A Petrov-Galerkin Multiscale Hybrid-Mixed method for the Darcy Equation on Polytopes

Resumo

Let Ω⊂Rd,d∈ {2,3}, be an open, bounded, polygonal domain with a Lipschitzboundary∂Ω. We consider the second order elliptic problem defined by findingu∈H10(Ω)such that(A∇u,∇v)Ω= (f,v)Ωfor allv∈H10(Ω),(1)wheref∈L2(Ω) andA∈L∞(Ω)d×dis a symmetric, uniformly elliptic tensor in Ω andmay involve multiscale features. In this work, we modify the Multiscale Hybrid-Mixed(MHM) method [1] to propose a new multiscale finite element method to approximate (1).Its construction starts from a Petrov-Galerkin formulation for the Lagrange multipliervariable searched in a polynomial space enriched with residual functions. As a result,jumping terms are added to the original MHM method. These extra terms preserve theaccuracy and the overall properties of the original MHM method, while yield underlyingsymmetric positive definite linear systems. Some notations are needed at that point. Westart by introducingP, a collection of closed, bounded, disjoint polytopes, denotedK,such that ̄Ω =∪K∈PK. The shape of the polytopesKis, a priori, arbitrary, but we willsuppose they satisfy a minimal angle condition (see [2]). For eachK∈P,nKdenotesthe unit outward normal to∂K. We also introduce∂P, the set of boundaries∂K, withK∈P, andEthe following set of faces ofP,E:={E=K∩K′orK∩∂Ω :K,K′∈P,and it is not reduced to ad−1 variety}. [...]