On the time discrete approximation of the Brinkman-Forchheimer equations

Abstract

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<t T,where w = u 1 −u 2 , with u i representing the solution of (1) with data b i , i , and i = 1, 2. In this work, we consider a time discretization of (5) using the fully implicit Euler schemewhere we have assumed that the external force f is independent of time, while k is the time step. One of our goals in this work is to derive the discrete counterpart of (7). To the best of our knowledge, the question of investigating the continuous dependence of solutions by con...

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