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Flux norm approach to finite-dimensional homogenization approximations with non-separated scales and high contrast. (English) Zbl 1229.35009

The authors consider elliptic or vectorial equations written in a divergence form as
\[ -\text{div}(a(x)\nabla u(x))=f(x) \quad\text{or}\quad -\text{div}(C(x):\varepsilon (u))=b(x) \]
and posed in a bounded and smooth domain \(\Omega \) of \(\mathbb R^d\), \(d\geq 2\). The source terms \(f\) and \(b\) respectively belong to \( L^2(\Omega )\) and \((L^2(\Omega ))^d\). Homogeneous Dirichlet boundary conditions are imposed on \(\partial \Omega \). In the elliptic case, the matrix \(a_{ij}\) is supposed to be symmetric and uniformly elliptic in \(\Omega \). In the second case, the fourth-order tensor \(C\) is supposed to be uniformly elliptic with \(C_{ijkl}\in L^\infty(\Omega )\). In the vectorial equation, \(\varepsilon \) denotes the linearized deformation tensor of elasticity.
The main purpose of the paper is to build finite-dimensional approximations of the solutions of these problems. The main idea for the construction of these approximations is the Weyl-Helmholtz decomposition of a function \(k\in (L^2(\Omega ))^d\) as \(k=k_{\text{pot}}+k_{\text{curl}}\), where \(k_{\text{pot}}\) (resp. \(k_{\text{curl}}\)) is the orthogonal projection of \(k\) on \(L_{\text{pot}}^{2}(\Omega )\) (resp. \(L_{\text{curl}}^2(\Omega )\)), \( L_{\text{pot}}^2(\Omega )\) (resp. \(L_{\text{curl}}^2(\Omega )\)) being the closure of \(\{\nabla f:f\in C_{0}^\infty(\Omega )\}\) (resp. \(\{\xi :\xi \in (C^\infty(\Omega ))^{d}\), \(\text{div}(\xi )=0\}\)) in \((L^2(\Omega ))^d\). The authors then introduce the flux norm \(\|\psi\| _{a\text{-flux}}:=\|(a\nabla \psi )_{\text{pot}}\|_{(L^2(\Omega ))^d}\) on \(H_0^1(\Omega )\).
For the first main result of the paper, the authors introduce the regular tesselation \(\Omega_h\) of \(\Omega \) of resolution \(h\) and the set \({\mathcal L}_0^h\) of piecewise linear functions on \(\Omega _h\). Considering each piecewise linear nodal basis element \(\varphi _k\) in \({\mathcal L}_0^h\) they introduce the solution \(\Phi _k\) of the elliptic scalar equation with source term \( \Delta \varphi _k\) and the space \(V_h:= \text{span}\{\Phi _k\}\). The first main result proves that
\[ \sup_{f\in L^2(\Omega )} \inf_{v\in V_h} \frac{\|u-v\|_{a\text{-flux}}}{\|f\|_{L^2(\Omega )}}\leq \text{Ch}, \]
where \(C\) depends only on \(\Omega \) and on the aspect ratios of the simplices of \(\Omega _h\). A similar result is proved for this elliptic equation with a perturbed reference space \(V_h^Q\) where \(Q\) is a symmetric, uniformly elliptic and divergence-free matrix, and for the vectorial equation. The main tools of the proof of these error estimates are the properties of the flux norm. Then the authors prove new inequalities concerning the elliptic operators which are here involved. They apply the previous results to non-conforming Galerkin approximations, getting a new version of Cordes condition. The paper ends with a short application of this method to the derivation of homogenization results.

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
49J45 Methods involving semicontinuity and convergence; relaxation
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74Q05 Homogenization in equilibrium problems of solid mechanics
35J25 Boundary value problems for second-order elliptic equations
35J57 Boundary value problems for second-order elliptic systems
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