We provide a priori error estimates for variational approximations of the ground state energy, eigenvalue and eigenvector of nonlinear elliptic eigenvalue problems of the formWe focus in particular on the Fourier spectral approximation (for periodic problems) and on the P1 and P2 finite-element discretizations. Denoting by (u δ , λ δ ) a variational approximation of the ground state eigenpair (u, λ), we are interested in the convergence rates of u δ − u H 1 , u δ − u L 2 , |λ δ − λ|, and the ground state energy, when the discretization parameter δ goes to zero. We prove in particular that if A, V and f satisfy certain conditions, |λ δ − λ| goes to zero as u δ − u 2We also show that under more restrictive assumptions on A, V and f , |λ δ − λ| converges to zero as u δ − u 2 H 1 , thus recovering a standard result for linear elliptic eigenvalue problems. For the latter analysis, we make use of estimates of the error u δ − u in negative Sobolev norms.

Abstract.In this article, we provide a priori error estimates for the spectral and pseudospectral Fourier (also called planewave) discretizations of the periodic Thomas-Fermi-von Weizsäcker (TFW) model and for the spectral discretization of the periodic Kohn-Sham model, within the local density approximation (LDA). These models allow to compute approximations of the electronic ground state energy and density of molecular systems in the condensed phase. The TFW model is strictly convex with respect to the electronic density, and allows for a comprehensive analysis. This is not the case for the Kohn-Sham LDA model, for which the uniqueness of the ground state electronic density is not guaranteed. We prove that, for any local minimizer Φ 0 of the Kohn-Sham LDA model, and under a coercivity assumption ensuring the local uniqueness of this minimizer up to unitary transform, the discretized Kohn-Sham LDA problem has a minimizer in the vicinity of Φ 0 for large enough energy cut-offs, and that this minimizer is unique up to unitary transform. We then derive optimal a priori error estimates for the spectral discretization method.Mathematics Subject Classification. 65N25, 65N35, 65T99, 35P30, 35Q40, 81Q05.

Computation Fluid Dynamics (CFD) simulation has become a routine design tool for i) predicting accurately the thermal performances of electronics set ups and devices such as cooling system and ii) optimizing configurations. Although CFD simulations using discretization methods such as finite volume or finite element can be performed at different scales, from component/board levels to larger system, these classical discretization techniques can prove to be too costly and time consuming, especially in the case of optimization purposes where similar systems, with different design parameters have to be solved sequentially. The design parameters can be of geometric nature or related to the boundary conditions. This motivates our interest on model reduction and particularly on reduced basis methods. As is well documented in the literature, the offline/online implementation of the standard RB method (a Galerkin approach within the reduced basis space) requires to modify the original CFD calculation code, which for a commercial one may be problematic even impossible. For this reason, we have proposed in a previous paper, with an application to a simple scalar convection diffusion problem, an alternative non-intrusive reduced basis approach (NIRB) based on a two-grid finite element discretization. Here also the process is two stages: offline, the construction of the reduced basis is performed on a fine mesh; online a new configuration is simulated using a coarse mesh. While such a coarse solution, can be computed quickly enough to be used in a rapid decision process, it is generally not accurate enough for practical use. In order to retrieve accuracy, we first project every such coarse solution into the reduced space, and then further improve them via a rectification technique. The purpose of this paper is to generalize the approach to a CFD configuration.

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