We introduce a new variational method for the numerical homogenization of divergence form elliptic, parabolic and hyperbolic equations with arbitrary rough (L ∞ ) coefficients. Our method does not rely on concepts of ergodicity or scale-separation but on compactness properties of the solution space and a new variational approach to homogenization. The approximation space is generated by an interpolation basis (over scattered points forming a mesh of resolution H) minimizing the L 2 norm of the source terms; its (pre-)computation involves minimizing O(H −d ) quadratic (cell) problems on (super-)localized sub-domains of size O(H ln(1/H)). The resulting localized linear systems remain sparse and banded. The resulting interpolation basis functions are biharmonic for d ≤ 3, and polyharmonic for d ≥ 4, for the operator − div(a∇·) and can be seen as a generalization of polyharmonic splines to differential operators with arbitrary rough coefficients. The accuracy of the method (O(H) in energy norm and independent from aspect ratios of the mesh formed by the scattered points) is established via the introduction of a new class of higherorder Poincaré inequalities. The method bypasses (pre-)computations on the full domain and naturally generalizes to time dependent problems, it also provides a natural solution to the inverse problem of recovering the solution of a divergence form elliptic equation from a finite number of point measurements.

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We derive the effective viscosity of dilute suspensions of swimming bacteria from the microscopic details of the interaction of an elongated body with the background flow. An individual bacterium propels itself forward by rotating its flagella and reorients itself randomly by tumbling. Due to the bacterium's asymmetric shape, interactions with a prescribed generic (such as planar shear or straining) background flow cause the bacteria to preferentially align in directions in which self-propulsion produces a significant reduction in the effective viscosity, in agreement with recent experiments on suspensions of Bacillus subtilis.

The viscosity of a suspension of swimming bacteria is investigated analytically and numerically. We propose a simple model that allows for efficient computation for a large number of bacteria. Our calculations show that long-range hydrodynamic interactions, intrinsic to self-locomoting objects in a viscous fluid, result in a dramatic reduction of the effective viscosity. In agreement with experiments on suspensions of Bacillus subtilis, we show that the viscosity reduction is related to the onset of large-scale collective motion due to interactions between the swimmers. The simulations reveal that the viscosity reduction occurs only for relatively low concentrations of swimmers: Further increases of the concentration yield an increase of the viscosity. We derive an explicit asymptotic formula for the effective viscosity in terms of known physical parameters and show that hydrodynamic interactions are manifested as self-induced noise in the absence of any explicit stochasticity in the system.

Let Ω ⊂ R 2 be a simply connected domain, let ω be a simply connected subdomain of Ω, and set A = Ω \ ω. Suppose that J is the class of complex-valued maps on the annular domain A with degree 1 both on ∂Ω and on ∂ω. We consider the variational problem for the Ginzburg-Landau energy E λ among all maps in J . Because only the degree of the map is prescribed on the boundary, the set J is not necessarily closed under a weak H 1 -convergence. We show that the attainability of the minimum of E λ over J is determined by the value of cap(A)-the H 1 -capacity of the domain A. In contrast, it is known, that the existence of minimizers of E λ among the maps with a prescribed Dirichlet boundary data does not depend on this geometric characteristic. When cap(A) π (A is either subcritical or critical), we show that the global minimizers of E λ exist for each λ > 0 and they are vortexless when λ is large. Assuming that λ → ∞, we demonstrate that the minimizers of E λ converge in H 1 (A) to an S 1 -valued harmonic map which we explicitly identify. When cap(A) < π (A is supercritical), we prove that either (i) there is a critical value λ 0 such that the global minimizers exist when λ < λ 0 and they do not exist when λ > λ 0 , or (ii) the global minimizers exist for each λ > 0. We conjecture that the second case never occurs. Further, for large λ, we establish that the minimizing sequences/minimizers in supercritical domains develop exactly two vortices-a vortex of degree 1 near ∂Ω and a vortex of degree −1 near ∂ω.