Highly Oscillating Conductivities with Jumps

We study the regularized, oscillatory, multiscale, quasilinear, elliptic problem in divergence form − div(Qη(x, x , u)∇u) = f, in Ω, (0.1) u = g, on ∂Ω, (0.2) assuming that the conductivity has the form Qη(x, y, r) = a−(x) + (a+(x)− a−(x))βη(r − ψ(x, y)), (0.3) for some domain Ω ⊂ Rn, n ≥ 1, a regularization βη of the Heaviside step function and a locally periodic function ψ(x, y), Y -periodic in y. The equation is studied qualitatively as well as quantitatively in terms of the limit processes η → 0 (loss of regularity) and → 0 (highly oscillatory problem data). Elementary theoretical results for the equation are accompanied by the construction of numerical methods based on the Finite Element Method (FEM) and the Finite Element Hetereogenous Multiscale Method (FE-HMM) of approximating solutions to the considered equations for different parameter regimes. The numerical experiments suggest that the devised solvers exhibit the expected convergences for the test cases studied. The theoretical study shows that as η → 0 solutions to the parameterized equation converges weakly in H1(Ω) to some function solving an equation where an extra term emerges, describing the interaction of the solutions with the level set function ψ, a phenomenon not present in the case where ψ ≡ 0. Furthermore, the empirical evidence points in the direction that depending on the structure of the obstacle ψ solutions u ,η converges to a function u =0,η which does not solve an equation of the same structural form as u ,η does.