We follow up on our previous work [C. Le Bris, F. Legoll, and A. Lozinski, Chinese Ann. Math. Ser. B, 34 (2013), pp. 113-138], where we have studied a multiscale finite element method (MsFEM) in the vein of the classical Crouzeix-Raviart FEM that is specifically adapted for highly oscillatory elliptic problems. We adapt the approach to address here a multiscale problem on a perforated domain. An additional ingredient of our approach is the enrichment of the multiscale finite element space using bubble functions. We first establish a theoretical error estimate. We next show that, for the problem we consider, the approach we propose outperforms all dedicated existing variants of the MsFEM we are aware of.

The concern of the present work is the introduction of a very efficient Asymptotic Preserving scheme for the resolution of highly anisotropic diffusion equations. The characteristic features of this scheme are the uniform convergence with respect to the anisotropy parameter 0 < ε << 1, the applicability (on cartesian grids) to cases of non-uniform and non-aligned anisotropy fields b and the simple extension to the case of a non-constant anisotropy intensity 1/ε. The mathematical approach and the numerical scheme are different from those presented in the previous work , arXiv:1008.3405v1] and its considerable advantages are pointed out.

Abstract. The present paper introduces an efficient and accurate numerical scheme for the solution of a highly anisotropic elliptic equation, the anisotropy direction being given by a variable vector field. This scheme is based on an asymptotic preserving reformulation of the original system, permitting an accurate resolution independently of the anisotropy strength and without the need of a mesh adapted to this anisotropy. The counterpart of this original procedure is the larger system size, enlarged by adding auxiliary variables and Lagrange multipliers. This Asymptotic-Preserving method generalizes the method investigated in a previous paper [P. Degond, F. Deluzet, and C. Negulescu, Multiscale Model. Simul., 8(2), 2009/10] to the case of an arbitrary anisotropy direction field.Key words. Anisotropic diffusion, asymptotic preserving scheme, finite element method.AMS subject classifications. 65N30. IntroductionAnisotropic problems are common in the mathematical modeling of physical problems. They occur in various fields of applications such as flows in porous media [3,23] [43] and so on, the list being not exhaustive. The initial motivation for this work is closely related to magnetized plasma simulations such as atmospheric plasma [27,29], internal fusion plasma [4,13] or plasma thrusters [1]. In this context, the medium is structured by the magnetic field, which may be strong in some regions and weak in others. Indeed, the gyration of the charged particles around magnetic field lines dominates the motion in the plane perpendicular to magnetic field. This explains the large number of collisions in the perpendicular plane while the motion along the field lines is rather undisturbed. As a consequence the mobility of particles in different directions differs by many orders of magnitude; this ratio can be as huge as 10 10 . On the other hand, when the magnetic field is weak the anisotropy is much smaller. As the regions with weak and strong magnetic field can coexist in the same computational domain, one needs a numerical scheme which gives accurate results for a large range of anisotropy strengths. The relevant boundary conditions in many fields of application are periodic (for instance in simulations