Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization

¹ Corresponding author. California Institute of Technology, Computing and Mathematical Sciences, MC 9-94, Pasadena, CA 91125, USA
owhadi@caltech.edu
² Shanghai Jiaotong University, Institute of Natural Sciences and Department of Mathematics, Key Laboratory of Scientific and Engineering Computing (Shanghai Jiao Tong University), Ministry of Education, 800 Dongchuan Road, Shanghai 200240, P.R. China
lzhang2012@sjtu.edu.cn
³ Pennsylvania State University, Department of Mathematics, University Park, PA, 16802, USA
berlyand@math.psu.edu

Received: 3 August 2013

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 L² 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.