore, the discrete diffusion limit converges in the Sobolev space H1 to the continuous one if the boundary data is isotropic. With anisotropic boundary data, a boundary layer occurs, and convergence holds in the broken Sobolev space H with s < 1/2 only © 2010 Society for Industrial and Applied Mathematics.

Surface Reconstruction and Image Enhancement via $L^1$-Minimization

Dobrev, Veselin; Guermond, Jean-Luc; Popov, Bojan (SIAM Journal on Scientific Computing, Society for Industrial & Applied Mathematics (SIAM), 2010-01) [Article]

A surface reconstruction technique based on minimization of the total variation of the gradient is introduced. Convergence of the method is established, and an interior-point algorithm solving the associated linear programming problem is introduced. The reconstruction algorithm is illustrated on various test cases including natural and urban terrain data, and enhancement oflow-resolution or aliased images. Copyright © by SIAM.

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