Alexandre Ern

Thermal diffusion effects in hydrogen-air and methane-air flames

The influence of thermal diffusion on the structure of hydrogen-air and methane-air flames is investigated numerically using complex chemistry and detailed transport models. All the transport coefficients in the mixture, including thermal diffusion coefficients, are evaluated using new algorithms which provide, at moderate computational costs, accurate approximations derived rigorously from the kinetic theory of gases. Our numerical results show that thermal diffusion is important for an accurate prediction of flame structure. §E-mail address: ern@cermics.enpc.fr ∥E-mail address: giovangi@cmapx.polytechnique.fr

A Posteriori Control of Modeling Errors and Discretization Errors

We investigate the concept of dual-weighted residuals for measuring model errors in the numerical solution of nonlinear partial differential equations. The method is first derived in the case where...

A hybrid high-order locking-free method for linear elasticity on general meshes

In this work we propose a novel Hybrid High-Order method for the incompressible Navier– Stokes equations based on a formulation of the convective term including Temam’s device for stability. The proposed method has several advantageous features: it supports arbitrary approximation orders on general meshes including polyhedral elements and non-matching interfaces; it is inf-sup stable; it is locally conservative; it supports both the weak and strong enforcement of velocity boundary conditions; it is amenable to efficient computer implementations where a large subset of the unknowns is eliminated by solving local problems inside each element. Particular care is devoted to the design of the convective trilinear form, which mimicks at the discrete level the non-dissipation property of the continuous one. The possibility to add a convective stabilisation term is also contemplated, and a formulation covering various classical options is discussed. The proposed method is theoretically analysed, and an energy error estimate in h (with h denoting the meshsize) is proved under the usual data smallness assumption. A thorough numerical validation on two and three-dimensional test cases is provided both to confirm the theoretical convergence rates and to assess the method in more physical configurations (including, in particular, the well-known twoand three-dimensional lid-driven cavity problems).Abstract We devise an arbitrary-order locking-free method for linear elasticity. The method relies on a pure-displacement (primal) formulation and leads to a symmetric, positive definite system matrix with compact stencil. The degrees of freedom are vector-valued polynomials of arbitrary order k ⩾ 1 on the mesh faces, so that in three space dimensions, the lowest-order scheme only requires 9 degrees of freedom per mesh face. The method can be deployed on general polyhedral meshes. The key idea is to reconstruct the symmetric gradient and divergence inside each mesh cell in terms of the degrees of freedom by solving inexpensive local problems. The discrete problem is assembled cell-wise using these operators and a high-order stabilization bilinear form. Locking-free error estimates are derived for the energy norm and for the L 2 -norm of the displacement, with optimal convergence rates of order ( k + 1 ) and ( k + 2 ) , respectively, for smooth solutions on general meshes. The theoretical results are confirmed numerically, and the CPU cost is evaluated on both standard and polygonal meshes.

Continuous interior penalty hp-finite element methods for advection and advection-diffusion equations

A continuous interior penalty hp-finite element method that penalizes the jump of the discrete solution across mesh interfaces is introduced. Error estimates are obtained for first-order and advection-dominated transport operators. The analysis relies on three technical results that are of independent interest: an hp- inverse trace inequality, a local discontinuous to continuous hp-interpolation result, and hp-error estimates for continuous L2-orthogonal projections.

Discrete functional analysis tools for Discontinuous Galerkin methods with application to the incompressible Navier-Stokes equations

Two discrete functional analysis tools are established for spaces of piecewise polynomial functions on general meshes: (i) a discrete counterpart of the continuous Sobolev embeddings, in both Hilbertian and non-Hilbertian settings; (ii) a compactness result for bounded sequences in a suitable Discontinuous Galerkin norm, together with a weak convergence property for some discrete gradients. The proofs rely on techniques inspired by the Finite Volume literature, which differ from those commonly used in Finite Element analysis. The discrete functional analysis tools are used to prove the convergence of Discontinuous Galerkin approximations of the steady incompressible Navier--Stokes equations. Two discrete convective trilinear forms are proposed, a non-conservative one relying on Temams device to control the kinetic energy balance and a conservative one based on a nonstandard modification of the pressure.

Intrusive Galerkin methods with upwinding for uncertain nonlinear hyperbolic systems

This paper deals with stochastic spectral methods for uncertainty propagation and quantification in nonlinear hyperbolic systems of conservation laws. We consider problems with parametric uncertainty in initial conditions and model coefficients, whose solutions exhibit discontinuities in the spatial as well as in the stochastic variables. The stochastic spectral method relies on multi-resolution schemes where the stochastic domain is discretized using tensor-product stochastic elements supporting local polynomial bases. A Galerkin projection is used to derive a system of deterministic equations for the stochastic modes of the solution. Hyperbolicity of the resulting Galerkin system is analyzed. A finite volume scheme with a Roe-type solver is used for discretization of the spatial and time variables. An original technique is introduced for the fast evaluation of approximate upwind matrices, which is particularly well adapted to local polynomial bases. Efficiency and robustness of the overall method are assessed on the Burgers and Euler equations with shocks.

Stabilized Galerkin approximation of convection-diffusion-reaction equations: discrete maximum principle and convergence

We analyze a nonlinear shock-capturing scheme for H1-conforming, piecewise-affine finite element approximations of linear elliptic problems. The meshes are assumed to satisfy two standard conditions: a local quasi-uniformity property and the Xu-Zikatanov condition ensuring that the stiffness matrix associated with the Poisson equation is an M-matrix. A discrete maximum principle is rigorously established in any space dimension for convection-diffusion-reaction problems. We prove that the shock-capturing finite element solution converges to that without shock-capturing if the cell Peclet numbers are sufficiently small. Moreover, in the diffusion-dominated regime, the difference between the two finite element solutions super-converges with respect to the actual approximation error. Numerical experiments on test problems with stiff layers confirm the sharpness of the a priori error estimates

Discontinuous Galerkin Methods for Friedrichs’ Systems

This work presents a unified analysis of Discontinuous Galerkin methods to approximate Friedrichs’ systems. A general set of boundary conditions is identified to guarantee existence and uniqueness of solutions to these systems. A formulation enforcing the boundary conditions weakly is proposed. This formulation is the starting point for the construction of Discontinuous Galerkin methods formulated in terms of boundary operators and of interface operators that mildly penalize interface jumps. A general convergence analysis is presented. The setting is subsequently specialized to two-field Friedrichs’ systems endowed with a particular 2×2 structure in which some of the unknowns can be eliminated to yield a system of second-order elliptic-like PDE’s for the remaining unknowns. A general Discontinuous Galerkin method where the above elimination can be performed in each mesh cell is proposed and analyzed. Finally, details are given for four examples, namely advection–reaction equations, advection–diffusion–reaction equations, the linear elasticity equations in the mixed stress–pressure–displacement form, and the Maxwell equations in the so-called elliptic regime.

Guaranteed and robust discontinuous Galerkin a posteriori error estimates for convection-diffusion-reaction problems

We propose and study a posteriori error estimates for convection-diffusion-reaction problems with inhomogeneous and anisotropic diffusion approximated by weighted interior-penalty discontinuous Galerkin methods. Our twofold objective is to derive estimates without undetermined constants and to analyze carefully the robustness of the estimates in singularly perturbed regimes due to dominant convection or reaction. We first derive locally computable estimates for the error measured in the energy (semi)norm. These estimates are evaluated using H(div,@W)-conforming diffusive and convective flux reconstructions, thereby extending the previous work on pure diffusion problems. The resulting estimates are semi-robust in the sense that local lower error bounds can be derived using suitable cutoff functions of the local Peclet and Damkohler numbers. Fully robust estimates are obtained for the error measured in an augmented norm consisting of the energy (semi)norm, a dual norm of the skew-symmetric part of the differential operator, and a suitable contribution of the interelement jumps of the discrete solution. Numerical experiments are presented to illustrate the theoretical results.

Nonlinear diffusion and discrete maximum principle for stabilized Galerkin approximations of the convection-diffusion-reaction equation

We investigate stabilized Galerkin approximations of linear and nonlinear convectiondiffusion-reaction equations. We derive nonlinear streamline and cross-wind diffusion methods that guarantee a discrete maximum principle for strictly acute meshes and first order polynomial interpolation. For pure convectiondiffusion problems, the discrete maximum principle is achieved using a nonlinear cross-wind diffusion factor that depends on the angle between the discrete solution and the flow velocity. For convectiondiffusion-reaction problems, two methods are considered: residual based, isotropic diffusion and the previous nonlinear cross-wind diffusion factor supplemented by additional isotropic diffusion scaling as the square of the mesh size. Practical versions of the present methods suitable for numerical implementation are compared to previous discontinuity capturing schemes lacking theoretical justification. Numerical results are investigated in terms of both solution quality (violation of maximum principle, smearing of internal layers) and computational costs.