A higher-order multiscale method for second order elliptic equations

Aboubacar Konaté (1)

1 LJLL - Laboratoire Jacques-Louis Lions

The aim of this work is to study a new discontinuous Galerkin (dG) discretization to the multiscale method introduced in [AB06] for solving a elliptic equation with parameters varying at a very small space scale. This is motivated by the fact that in some applications (for example in transport flow), particulary when the parameters are discontinuous or when the geometry is complex (non-conformities, faults, ...), dG discretizations are more suitable than those based on finite volume or continuous finite elements. Using standard methods when the parameters are varying at a very small space scale is demanding in term of computing times and in term of computer memory. Roughly speaking, multiscale methods consist in building basis functions which take into account the variation of parameters which leads to better balance between accuracy and computing times. We introduce a new Dirichlet-Neumann boundary condition to the so-called cell problems in order to reduce the resonance error (sometimes to remove it completely). An error estimate is established where the parameters are assumed to be periodic. Numerical illustrations are made both in periodic and non-periodic case.

Mots clés: Discontinuous Galerkin Element Method; Multiscale methods; Homogenization

Domaines: Analyse numérique [math.NA]

Aboubacar Konaté. A higher-order multiscale method for second order elliptic equations. 2018. ⟨hal-01701349⟩