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Guermond, Jean-Luc (19)

Popov, Bojan (8)Bonito, Andrea (4)Minev, Peter D. (3)Pasquetti, Richard (3)View MoreJournalComputer Methods in Applied Mechanics and Engineering (3)Mathematics of Computation (3)SIAM Journal on Numerical Analysis (2)Communications in Mathematical Sciences (1)Comptes Rendus Mathematique (1)View MoreKAUST Grant NumberKUS-C1-016-04 (18)PublisherElsevier BV (6)Springer Nature (5)Society for Industrial & Applied Mathematics (SIAM) (4)American Mathematical Society (AMS) (3)International Press of Boston (1)SubjectFinite elements (6)Conservation equations (3)Entropy (3)Entropy viscosity (3)Parabolic regularization (3)View MoreTypeArticle (16)Book Chapter (3)Year (Issue Date)2014 (4)2013 (6)2012 (1)2011 (2)2010 (5)View MoreItem AvailabilityMetadata Only (19)

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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.

Convergence analysis of a class of massively parallel direction splitting algorithms for the Navier-Stokes equations in simple domains

Guermond, Jean-Luc; Minev, Peter D.; Salgado, Abner J. (Mathematics of Computation, American Mathematical Society (AMS), 2012) [Article]

We provide a convergence analysis for a new fractional timestepping technique for the incompressible Navier-Stokes equations based on direction splitting. This new technique is of linear complexity, unconditionally stable and convergent, and suitable for massive parallelization. © 2012 American Mathematical Society.

Approximation of the eigenvalue problem for the time harmonic Maxwell system by continuous Lagrange finite elements

Bonito, Andrea; Guermond, Jean-Luc (Mathematics of Computation, American Mathematical Society (AMS), 2011) [Article]

We propose and analyze an approximation technique for the Maxwell eigenvalue problem using H1-conforming finite elements. The key idea consists of considering a mixed method controlling the divergence of the electric field in a fractional Sobolev space H-α with α ∈ (1/2, 1). The method is shown to be convergent and spectrally correct. © 2011 American Mathematical Society.

A new class of fractional step techniques for the incompressible Navier–Stokes equations using direction splitting

Guermond, Jean-Luc; Minev, Peter D. (Comptes Rendus Mathematique, Elsevier BV, 2010-05) [Article]

A new direction-splitting-based fractional time stepping is introduced for solving the incompressible Navier-Stokes equations. The main originality of the method is that the pressure correction is computed by solving a sequence of one-dimensional elliptic problems in each spatial direction. The method is very simple to program in parallel, very fast, and has exactly the same stability and convergence properties as the Poisson-based pressure-correction technique, either in standard or rotational form. © 2010 Académie des sciences.

A Second-Order Maximum Principle Preserving Lagrange Finite Element Technique for Nonlinear Scalar Conservation Equations

Guermond, Jean-Luc; Nazarov, Murtazo; Popov, Bojan; Yang, Yong (SIAM Journal on Numerical Analysis, Society for Industrial & Applied Mathematics (SIAM), 2014-01) [Article]

© 2014 Society for Industrial and Applied Mathematics. This paper proposes an explicit, (at least) second-order, maximum principle satisfying, Lagrange finite element method for solving nonlinear scalar conservation equations. The technique is based on a new viscous bilinear form introduced in Guermond and Nazarov [Comput. Methods Appl. Mech. Engrg., 272 (2014), pp. 198-213], a high-order entropy viscosity method, and the Boris-Book-Zalesak flux correction technique. The algorithm works for arbitrary meshes in any space dimension and for all Lipschitz fluxes. The formal second-order accuracy of the method and its convergence properties are tested on a series of linear and nonlinear benchmark problems.

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.

An optimal L1-minimization algorithm for stationary Hamilton-Jacobi equations

Guermond, Jean-Luc; Popov, Bojan (Communications in Mathematical Sciences, International Press of Boston, 2009) [Article]

We describe an algorithm for solving steady one-dimensional convex-like Hamilton-Jacobi equations using a L1-minimization technique on piecewise linear approximations. For a large class of convex Hamiltonians, the algorithm is proven to be convergent and of optimal complexity whenever the viscosity solution is q-semiconcave. Numerical results are presented to illustrate the performance of the method.

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.

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.

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.

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