Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization. (11th March 2014) - Owhadi, Houman
Abstract : 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 higher-order 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.