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Data‐sparse numerical models for SNOM tips
Author(s) -
Hafner Christian,
Hiptmair Ralf,
Souzangar Pegah
Publication year - 2016
Publication title -
international journal of numerical modelling: electronic networks, devices and fields
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.249
H-Index - 30
eISSN - 1099-1204
pISSN - 0894-3370
DOI - 10.1002/jnm.2178
Subject(s) - schur complement , complement (music) , finite element method , rotational symmetry , matrix (chemical analysis) , sparse matrix , boundary (topology) , computer science , algorithm , representation (politics) , field (mathematics) , matrix completion , rank (graph theory) , computational science , mathematics , mathematical analysis , geometry , physics , combinatorics , pure mathematics , materials science , law , chemistry , composite material , biochemistry , quantum mechanics , political science , gaussian , thermodynamics , eigenvalues and eigenvectors , complementation , politics , gene , phenotype
Summary We propose a compressed non‐local surface‐impedance‐type boundary condition for the efficient numerical modeling of large geometrically persistent parts in multi‐scale electromagnetic simulations. The underlying compressed model is an approximate Schur complement of finite element Galerkin matrix. Our approach relies on local low‐rank representation in the framework of the ℋ ‐matrix storage format. We discuss two ways to build ℋ ‐matrix approximations of Schur‐complement matrices: adaptive cross approximation and ℋ ‐arithmetics. Profound numerical tests and studies of accuracy are carried out for an axisymmetric setting, employing the open source library AHMED by M. Bebendorf. To demonstrate the use of our method, we build the compressed model for an axisymmetric scanning near‐field optical microscopy tip. The model can be embedded in three‐dimensional tip‐sample simulations. Copyright © 2016 John Wiley & Sons, Ltd.