15
views
0
recommends
+1 Recommend
0 collections
    0
    shares
      • Record: found
      • Abstract: found
      • Article: found
      Is Open Access

      Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization

      Preprint
      , ,

      Read this article at

      Bookmark
          There is no author summary for this article yet. Authors can add summaries to their articles on ScienceOpen to make them more accessible to a non-specialist audience.

          Abstract

          We introduce a new variational method for the numerical homogenization of divergence form elliptic, parabolic and hyperbolic equations with arbitrary rough (\(L^\infty\)) 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 \(\mathcal{O}(H^{-d})\) quadratic (cell) problems on (super-)localized sub-domains of size \(\mathcal{O}(H \ln (1/ H))\). The resulting localized linear systems remain sparse and banded. The resulting interpolation basis functions are biharmonic for \(d\leq 3\), and polyharmonic for \(d\geq 4\), for the operator \(-\diiv(a\nabla \cdot)\) and can be seen as a generalization of polyharmonic splines to differential operators with arbitrary rough coefficients. The accuracy of the method (\(\mathcal{O}(H)\) 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 Poincar\'{e} 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.

          Related collections

          Author and article information

          Journal
          2012-12-04
          2013-06-11
          Article
          1212.0812
          7a802cf1-7e64-47a3-9c54-9aa2966267b2

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

          History
          Custom metadata
          41A15, 34E13
          ESAIM: Mathematical Modelling and Numerical Analysis. Special issue (2013)
          math.NA math.AP

          Analysis,Numerical & Computational mathematics
          Analysis, Numerical & Computational mathematics

          Comments

          Comment on this article