The Schrödinger equation is a model for many physical processes in quantum physics. It is a singularly perturbed differential equation where the presence of the small reduced Planck's constant makes the classical numerical methods very costly and inefficient. We design two new schemes. The first scheme is the nonstandard finite volume method, whereby the perturbation term is approximated by nonstandard technique, the potential is approximated by its mean value on the cell and the complex dependent boundary conditions are handled by exact schemes. In the second scheme, the deficiency of classical schemes is corrected by the inner expansion in the boundary layer region. Numerical simulations supporting the performance of the schemes are presented.

On the time discrete approximation of the Brinkman-Forchheimer equations J. Djoko Kamdem * † Communicated by J. BanasiakIn this work, we study the structural stability of the fully implicit Euler scheme for the Brinkman-Forchheimer equations. More precisely, we consider the time discretization scheme of the unsteady Brinkman-Forchheimer equations, and we prove the existence of solutions. Moreover, we derive some a priori estimates of the discrete in time solutions. Next, with the aid of the discrete Gronwall lemma, we show that the numerical solutions depend continuously on the Brinkman and the Forchheimer coefficient. Copyright © 2011 John Wiley & Sons, Ltd.Keywords: Brinkman-Forchheimer; discrete Gronwall lemma; monotone; implicit Euler scheme Introduction) be an open bounded set with boundary * regular enough to allow the use of divergence theorem. We consider the Brinkman-Forchheimer model where |·| denotes the Euclidean vector norm |u| 2 = u·u, u 0 is the initial velocity of the fluid, f represents the body forces acting on the fluid, is the Brinkman coefficient, a>0 is the Darcy coefficient, b>0 is the Forchheimer coefficient, p is the pressure and ∈ [1, 2]. The model under consideration has been derived using mixture theory in [1]; specific conditions concerning the applicability of the model are also highlighted. For the mathematical setting of the problem (1) we denote by H m ( ) and · k the classical Sobolev space and its norm, respectively, and by (a, b) the real L 2 inner product of a and b. We consider the following spaces:where n is the outward normal on * . The space V is equipped with the scalar product and the normand H is endowed with the scalar product and norm of L 2 ( ) d . We denote by P the orthogonal projection of L 2 ( ) d onto H, by D(A) the domain of the Stokes operator A =−P . It can be shown [2,3] thatwith A : V → V a linear continuous operator such that each injection being continuous:where 1 >0 is the first eigenvalue value of the Stokes operator A. In the notation adopted, system (1) can be rewritten as the functional evolution equationwithIt is well known [4,5] that for ∈ [1 , 2] and u 0 ∈ D(A), the weak solution of the Brinkman-Forchheimer equations (5) is smooth, that is; for any T>0,Moreover, one has the following a priori estimates for the velocity:where K is a positive constant depending on the initial data and the parameters of (1). It is also proved in [4,5] that the unique weak solution of (1) depends continuously on the coefficients that is: one can find K 1 ,K 2 positive constant such that for all t, with 0

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