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Data Sparse Numerical Models for SNOM Tips

by C. Hafner and R. Hiptmair and P. Souzangar

(Report number 2015-26)

Abstract
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 $\mathcal{H}$-matrix storage format. We discuss two ways to build $\mathcal{H}$-matrix approximations of Schur-complement matrices: adaptive cross approximation (ACA) and $\mathcal{H}$-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 SNOM tip. The model can be embedded in three dimensional tip-sample simulations.

Keywords: Maxwell's equations, finite elements, matrix compression, H-matrices, ACA, AHMED library

@Techreport{HHS15_616,
  author = {C. Hafner and R. Hiptmair and P. Souzangar},
  title = {Data Sparse Numerical Models for SNOM Tips},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2015-26},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2015/2015-26.pdf },
  year = {2015}
}

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